A thematic trimester in arithmetic will take place in ENS de Lyon and Université Lyon 1 from April 23, 2018 to June 29, 2018. This trimester is divided in two parts with different focus, with a conference taking place from May 22 to May 25.

The first part of the trimester focuses on the theory of algebraic groups, and particularly the Grothendieck-Serre conjecture on locally trivial homogeneous principal spaces. The conjecture was proved by Fedorov and Panin, following work of Colliot-Thélène, Nisnevich, Ojanguren, Ragunathan, Stavrowa, Vavilov...In the arithmetic case, Fedorov proved recently a significant special case of the conjecture. The study of this problem uses, among others, approximation techniques in algebraic groups, patching techniques, and affine grassmanians.

The second part of trimester focuses mainly on the question of the geometrization of the local Langlands correspondence. This problem was born from three major advances in arithmetic geometry: the introduction of the theory of perfectoid spaces by Scholze, the work of V. Lafforgue on the Langlands correspondence for function fields, and the introduction by Fargues and Fontaine of the fundamental curve of p-adic Hodge theory. We will take stock of the latest advances.

This conference has been held annually since 1978 with support of the Real Analysis Exchange (https://www.stolaf.edu/analysis/), and is considered to be the premier conference of its type by members of the real analysis community. The Symposium will visit Russia for the first time this summer! We have defined real analysis in a wide manner which encompasses trigonometric series, many parts of harmonic analysis as well as parts of dynamical systems, measure theory, the traditional theory of integration and properties of sets and functions. The goals of the Symposium are to provide a forum for researchers to present their newest results, and to provide ample time for participants to collaborate.

The canonical example of an arithmetic lattice is the modular group PSL(2, Z), whose deep connections with geometry and number theory (among many other areas) have been of profound interest for well over a century. Geometric invariants of the modular surface—the quotient of the complex upper half-plane by PSL(2, Z)—are typically paired with objects of equally deep interest in number theory. For example, its volume in its metric of constant curvature ­-1 is naturally related to a special value of the Riemann zeta function, and the lengths of its closed geodesics are intimately related to class numbers and regulators of real quadratic fields. More generally, rigidity phenomena (Weil, Mostow, etc.) imply that similar connections exist between number theory and the geometry of higher-dimensional hyperbolic manifolds.

The primary focus during the workshop will be to introduce the participants to problems at the interface of geometry and number theory that are currently attracting significant interest, and provide them with the tools necessary to make progress on some open questions. General areas to be discussed include the Laplace eigenvalue spectra and geodesic length spectra of hyperbolic 2- and 3-manifolds, growth of the systole of a hyperbolic manifold, and the ‘realization problem’ for trace fields of hyperbolic 3-manifolds. The number theoretic techniques that we will use to address these problems make use of quaternion algebras over number fields, Mahler measures of algebraic integers, and classical results from multiplicative number theory. The ultimate goal of this workshop is to start a dialogue between young mathematicians from different backgrounds that will lead to new and long-lasting collaborations between fields that have a great deal to say to one another. No background in either subject is expected.

This meeting will discuss the recent developments in Schubert calculus and representation theory, bringing together scientists working in algebra, geometry and topology and mathematical physics with the aim of building interdisciplinary contacts. The meeting is free to attend; participants just have to cover their accommodation, travel and subsistence.

Each summer Clemson University will host a one-week Early Career Research Workshop (ECRW) in Coding Theory, Cryptography, and Number Theory, aimed at postdocs and junior faculty. Three prominent researchers that have demonstrated success in mentoring postdocs will be invited to lead research teams consisting of postdocs and junior faculty (and possibly advanced graduate students when appropriate.) This workshop will help postdocs and junior faculty foster new collaborations and expand their existing research programs, both essential ingredients to launching and maintaining a healthy research program.

We will fund 9 participants for the ECRW with up to $500 for travel, per diem at $25/day for 6 days, and accommodations at $100/night for 5 nights. The funding is restricted to citizens or permanent residents of the US.

A school devoted to Topological Data Analysis and Persistent Homology will take place from Monday, June 11 to Friday June 15, 2018, at the Hotel Bellavista in Levico Terme (Trento, Italy), in the framework of the scientific activity of CIRM-FBK (cirm.fbk.eu/).

Topological Data Analysis (TDA) is a new research area that has recently attracted the attention of various scientific communities, both from pure and applied mathematics. The aim of this school is to provide a gentle introduction to TDA, starting from scratch, providing concrete applications and getting to the state of the art. Lectures are given by some of the most recognized specialists in this field and are addressed to a broad audience, mixing PhD students and young Post-Docs with more experienced researchers from different disciplines, among which Algebraic Topology and Algebraic Geometry.

The final program consists of three mini-courses delivered by John Harer (Duke University), Steve Oudot (INRIA Saclay) and Christopher J. Tralie (Duke University).

For details we refer to the updated official webpage of the school, published at the link

http://www.science.unitn.it/cirm/TDAPH2018.html

Some funds to cover living expenses for young participants are available. The deadline for registration and application for financial support is April 30, 2018.

Looking forward to seeing you in Levico,

The organizers

Gilberto Bini and Claudio Fontanari

Aim of the workshop is to put together researchers of different areas, in particular Mathematics and Engineering, and give them the opportunity to discuss, in a friendly atmosphere, recent developments in computational and theoretical methods in several fields of Computational Mathematics and Engineering. In particular, the workshop will be focused on numerical and theoretical aspects of the following topics:

• Mathematical models and simulation in Biology, Medicine and Engineering;
• Innovative numerical methods for ODEs and PDEs;
• Continuous and discrete dynamical systems: theory and numerical methods;
• Fractional calculus: theory, applications and numerical methods;
• Optimal control problems.

The aim of this programme is to provide the series of lectures on computational aspects of algebraic geometry. The lectures will be accompanied with computational Magma sessions. The programme will include:

• Mini course on GPU programming and applications (by Zvonimir Bujanović)
• Computing the Picard lattice of a genus two K3 surface (by Dino Festi)
• Mini course on Elliptic surfaces (by Bartosz Naskręcki)

The next Workshop in Geometric Topology will be held at Calvin College in Grand Rapids, Michigan, June 14 - 16, 2018. The principal speaker will be Professor Andras Stipsicz of the Alfréd Rényi Institute of Mathematics in Budapest. He will give a series of three lectures on the following topics: Lecture 1: Invariants of knots and links Lecture 2: Knot Floer homology Lecture 3: The Upsilon function of knots

In addition to the main lectures, there will be contributed talks by participants as well as ample time for less formal interaction between participants.

NSF funds will allow the workshop to provide travel support for graduate students and researchers without other means of support. The deadline for travel support requests is May 1, 2018.

This week-long workshop is for late-stage graduate students and early-stage postdocs in number theory to learn and develop strategies for giving clear, motivated, and informative talks. While some of the workshop will focus on basic skills like good boardwork and eye contact, the majority of the workshop will be focused on how to effectively convey mathematical ideas to people outside of one's particular specialty in number theory. There will be multiple small breakout sessions where faculty mentors (including the organizers) will give feedback on talks and discuss concrete strategies for preparing and giving talks.

The aim of this workshop is to discuss possible future directions in the study of effective methods in Diophantine number theory.

Classical effective methods in Diophantine number theory include Baker's method, Runge's method, Chabauty's method and the hypergeometric method. These famous methods have governed the direction of the research for decades. Around the end of the previous century it seemed that these methods had reached their limits. However, the past few years have seen important new developments by combining the existing approaches with methods from algebraic geometry. We mention a few of these developments: methods using modularity of elliptic curves; effective results of Shafarevich type; effective methods for finding integral points of curves and varieties over number fields and finitely generated domains; effective methods for points on subvarieties of tori.

Each of these recent developments revitalized the research on effective results in Diophantine number theory. Further, in many cases the best results are obtained by combining new methods, especially insights from algebraic or arithmetic geometry, with classical methods, such as Baker's method.

The goal of the proposed workshop is to bring together researchers interested in using these new methods, to share their expertise and try to figure out how these new methods can be combined to provide even more powerful tools to prove effective results in Diophantine number theory.

This international conference provides an opportunity for dissemination of important and recent results in the field of Applied Mathematics and Mechanics, and their Applications to Physics, Engineering Sciences, Biology and Economy. The conference will bring together mathematicians, interdisciplinary scientists and engineers from all over the world. It is a continuation of the ETAMM2016 Conference which held in 2016 in Perpignan, France. The conference ETAMM2018 will be held from June 18 to June 22, 2018 in Krakow and is organized by the Jagiellonian University which is the oldest university in Poland. The conference will consist of invited plenary talks, keynotes lectures, minisymposia and contributed short talks on recent results and future challenges in the field. The abstract of accepted presentation will be published in the conference abstract book. Selected and peer review papers will be included in a book of proceedings and to be published in special issues of the international journals.

Topics. Applied Mathematics: Biomathematics, Dynamical Systems, Finite Element Methods, Fixed Point Theories, Hemivariational Inequalities, Multi-Level Optimization and Design, Nonconvex and Nonsmooth Analysis and Optimization, Nonlinear Inclusions and Operator Theory, Numerical Analysis and Large Scale Computation, Numerical Methods for ODE and PDE, Optimal Control and Simulations, Partial Differential Equations,

Mechanics: Biomechanics, Buckling Analysis of Large Deformation Structures, Computational Mechanics, Constitutive Laws for Solids, Contact Mechanics, Electronic, Magnetic, Thermoelectric, and Smart Materials, Finite Deformation Elasto-Plasticity, Fluid Structure Interaction, Mathematical Theory and Methods for Phase Transitions and Hysteresis, Non-Newtonian Fluids, Optimization and Design of Structures, Structural Dynamics, Wear, Adhesion, Damage.

Géométrie Algébrique en Liberté, also known as GAeL, is an annual workshop organized by and for young algebraic geometers. The aim of this workshop is to gather young mathematicians in this field of research and give them an opportunity to discuss freely without concerns that their questions or viewpoints might be silly: hence the name of the workshop.

Next edition, GAeL XXVI will be held in Strasbourg (France), from Monday 18th until Friday 22nd of June. It will feature three minicourses by Paolo Cascini, Melody Chan and by Ben Moonen. There will be also contributed talks by participants and a poster session.

This meeting will be loosely structured around the Brauer Group and related topics. In the tradition of past Pingree Park Brauer group meetings, we are happy to welcome families, graduate students and young researchers, and hope to provide ample time to informally discuss mathematics and explore the beautiful Rocky Mountains.

The Homotopy Theory Summer Berlin 2018 will consist of two workshops with introductory summer schools on algebraic K-theory, equivariant and motivic homotopy theory. It is supported by the Berlin Mathematical School, the DFG Priority Program 1786 "Homotopy theory and algebraic geometry", and the RCN Frontier Research Group "Motivic Hopf equations". The list of confirmed participants includes:

• Stefan Schwede (Summer school lecturer)
• Florian Strunk (Summer school lecturer)
• Georg Tamme (Summer school lecturer)
• Marco Varisco (Summer school lecturer)
• Alexey Ananyevskiy
• Ben Antieau
• Aravind Asok
• Tom Bachmann
• Jean Fasel (tentative)
• Grigory Garkusha
• Bertrand Guillou
• Jeremiah Heller
• Marc Hoyois
• Dan Isaksen
• Akhil Mathew
• Haynes Miller (tentative)
• Fabien Morel (tentative)
• Thomas Nikolaus (tentative)
• Justin Noel
• Kyle Ormsby (tentative)
• Ivan Panin
• Birgit Richter (tentative)
• Christian Schlichtkrull
• Markus Spitzweck
• Vesna Stojanoska
• Kirsten Wickelgren
• Ben Williams
• Glen Wilson
• Inna Zakharevich

We look forward to seeing many of you in Berlin during this event!

The organizers:

Andrew Blumberg, Teena Gerhardt, Mike Hill, Marc Levine, Holger Reich, Oliver Röndigs, Paul Arne Østvær

Materials science has provoked and contributed to the emergence of various Nano materials, biomaterials, electronic, optical, magnetic materials, Surface engineering, Environmental and Green Materials, Biosensor and Bio-electronic Materials, Carbon Nano Structures and Graphene, Energy Harvesting Materials, Metals and Metallurgy and design of complicated structures through the innovation of technology by the advancements in the study of materials science. This meeting is also providing a platform to the companies and/or institutions to present their services, products, innovations and research results. For More Deatails Please visit: https://advancedmaterials.materialsconferences.com/

Organisers

1. Guido Kings, Regensburg
2. Ramdorai Sujatha, Vancouver
3. Eric Urban, New York
4. Otmar Venjakob, Heidelberg

This is the eighth annual joint workshop between Boston University (Boston, MA, USA) and Keio University (Yokohama, JP). The purpose of these workshops is to expose graduate students, junior faculty and researchers to active areas of research in areas of mathematics represented by BU and Keio faculty. This year's program will focus on dynamical systems. All talks are intended to be accessible to advanced graduate students with an interest in dynamical systems. There is limited funding available to cover the local expenses of graduate students and postdocs from outside the Boston area. Please see the workshop webpage for funding details.

L-functions are among the key unifying themes of modern number theory: seemingly ubiquitous, one finds them naturally associated to number fields, automorphic forms, abelian varieties, Artin representations, and more.

Central analytic questions about L-functions include their moments, size, non-vanishing, distribution of zeros and have been key to expanding our understanding of the distribution of prime numbers, arithmetic statistics, equidistribution of special points and periods, Diophantine equations, random matrix theory, and quantum chaos. In turn, exciting progress on L-functions is being made using and often combining tools ranging from as far as exponential sums, oscillatory integrals, complex and p-adic analysis to spectral analysis, trace formulas, and algebraic geometry.

This summer school aims to introduce graduate students and young postdocs to a variety of approaches and techniques in the analytic theory of L-functions. The school will emphasize the thinking and intuition underlying various approaches, their use in practice and adaptability to a range of situations, and the organic unity of the subject.

Speakers:

Valentin Blomer (Universität Göttingen)
Philippe Michel (Ecole Polytechnique Fédérale de Lausanne)
Djordje Milićević (Bryn Mawr College)
Caroline Turnage-Butterbaugh (Duke University)
Matthew Young (Texas A&M University)

This is an annual conference focused on exciting recent developments in number theory and related subjects, especially those occurring in the greater Asian regions or by mathematicians of Asian descent.

The purpose of this conference is to bring together experts working on torsion groups and Galois representations attached to elliptic curves and related areas.

Hội nghị sẽ đề cập đến nhiều hướng nghiên cứu chính của lý thuyết số và hình học đại số đang được phát triển mạnh tại châu Á. Ba chủ đề chính sẽ xoay quanh đa tạp Shimura, đối đồng điều p-adic và lý thuyết số trên trường hàm.

We are pleased to announce the "first edition" of the International Conference on Algebra and Related Topics (ICART 2018).

ICART 2018 will be held from the 2nd to the 5th of July 2018 at the Faculty of Sciences, Mohammed V University in Rabat, Morocco.

ICART 2018 will cover various research areas presented in the following three sessions:

• Applied and Computational Homology in Topology, Algebra and Geometry (ACHTAG)
• Homological Algebra, Modules, Rings and Categories (HAMRC)
• Linear and Multilinear Algebra and Function Spaces (LMAFS)

Please, see the website* http://www.fsr.ac.ma/icart2018/ to get more information including the list of Plenary Speakers.

Proceedings. http://fsr.um5.ac.ma/icart2018/proceedings.html
The proceedings of ICART2018 will be published in :
• A special issue of "Algebra Colloquium" for the sessions "Homological Algebra, Modules, Rings and Categories" and "Applied and Computational Homology in Topology, Algebra and Geometry".
• A volume in "Contemporary Mathematics" for the session "Linear and Multilinear Algebra and Function Spaces".

Please keep the following dates in mind as you prepare to attend icart2018 :
• Submission of Abstracts: April 20th, 2018.
• Early Registration Fee Payment: April 30th, 2018.
• Late Registration Fee Payment: Mai 31st, 2018.

The second edition of the summer school ALGAR is dedicated to the study of sums of squares in fields. In four lecture series and a few special talks we introduce the audience to the necessary algebraic, arithmetic and geometric methods that yield to beautiful and deep results on the length of sums of squares in certain types of fields.

Sums of squares have been studied since antiquity. The development of classical algebraic number theory was accompanied by famous achievements by Fermat, Euler, Lagrange and Gauss on the representation of natural numbers as sums of squares.

Writing polynomials with real coefficients as sums of squares is a way of showing that they are positive definite. This is also relevant in applied mathematics, for example in linear optimisation. In his 17th problem Hilbert asked whether any positive definite rational function with real coefficients in n variables can be written as a sum of squares of rational functions. He had shown this for n=1 and n=2. The general positive answer was obtained by Artin in 1927, but its proof is not constructive and does not yield directly an upper bound on the number of squares that is required. However, in 1967 Pfister showed that a representation as a sum of 2^n squares of rational functions with real coefficients is always possible.

The Pythagoras number of a commutative ring is defined as the smallest number p such that every sum of squares in the ring is equal to a sum of p squares. It can be very difficult to determine the Pythagoras number of a given ring. In particular, it is very hard to show that a certain sum of squares cannot be written with fewer terms. In the summer school, we will restrict mostly to fields, with a particular focus on rational function fields, function fields of real surfaces and function fields over the rational numbers.

The aim of the event is to survey some of the methods that are used in the study of sums of squares and thereby to illustrate various relations of this active research area to other branches of mathematics, obviously including algebraic geometry and number theory, but also analysis and topology, and furthermore logic and model theory.

This is the third conference of the series Glances@Manifolds. The main topics this year are: arithmetic groups, homogeneous manifolds, contact and symplectic manifolds, cobordism and group actions on manifolds. During the conference aside from the main lectures we intend to hold a large poster session and two small courses. The first one by Taras E.Panov on "Cobordism and group actions on manifolds". The Second one by Andrei Rapinchuk on "Arithmetic and Zariski-dense subgroups". More information can be found on the oficial website of the conference (above).

Part of the Semester Programme "Homotopy Harnessing Higher Structures". This programme will highlight four related themes: the new algebraic topology of differentiable manifolds, derived representation theory and equivariant homotopy theory, the interplay between arithmetic geometry and stable homotopy theory, and the analysis of foundations in these new contexts "the homotopy theory of homotopy theory". This particular workshop will focus on the last of these themes.

Since its introduction in the 70’s, the Langlands program is now the center of major developments with striking arithmetic applications. Since recently and in particular with the works of P. Scholze, a p-adic version of this program has emerged which concentrates a large number of arithmeticians.

This conference organised inside our ANR project aims to gather mathematicians interested in the last developments of the Langlands program and in particular the young researchers. Through about 20 talks, we aim to present the more interesting recent results and cover a large field of subjects.​

The costs of the stay will be covered by the CIRM and ANR: young colleagues who would need some funds for travelling should write to us.

This is a conference that will focus on new and developing links between operator K-theory and representation theory. More specifically, the meeting will consider the following themes:

• recent advances in operator K-theory and representation theory for affine Hecke algebras, with applications to the ABPS conjecture for p-adic groups, the Baum-Connes conjecture and the local Langlands correspondence,
• new constructions linking KK-theory, the Hecke algebras associated to arithmetic groups and automorphic forms,
• new approaches to tempered representation theory for real groups via operator algebras and noncommutative geometry,
• progress on the 'Mackey bijection' between the irreducible representations of real groups and the irreducible representations of associated Cartan motion groups.

We believe that the time is ripe to expect fruitful interaction among the above mentioned research areas. We tried to push the synergistic aspects of the meeting further by bringing together a balance of experts who work in representation theory, harmonic analysis, and automorphic forms with experts in noncommutative geometry and operator K-theory. Speakers will be encouraged to present open problems accessible across boundaries, and touch upon possible new points of contact between the different subjects represented at the meeting.

The aim of the summer school is to provide foundational knowledge necessary for carrying out independent research on explicit and/or computational aspects of Galois representations. It is especially aimed at PhD candidates, but more advanced researchers as well as advanced Master students are welcome. Women are explicitly encouraged to participate in the Summer School!

In the last decades, several notions of entropy have been introduced in different areas of Mathematics. The aim of this workshop is to gather mathematicians working on this topic from different points of view, in order to understand the present state of the art and to discover new connections.

There will be three mini-courses consisting of two lectures, and six plenary talks (see the schedule). Moreover, the participants are invited to propose contributed talks. The workshop will be closed by an Open problem Session.

The number of participants is limited to 50. There is no registration fee for the workshop, but registration is compulsory. Registration is possible until the 1st of May 2018, or until the expected number of participants is reached.

Invited Speakers:

Helge Glöckner

Michael Megrelishvili

Luigi Salce

Andrea Sambusetti

Manuel Sanchis

Klaus Schmidt

Pablo Spiga

Luchezar Stojanov

Benjamin Weiss

Organizing Committee

Federico Berlai, Ilaria Castellano, Anna Giordano Bruno, Menachem Shlossberg, Daniele Toller, Nicolò Zava, Fabio Zuddas.

Derived categories, introduced by Verdier in the 60s, are nowaday a widespread tools for studying the geometry of smooth projective varieties, and that of their moduli spaces of stable sheaves. However the majority of techniques employed in this investigation rely heavily on Hodge theory and Torelli theorems and so can be applied only to varieties defined over the complex numbers, or, more generally (thanks to the Lefschetz principle), to varieties over algebraically closed fields of characteristic 0. Till very recently the more general setting of varieties defined over (possibly finite) fields of positive characteristic had been barely explored. This workshop aims at bringing together international experts -- in both the fields of derived categories, and of positive characteristic algebraic geometry -- in order to build a bridge between the two fields, disseminate the latest research techniques, promote mutual learning, forging new research connections and encourage the discussion on recent development and open problems.

Women in Mean Curvature Flow

July 5, 2018

organized by Kathlynn Marie Simotas (PUWMAS) and Christina Sormani (CUNY)

All people who have taken a course using differential geometry are welcome to attend the talks. If you wish to attend, please contact Christina Sormani (sormanic at gmail) and send your name, your background (undergrad, grad, postdoc etc) and your field (physics, math, engineering etc). On July 7-9 there is an international conference on Minimal Surfaces and Mean Curvature Flow that we encourage everyone to register for and attend as well.

Schedule: (location TBA please register)

Lu Wang (U Wisconsin Madisson)* 3:00-4:00 pm “Mean Curvature Flow and Self Shrinkers with Conical Ends”

Ling Xiao (U Connecticut Storrs)* 5:00-6:00 pm “Translating Solitons and Convexity”

Dinner 7:00-9:00 pm TBA

This is a four day school at the PhD level. It will consist of lecture series by Mohamed Abouzaid (Columbia), , Joana Cirici (Barcelona), Craig Westerland (University of Minnesota), Hisham Sati (NYU-Abu Dhabi), Paolo Salvatore (Roma Tor Vergata) and a final speaker (tba).

Main topics of the workshop will include recent modularity theorems, families of Galois representations, results on lifting and derived methods.

The Canadian Number Theory Association (CNTA) was founded in 1987 at the International Number Theory Conference at Laval University (Quebec), for the purpose of enhancing and promoting learning and research in number theory, particularly in Canada. To advance these goals, the CNTA organizes bi-annual conferences that showcase new research in number theory, with the aim of exposing Canadian and international students and researchers to the latest developments in the field. The CNTA meetings are among the largest number theory conferences world-wide.

The focus of this research school is on three major advances that have emerged lately in the interface between homotopy theory and arithemic: cohomological methods in intersection theory, with emphasis on motivic sheaves; homotopical obstruction theory for rational points and zero cycles; and arithmetic curve counts using motivic homotopy theory. The emergence of homotopical methods in arithmetic represents one of the most important and exciting trends in number theory, and the lectures will give a gentle introduction to this highly technical area.

The three main lecture course topics are:

Cohomological methods in intersection theory Lecturer: Denis-Charles Cisinski (Regensburg) Assistant: Chris Lazda (Amsterdam)

Homotopical manifestations of rational points and algebraic cycles Lecturer: Tomer Schlank (HUJ) Assistant: Ambrus Pal (Imperial)

Arithmetic enrichments of curve counts Lecturer: Kirsten Wickelgren (Georgia Tech) Assistant: Frank Neumann (Leicester)

The lecture courses will be supplemented by tutorial sessions and guest lectures by Paul Arne Østvær (Oslo), Jon Pridham (Edinburgh) and Vesna Stojanovska (Illinois at Urbana-Champaign)

The Young Topologists Meeting is an annual event that has previously been organised by the EPFL, Switzerland; the University of Copenhagen; and jointly by Stockholm University and the Royal Institute of Technology. The YTM 2018 will take place in Copenhagen July 9–13, 2018. The meeting is intended as an opportunity for graduate students, recent PhDs, and other junior researchers in topology to meet each other and share their work. In addition to short talks by the participants, the program for the meeting will include lectures by Kathryn Mann (Brown University) and Mike Hill (UCLA).

This will be a two-week school, following the lead of the previous two ones: AGRA I in Santiago, Chile, 2012 and AGRA II in Cusco, Peru 2015. The main aim of the series is the development of number theory (in the broadest sense) as an area of research in South America.

This is to announce that there will be a Summer School on K-theory in La Plata, Argentina during the week of July 16-20. The courses in the school will be targeted to advanced undergrad students, graduate students and postdocs. Funding is available, for US applicants in particular; the funding application forms are now available through our webpage: http://mate.dm.uba.ar/~gtartag/ktheoryconference.html Applicants are encouraged to apply for financial support in March.

The School will be followed by a K-theory Conference in Buenos AIres July 23-27; funding is also available to attend it. The combined events have been accepted as an official ICM2018 Satellite.

The following people have agreed to give courses at the School:

• Arthur Bartels, Münster, Germany.

• Inna Zakharevich, Cornell, USA.

• Andreas Thom, Dresden, Germany.

• Gonçalo Tabuada, MIT, USA.

• Beatriz Abadie, Universidad de la República, Montevideo, Uruguay.

Further information about the conference is available at the web site mate.dm.uba.ar/~gtartag/ktheoryconference.html On behalf of the organizing committee, Chuck Weibel

This Workshop is the latest edition of a series organized periodically and jointly with the colleagues from Genova and Milano since 2008. Previous editions took place in Torino (September 2008, April 2010, June 2012, October 2014, December 2017), Genova (March 2009, February 2011, March 2013, February 2016, February 2017) and Milano (November 2009, November 2011, January 2014, September 2015).

These workshops are intended to give both an overview on some recent results on Commutative Algebra and Algebraic Geometry, and the opportunity to researchers working on similar topics to compare recent results and techniques. The outcome is to strengthen past and present collaborations, to give young researchers the opportunity to present their results, and to meet leading researchers in the aforementioned topics that have a visiting position at a Mathematical Department of a University in one of the three cities, or invited for the event. Similar joint seminars are well consolidated events in the international mathematical world.

The scientific committee of the present workshop consists of A. Alzati (Milano), G. Casnati (Torino), F. Galluzzi (Torino), R. Notari (Milano), M.E. Rossi (Genova), M. Varbaro (Genova). The workshop is supported by Dipartimento di Matematica, Politecnico di Milano and by Dipartimento di Matematica, Università degli Studi di Milano.

Up to now, the following speakers already confirmed their participation:

• V. Antonelli (Politecnico di Torino)
• L. Chiantini (Università degli Studi di Siena)
• A. De Stefani (KTH Stockholm)
• A. Hochenegger (Università degli Studi di Milano)
• S. Mongodi (Politecnico di Milano)
• P. Saracco (Università degli Studi di Torino)

A social dinner is planned on July 17.

Please send an e-mail to paolo.lella at polimi.it if you plan to attend the workshop and/or to join the dinner.

Organisers

1. Karim Belabas, Bordeaux
2. Bjorn Poonen, Cambridge MA
3. Fernando Rodriguez Villegas, Trieste

Young Researchers in Mathematics is an annual conference bringing together communities of PhD students, postdocs, and other young researchers from all areas of mathematics. There will be a mixture of invited speakers, workshops, and contributed talks from a number of different topics within mathematics and related industry, as well as opportunities to meet and socialise with peers. This summer, we look forward to meeting you at the University of Southampton, 23 July-26 July 2018.

Automorphic forms, Galois representations and L-functions, and the interplay among them, have been at the heart of numerous major advances in number theory over the last few decades, from their relevance to long-standing problems such as Fermat's Last Theorem and the Birch and Swinnerton-Dyer Conjecture to their role in the evolution of new research directions such as the the p-adic Langlands program and the theory of perfectoid spaces. The conference will focus on recent developments, with topics that include the Langlands program, special values of L-functions, Shimura varieties and p-adic Hodge theory.

The ICM2018 Satellite Conference "Braid groups, configuration spaces and homotopy theory" will take place in Salvador de Bahia, Brazil, at the Federal University of Bahia, from the 23rd to the 27th of July 2018. This meeting will focus on braid groups, configuration spaces and homotopy theory, and the interplay between them. There will be around 20 plenary talks given by invited speakers, as well as shorter talks given by the participants.

More information about the meeting may be found on the website of the conference: https://braids2018.ufba.br

For those participants who wish to present work, the deadline for registration and for submitting their work is the 2nd of March 2018. For other participants, the deadline for registration is the 31st of May 2018.

On behalf of the organisers, Daciberg Gonçalves, John Guaschi and Oscar Ocampo.

When: July 31 - August 3, 2018
Where: The University of Southern California
What: The theme of the summer school is ``Computations in stable motivic homotopy theory." For more precise information see the webpage of the summer school.
Who: The intended audience is graduate students and postdocs interested in the subject. Limited funding is available to support travel to and housing at the summer school. Please register at the webpage for the summer school to request funding. Priority for funding requests will be given to those applications complete before July 1, 2018.
Organizers: Aravind Asok and Oliver Röndigs

Satellite conferences will appear later with their own entries.

The International Quantitative Research and Applications Conference 2018 (IQRAC2018) is an international conference on mathematical sciences and their applications organized jointly by the Institute for Mathematical Research (INSPEM) of Universiti Putra Malaysia and the Malaysian Academy of Mathematical Scientists (AISMM). IQRAC2018 will feature keynote, plenary and oral contributed talks. The conference invites academicians, researchers, practitioners and students to participate and take opportunity to exchange information on the latest research findings in the field of mathematical sciences and their applications. The areas for presentations may include but not limited to topics and their applications in the broad area of analysis, algebra, graph theory, number theory, topology, statistics and probability, operations research, numerical analysis, cryptography, mathematics education and ethno mathematics. Prospective authors are invited to submit abstracts for their presentations in their chosen topics.

The aim of this workshop is to provide a researcher-tailored environment for groups of collaborators to work on ongoing projects related to the arithmetic of curves. Accommodation and meals are provided by Baskerville Hall.

Part of the Semester Program "Homotopy Harnessing Higher Structures." This programme will highlight four related themes: the new algebraic topology of differentiable manifolds, derived representation theory and equivariant homotopy theory, the interplay between arithmetic geometry and stable homotopy theory, and the analysis of foundations in these new contexts "the homotopy theory of homotopy theory". This workshop will focus on equivariant homotopy theory and its applications.

The aim of the summer school is to provide an introduction to the theme of “the (Iwasawa) main conjecture”, one of the deepest and most beautiful known result about the arithmetic of cyclotomic fields. It is the simplest example of a vast array of subsequent, unproven “main conjectures” in modern arithmetic geometry involving the arithmetic behaviour of motives over p-adic Lie extensions of number fields. These main conjectures are concerned with what one might loosely call the exact formulae of number theory which conjecturally link the special values of zeta- and L-functions to purely arithmetic expressions (the most basic example being the class number formula). Iwasawa himself not only discovered the main conjecture but proved an important theorem which implies it in all known numerical cases. In this workshop, we follow this approach to the main conjecture via Iwasawa’s theorem, and - if time permits - complete its proof by the ingenious arguments using Euler systems, due to Kolyvagin, Rubin and Thaine. This treatment also gives a very simple proof of the existence of the p-adic analogue of the Riemann zeta function using Coleman power series and cyclotomic units. The summer school is intended for students with interest in algebraic number theory and p-adic methods.

Analytic number theory is the branch of mathematics in which ideas and methods of real and complex analysis are brought to bear on problems about integer numbers. In particular, analytic techniques have led to major advances in number theory and helped to solve several important and difficult questions about integers. One of the crowning achievements of analytic number theory was the proof of the prime number theorem by using properties of the Riemann zeta function. In the last few years, analytic number theory has flourished and we have seen an upsurge of activity worldwide related to analytic number theory, prime number theory, and solutions to equations.

In this research school young researchers will learn key ideas and techniques of the field and about the most recent results and future directions of the field. It will benefit researchers having a background in number theory as well as those from parallel areas such as random matrix theory, Diophantine geometry and function field arithmetic. This research school will focus on three related topics and developments in analytic number theory:

• (i) classical analytic number theory, prime number theory and its recent developments
• (ii) Diophantine geometry and Hardy-Littlewood circle method
• (iii) analytic number theory in the function field context

These are active research areas where techniques from analysis (real, complex and Fourier analysis) plays an important role in trying to solve questions about integer numbers. Six experts in analytic number theory and its applications will conduct three mini-courses. To complement the mini-courses, distinguished speakers with substantial reputations in prime number theory, elliptic curves, additive combinatorics and random matrix theory will give an overview and research lectures on recent advances in the field.

In just the past few years, there have been a number of significant advances in building explicit links between tropical geometry and the algebraic geometry of moduli spaces, especially for curves and abelian varieties, using skeletons of nonarchimedean analytic spaces as a bridge between the two. These links explain many older correspondence theorems, from the beginnings of tropical enumerative geometry, and are being used to establish new correspondences and to prove new enumerative results of classical flavor, e.g. for counting curves with fixed invariants, and to prove degeneration formulas for logarithmic Gromov–Witten invariants as part of a broader program linking tropical geometry to mirror symmetry. At the same time, new algebraic foundations are emerging for scheme-theoretic tropical geometry, built out of the idempotent semirings studied by Brazilian mathematician and theoretical computer scientist Imre Simon.

We plan to use the gathering of a vast number of mathematicians for the 2018 ICM in Rio de Janeiro as an opportunity to bring together international and local experts working on tropical geometry, moduli spaces, and nonarchimedean analytic geometry for a conference that will include research talks by leading experts along with mini-courses to help graduate students and early career mathematicians enter this young and very promising area of mathematics.

There will be limited funding for the workshop, with a preference for young participants and participants from Latin America and developing countries.

Focus points

• Group actions: Mori Dream Spaces, $T$-varieties, also toric varieties, homogeneous spaces, contact Fano manifolds, Cremona groups, actions of finite groups, $\mathbb{G}_a$ and $\mathbb{G}_m$ actions on affine varieties,
• Arithmetic: arithmetic aspects of differential equations, $p$-adic cohomologies, crystals, automorphic forms, Calabi-Yau varieties, arithmetic aspects of mirror symmetry, finding rational points on manifolds,
• Parametrizing varieties: Hilbert scheme of points, rational curves on manifolds, secant varieties, tensor ranks, Waring ranks and related notions with their applications to complexity theory, engineering and quantum physics

See website.

Teachers:

Frits Beukers (Utrecht)
Duco van Straten (Mainz)
Kiran S. Kedlaya (UC San Diego)

This is a conference in number theory and arithmetic geometry specially addressed to female mathematicians working in these areas. However, anybody interested in these topics is kindly invited to attend. Invited Speakers Include

Kathrin Bringmann (U Köln)
Miranda Cheng (U Amsterdam)
Jessica Fintzen (U Michigan/U Cambridge)
Özlem Imamoglu (ETH Zurich)
Judith Ludwig (U Bonn)
Jasmin Matz (Hebrew U)
Kathrin Maurischat (U Heidelberg)
Anke Pohl (U Jena)
Christelle Vincent (U Vermont)

Whilst in Manchester, Paul Erdős co-authored two formative papers in Ramsey theory:

• 'A combinatorial theorem in geometry' (with G. Szekeres, 1935), giving a new proof of Ramsey's theorem.
• 'On some sequences of integers' (with P. Turán, 1936), laying the foundation for density results over arithmetic sets.

Some 80 years later, we would like to commemorate this and subsequent discoveries in additive combinatorics, continuing the celebration of the return of the number theory group to Manchester initiated by Diophantine Problems.

The central topic of the conference concerns the existence of structure within large arithmetic sets (broadly interpreted). In particular:

• Density bounds for sets lacking arithmetic configurations (Roth's theorem, Szemerédi's theorem, the cap-set problem and the polynomial method).
• Existence of arithmetic configurations in relatively dense sets (Szemerédi's theorem in the primes, combinatorial theorems in sparse (pseudo)random sets).
• Partition regularity of arithmetic configurations (monochromatic sums and products, regularity of non-linear equations).
• Applications of higher-order Fourier analysis to all of the above, including counting solutions to equations in arithmetic sets of interest.

Enquiries can be sent to the organisers at arithmeticramsey@gmail.com.

This is a Summer School of the SFB/TRR 45 Bonn-Essen-Mainz financed by the DFG (Deutsche Forschungsgemeinschaft). It will take place from September 10th until September 14th, 2018 at the Università degli studi di Salerno (Italy).

This summer school is intended for advanced master students, PhD students, and young researchers in algebra, number theory and geometry.

The school consists of four mini-courses of four 1-hour classes each, plus three research talks. During the week, two sessions for questions of the students to the lecturers of the mini-courses are also scheduled.

Lecturers: Jarod Alper, Ekaterina Amerik, Michel Brion, Jason Starr.

Research speakers: Olivier Benoist, Simone Diverio, Erwan Rousseau.

Dear All, We are writing to announce the Dragon Applied Topology Conference that will take place in Swansea, UK 11-14 September 2018. The conference webpage (still under construction) can be found here: https://sites.google.com/view/dragon-applied-topology The aim of the conference is to bring together applied and theoretical academic researchers, and industry practitioners, working with topological data analysis. Registration will open in early March. If you have any questions, please let us know, Hope to see you in Swansea, all the best, pawel dlotko

Dear mathematicians,

Vladimir Voevodsky was an enormously creative and wide ranging mathematician whose insight into topology and homotopy theory greatly advanced the fields of algebraic geometry, motivic homotopy theory, homotopy type theory, and univalent foundations. We invite you to attend the Vladimir Voevodsky Memorial Conference, September 11-14, 2018, in Princeton at the Institute for Advanced Study, to help us honor his contributions. The speakers are:

Benedikt Ahrens - University of Birmingham
Joseph Ayoub - University of Zurich
Pierre Deligne - Institute for Advanced Study
Eric Friedlander - University of Southern California
Daniel Grayson - University of Illinois, Urbana-Champaign/Institute for Advanced Study
Michael Hopkins - Harvard University
André Joyal - Universite du Quebec a Montreal
Mikahil Kapranov - Institute for Advanced Study
Daniel Licata - Wesleyan University
Alexander Merkurjev - University of California, Los Angeles
Fabien Morel - Mathematisches Institut der Universitat Munchen
Emily Riehl - Johns Hopkins University
George Shabat - Russian State University for the Humanities
Michael Shulman - University of San Diego
Alexander Vishik - The University of Nottingham
Claire Voisin - College de France
Inna Zakharevich - Cornell University

One of the lectures will be a public lecture, followed by a public reception. There will also be a conference banquet on the final evening in the Institute's own fine dining hall.

You are welcome to register for the conference and for the dinner on the web site at http://www.math.ias.edu/vvmc2018 .

On behalf of the organizing committee,

Thierry Coquand
Dan Grayson
Marc Levine
Charles Weibel

The conference will showcase theoretical and computational advances in the general field of discrete mathematics. It is open to researchers working with mathematical structures and abstract constructs, and to those involved in the theory and practise of discrete algorithmic computing. The purpose of this event is to highlight progress in the field through the development of novel theories, methodologies and applications accordingly, and to inspire future work.

Yerevan State University, joint with the Institute of Mathematics of the Armenian National Academy of Sciences, is organizing the next international conference in Harmonic Analysis and Approximations on September 16 - 22, 2018. The conference will be held at the Yerevan State University guesthouse, Tsaghkadzor (Armenia).

The aim of the conference is to bring together young and prominent researchers to present latest advances and problems in the field, and to provide a scientific forum to discuss and communicate on the issues and challenges in the field of Harmonic Analysis, Approximations and related fields.

UMI-SIMAI-PTM Mathematical Meeting is a joint initiative of the Polish Mathematical Society (Polskie Towarzystwo Matematyczne), the Italian Mathematical Union (Unione Matematica Italiana) and the Italian Society of Industrial and Applied Mathematics (Società Italiana de Matematica Applicata e Industriale).

The meeting aims at continuation of the tradition of bilateral meetings held in the last years by the Polish Mathematical Society together with other national societies. Mathematicians from other countries are also cordially invited to participate. The meeting will be hosted by the Faculty of Mathematics and Computer Science of the University of Wrocław and the Faculty of Mathematics of Wrocław University of Science and Technology. Its program will cover topics pertaining to mathematical research conducted in Italy and Poland with a special focus on the applications. It will start on Monday morning September 17, 2018 and end its Scientific Program in the afternoon on Thursday, September 20, 2018. Thursday night and two following days (Friday and Saturday) may be devoted to social activities up to a personal choice (excursions, sightseeing, town's cultural events, shopping), as well as for individually organized scientific activities.

The aim of this conference is to bring together researchers as well as students interested - in a broad sense - in any of the two topics. Mathematically speaking, the main goal of the conference is a better understanding of the interplay between low-dimensional topology and (higher) representation theory with its corresponding applications in link homologies and categorification. Another main goal is to get young researcher in these flourishing and rapidly growing fields together helping them to develop new projects and ideas in an inspiring, international atmosphere.

In addition to bringing together mathematicians and cryptographers to discuss open questions, this conference will celebrate Alice Silverberg's 60th birthday.

Confirmed lecturers:

Ana-Maria Castravet (Northeastern University), title: Mori Dream Spaces and Blow-ups,
Michel Brion (Institut Fourier, Grenoble), title: Automorphism groups in algebraic geometry,
Zach Teitler (Boise University), title: Waring rank.

Organizers: Jarosław Buczyński (Institute of Mathematics, Polish Academy of Sciences), Nathan Ilten (Simon Fraser University), Tomasz Mańdziuk (University of Warsaw), Jarosław Wiśniewski (University of Warsaw)

The titles of lecture series are provisional.

The workshop (Thursday-Saturday) will be devoted to more advanced research topics presented by invited speakers and participants. Afternoons will be devoted to discussions groups. Possible subjects: Mori Dream Spaces, torus actions, homogeneous spaces, contact Fano manifolds, Hilbert schemes, rational curves on manifolds, secant varieties, tensor and Waring ranks with their applications.

The event is supported by the Banach Center, the Targeted Grant to Institutes of Simons Foundation and by Polish Ministry of Science and Higher Education (MNiSW).

It is a part of the Simons Semester VAT (Varieties and Transformations).

This Arbeitsgemeinschaft, organized by Ayoub, Klingler and Pepin Lehalleur, will be devoted to Ayoub's proof of the Conservativity conjecture.

Part of the Semester Program "Homotopy Harnessing Higher Structures." This programme will highlight four related themes: the new algebraic topology of differentiable manifolds, derived representation theory and equivariant homotopy theory, the interplay between arithmetic geometry and stable homotopy theory, and the analysis of foundations in these new contexts "the homotopy theory of homotopy theory". This workshop will focus on derived algebraic geometry and the interplay with stable homotopy theory.

Neuroscience is nowadays one of the most collaborative and active scientific research fields. Differential equations are ubiquitous in the modeling of such phenomena and, consequently, nonlinear dynamics and dynamical systems techniques become fundamental sources of new tools to study neuroscience models. The aim of this Second Workshop on Neurodynamics is to present an overview of successful achievements in this rapidly developing collaborative field by putting together different types of applications of nonlinear dynamics (geometrical tools in dynamical systems, numerical methods, computational schemes, ...) to different problems in neuroscience (mononeuronal dynamics, network activity, cognitive problems,...). The final goal is spreading together mathematical methodology and neuroscience challenges and stimulating future cross-collaborations among participants, being Mathematical Neuroscience the generic topic for NDy'18.

Confirmed speakers are

• Edoardo Ballico (Università degli Studi di Trento)
• Gunnar Floystad (University of Bergen)
• Andre Galligo (Université Nice Sophia Antipolis)
• Steve Kleiman (MIT)
• Maria Grazia Marinari (Università degli Studi di Genova)
• Teo Mora (Università degli Studi di Genova)
• Bernard Mourrain (Inria Sophia Antipolis Méditerranée)
• Werner Seiler (Universität Kassel)
• Lea Terracini (Università degli Studi di Torino)

17th Edition of International Conference and Exhibition on Pharmaceutics & Novel Drug Delivery Systems October 04-06, 2018 Moscow, Russia Dear all, Greetings from Pharmaceutics 2018 Hope you are doing well. It is our pleasure to announce the 17th Edition of International Conference and Exhibition on Pharmaceutics & Novel Drug Delivery Systems to be held on 04-06 October 2018 in Moscow, Russia of Euroscicon ltd by a world class scientific events organizer. The Pharmaceutics 2018 conference has a long history and solid tradition. It has been held in different countries and cities associated to major research centers, where activity is connected with the development of innovative technology and research. For more details about the conference PS: https://novel-drugdelivery-systems.euroscicon.com/ And for any queries mail us at: pharmaceutics@eurosciconmeetings.org Christinerosepharmaceutics2018@outlook.com
Target Audience: We invite directors, presidents & CEO’s from companies, academic groups, directors from pharmaceutical companies. Laboratory scientists who identifies, quantifies, analyzes or tests the chemical or biological properties of compounds or molecules or who manages these laboratory scientists. chemical researcher, molecular diagnostics, clinical laboratories, health Care Industry, quantitative analysts, qualitative analysts, editorial board members, students, faculty members. chemical instrument vendors, professors and students from academia in the field of pharmaceutics sciences. Delegates from various pharma & instrumental companies from all over the world Conference start time: 9:00 AM Conference end time: 6:00 PM Conference type: Seminar/webinar/news/conference/education/ science and technology Your valuable presence will enhance the scientific essence of the conference and help to make it a grand success. Kindly confirm your interest and participation in the 2018 conference by submitting abstract online or you can revert back to this mail. Hope you accept this invitation and for any further queries feel free to contact me. Looking forward to hear from you. Thanks & Regards, Christine Rose Program Director Pharmaceutics 2018 Euroscicon ltd., 40 Bloomsbury Way Lower Ground Floor London, United Kingdom WC1A 2SE

This conference will be the mid-project meeting of the A.N.R. project ECOVA "Cohomological study of algebraic varieties". The aim is to gather together algebraic and arithmetic geometers whose research focusses on cohomological aspects. We intend to cover the wide spectrum of the ECOVA project, including motivic, geometric and complex Hodge-theoretic, p-adic aspects, non-archimedean and arithmetic aspects.

ECC is an annual workshops dedicated to the study of elliptic curve cryptography and related areas. For more information, also about past editions of ECC, please see the main ECC website. The 22nd Workshop on Elliptic Curve Cryptography (ECC 2018) will take place on 2018, in Osaka, Japan. The aim of ECC is to bring together leading experts from academia, industry, and government, as well as young researchers and graduate students for the purpose of exchanging ideas and presenting their work. The ECC Workshops has invited presentations only. Presentations tend to give an overview on emerging or established areas of modern cryptography, often combined with new research findings.

The workshop is accompanied by a 2-days autumn school on elliptic curves aimed at getting graduate students involved in the area. The school will take place on November 17-18, 2018.

Part of the Semester Program "Homotopy Harnessing Higher Structures." This programme will highlight four related themes: the new algebraic topology of differentiable manifolds, derived representation theory and equivariant homotopy theory, the interplay between arithmetic geometry and stable homotopy theory, and the analysis of foundations in these new contexts "the homotopy theory of homotopy theory". This workshop will focus on the algebraic topology of manifolds.

Objective: To inspire and motivate researchers and students working in the areas of Topology and its Applications

Workshop Themes: • Topological Dynamical Systems (TDS) • Topological Data Analysis (TDA)

Invited talks and Contributory papers in the conference: • Topological Dynamical Systems • General Topology • Algebraic Top0logy • Differential Topology • Fuzzy Topology • Topological Groups • Linear Dynamics • Dynamics of Operators • Network topology • Topological Data Analysis • Iterated Function Systems • Applications of Topology

Invited Speakers • *A. N. Sharkovsky, Institute of Mathematics, National Academy of Science of Ukraine • *Andrei Tetenov, Gorno-Altaisk State University, Russia • *Anima Nagar, Indian Institute of Technology, Delhi, India • *Chris Good, School of Mathematics, University of Birmingham, UK • *Dan Burghelea, Ohio State University, USA • *Dominic Kwietniak, Jagiellonian University in Krakow, Poland • * Henk Bruin, University of Vienna, Austria • *Herbert Edlesbrunner, Institute of Science & Technology, Austria • *Hisao Kato, University of Tsukuba, Japan • *James Yorke, University of Maryland, USA • *Javier Camargo, Universidad Industrial de Santander, Colombia • *Jose S Canovas, Polytechnic University of Cartagena, Spain • *Karoly Simon, Budapest University of Technology and Economics, Hungary • *Kit Chan, Bowling Greenstate University, USA • *Krzysztof Lesniak, Nicolaus Copernicus University, Poland • *P G Patil, Karnatak University, Dharward, India • *Lubomir Snoha, Matej Bel University, Slovakia • *Parameswaran Sankaran, Institute of Mathematical Sciences, India • *Robert Devaney, Boston University, USA • *Roman Hric, Matej Bel University, Slovakia • *Saber Elaydi, Trinity University, USA • *Sergiy Kolyada, Institute of Mathematics, Ukraine • *Varadachariar Kannan, University of Hyderabad, India • *Vijay Natarajan, Indian Institute of Science, India • *Wlodzimierz .J. Charatonik, Missouri University of Science and Technology, USA • Bau Sen Du, Academia sinica, Taiwan • Felix Martinez Gimenez, Instituto de Matemàtica Pura y Aplicada,, Spain • Gunnar Carllson, Stanford University, USA • Jaroslav Smital, Mathematical Institute in Opava, Czech Republic • Jian Li, Shantou University, China • Jie Hua Mai, Shantou University, China • Joseph Auslander, University of Maryland, USA • Larry Wasserman, Carnegie Mellon University, USA • Luis Alseda, Universitat Autònoma de Barcelona, Spain, • Michal Malek, Mathematical Institute in Opava, Czech Republic • Michal Rams, Institute of Mathematics, Polish Academy of Sciences • Paul S Bourdon, University of Virginia • Pat Touhey, Misericordia University, USA • Peter Raith, Universität Wien,Oskar-Morgenstern-Platz 1, Austria • Tomasz Downarowicz, Wroclaw University of Technology, Poland • Vin de Silva, Pomona College, California, USA *confirmed speakers

In 2000, S. Fomin and A. Zelevinsky introduced Cluster Algebras as abstractions of a combinatoro-algebraic phenomenon that is found in many different areas throughout mathematics. Defined as a sub-algebra of the ambient field of rational functions in finitely many variables, it is generated by union of cluster variables. The cluster variables are distributed across clusters. The clusters arise from an original "seed'' by a process known as mutation. For example, given a regular polygon with n sides the triangulations of this polygon with non-crossing diagonals can be obtained from a given triangulation of this type by a sequence of diagonal flips. Combinatorial data at a given cluster may be defined in terms of a quiver or alternatively a skew-symmetric matrix and using this quiver or matrix, the other clusters may be obtained by mutations. The clusters can be visualized as a graph with vertices being the clusters and edges being the mutations.

As it turns out the coordinate rings of Grassmannians, Shubert varieties and homogenous spaces are cluster algebras. The initial motivation behind study of these algebras was to provide an algebraic and combinatorial framework for Lusztig's work on canonical bases but now it has gone far beyond that initial motivation. Cluster algebras now have connections to string theory, Poisson geometry, algebraic geometry, combinatorics, representation theory and Teichmuller theory, quantum groups, and quiver representations.

The aim of this program is to bring together experts and interested students and researchers in this area. There will be lectures and talks by the experts on various topics around the central theme of cluster algebras and their applications. The hope is that participants will benefit from this program both by gaining insight into a key theme pervading and unifying several important fields and by applying the new methods to their research.

FRG-supported conference on the interaction between symplectic topology and homotopy theory.

This programme will focus on the study of the Langlands functoriality conjecture, including the endoscopy theory and the cases beyond endoscopy.

For the endoscopy case, Langlands functoriality is established by using the trace formula approach in general. Some special cases can also be established by the converse theorem-integral representation approach, and the automorphic integral transform approach.

Based on those successful cases, several new approaches are proposed. The key idea is to construct certain appropriate automorphic kernel functions and study them in various ways in order to establish functorial transfers for automorphic forms in the relative general setting.

These bring us the following four topics of the program:

Refined structures and properties for endoscopy theory: local and global.
Various types of trace formulas, generalized Fourier transforms and Poisson summation formulas, and applications.
Explicit constructions of certain Langlands functorial transfers via integral transformations.
Extension of the existing theory in the Langlands program to covering groups.

The goal of the program is to bring together experts researching in automorphic kernel functions to foster interaction, collaboration and the exchange of ideas on the new approaches. It aims to develop those approaches that will provide us further insights and progress, and lead to an eventual resolution of the Langlands functoriality conjectures.

Birational Geometry and Moduli Spaces are two important areas of Algebraic Geometry that have recently witnessed a flurry of activity and substantial progress on many fundamental open questions. In this program we aim to bring together key researchers in these and related areas to highlight the recent exciting progress and to explore future avenues of research.

This program will focus on the following themes: Geometry and Derived Categories, Birational Algebraic Geometry, Moduli Spaces of Stable Varieties, Geometry in Characteristic p>0, and Applications of Algebraic Geometry: Elliptic Fibrations of Calabi-Yau Varieties in Geometry, Arithmetic and the Physics of String Theory

Derived algebraic geometry is an extension of algebraic geometry that provides a convenient framework for directly treating non-generic geometric situations (such as non-transverse intersections in intersection theory), in lieu of the more traditional perturbative approaches (such as the “moving” lemma). This direct approach, in addition to being conceptually satisfying, has the distinct advantage of preserving the symmetries of the situation, which makes it much more applicable. In particular, in recent years, such techniques have found applications in diverse areas of mathematics, ranging from arithmetic geometry, mathematical physics, geometric representation theory, and homotopy theory. This semester long program will be dedicated to exploring these directions further, and finding new connections.

This workshop will be on different aspects of Algebraic Geometry relating Derived Algebraic Geometry and Birational Geometry. In particular the workshop will focus on connections to other branches of mathematics and open problems. There will be some colloquium style lectures as well as shorter research talks. The workshop is open to all.

The last years have seen important developments in the understanding of the geometry of complex algebraic varieties, and more generally Kählerian varieties. This session will aim at gathering experts at an international scale on very active areas of research : birational geometry and Hodge theory, singular metrics, hyperbolicity, surfaces of general type. This will be an opportunity to encourage young researchers to get involved in these promising developments.

The workshop will survey several areas of algebraic geometry, providing an introduction to the two main programs hosted by MSRI in Spring 2019. It will consist of 6 expository mini-courses and 8 separate lectures, each given by top experts in the field.

The focus of the workshop will be the recent progress in derived algebraic geometry, birational geometry and moduli spaces. The lectures will be aimed at a wide audience including advanced graduate students and postdocs with a background in algebraic geometry.

The last decades have witnessed the emergence of a full ecosystem of open source software for (pure) mathematics, developed by overlapping international communities of researchers, teachers, engineers and amateurs. This ranges from specialized libraries (e.g. MPIR, LINBOX) to thematic systems (e.g. GAP, PARI, SINGULAR, xcas) to general purpose systems (e.g. Mathemagix, SAGE), via online databases (e.g. the OEIS, MATHHUB, or the LMFDB). This is part of the greater trend for Open and Reproducible Science, and is supported by the advancement of cross-discipline tools like the interactive computing environment JUPYTER. The involved communities are strong in Europe – specially so in France.

The main goal of this conference is community building and training :

• Bringing together the various communities of users and developers of the ecosystem of (open source) (pure) mathematics software.
• Give newcomers as well as experts an overview of this ecosystem : what the existing software systems are, what they can compute or solve, how they are developed and by whom, what the success stories and difficulties are.
• Train newcomers as well as experts on using this ecosystem to solve their own problems.
• Share perspectives and best practices, build a joint vision, and seek venues for tighter cooperation.
• Encourage participants to get involved, especially women and more junior researchers, in particular by showcasing role models.

This conference will also be the closing event of OpenDreamKit, an H2020 European E-Infrastructure project lead by the organizers, and the occasion to present its outcomes.

Following a long trend of highly productive workshops within the various communities (e.g. the SAGE Days series), this conference will consist of :

• five or six keynote talks delivering a variety of perspectives on the ecosystem.
• Hands-on tutorials run by experts of the various systems.
• Plenty of free time for interactions and collaborative work, self-organized through regular project planning sessions and progress reports

Matrix-Analytic Methods in Stochastic Models (MAM) conferences aim to bring together researchers working on the theoretical, algorithmic and methodological aspects of matrix-analytic methods in stochastic models and the applications of such mathematical research across a broad spectrum of fields, which includes computer science and engineering, telephony and communication networks, electrical and industrial engineering, operations research, management science, financial and risk analysis, bio-statistics, and evolution.

Overview

HIL2019 web imageThe purpose of this workshop is to bring focused attention to Hilbert’s 13th problem, and to the broader notion of resolvent degree. While Abel’s 1824 theorem–that the general degree n polynomial is only solvable in radicals for n<4

is well known, less well known is Bring’s 1786 proof that a general quintic is solvable in algebraic functions of only one variable. Hilbert conjectured that for a general sextic, one needs algebraic functions of two variables, and that for a general degree 7 polynomial, one needs algebraic functions of three variables. More generally, it is natural to expect that as n → ∞ , so does the minimal number of variables needed to solve the general degree n polynomial. In a celebrated theorem, Arnol’d and Kolmogorov proved that, at the level of continuous functions, there is no local obstruction to reducing the number of variables to one. Thus, a resolution of Hilbert’s problem must lie deeper. Resolvent degree was introduced by Brauer in order to provide a rigorous statement of these conjectures. While no progress has yet been made on these conjectures, the study of resolvent degree is receiving renewed attention and an influx of ideas from related fields, including:

the theory of essential dimension
uniformization of moduli spaces
braid monodromy
p-adic Hodge theory

This workshop will bring together young and established researchers who work in these fields to explore topics related to Hilbert’s conjectures, to facilitate interaction between subdisciplines, and to lay the groundwork for future progress. The workshop will be organized around mini-courses by experts which will be aimed at 1) conveying the methods that can be brought to bear from each area, and 2) formulating problems concerning resolvent degree that seem particularly tractable using these methods.

The theory of automorphic forms is one of the frontier areas in mathematics, which links diverse areas such as representation theory of real and p-adic groups, theory of L-functions, and modular forms.

The algebraic and analytic aspects of the theory of automorphic forms are at the basis of much of modern number theory. The algebraic theory of automorphic forms broadly comprises of the study of automorphic representations of adelic groups, their L-functions, and also the study of their local components. The theory of group representations for real, p-adic, and adelic groups, is an actively pursued area of research and plays a central role in modern number theory. In the analytic theory of automorphic forms, the study of L-functions is one of the most important topics, together with Fourier coefficients of modular forms.

The workshop will be from 25th of February to 1st of March, 2019, and the discussion meeting from 4th of March to 7th of March, 2019. The workshop will consist of the following 4 courses, each of 6 hours duration, with some extra time devoted to tutorials.

1. Prof. Anna von Pippich (TU Darmstadt, Germany): “Theory of Eisenstein series”. Pippich will review the theory of Eisenstein series, and their significance in the spectral theory of automorphic forms.
2. Prof. Ameya Pitale (University of Oklahoma, USA): “Siegel modular forms: Classical and adelic aspects”. Pitale will discuss Siegel modular forms, both from the classical viewpoint of functions on Siegel upper half spaces as well as the modern viewpoint of automorphic representations of the symplectic group.
3. Prof. Vincent Sécherre (Université de Versailles, France): “ℓ-Modular representations of p-adic groups”. Sécherre will discuss the theory of smooth representations of p-adic groups with coefficients in an algebraically closed field of characteristic ℓ, where ℓ is different from 0 and p.
4. Prof. Shunsuke Yamana (Kyoto University, Japan): “L-functions and theta correspondence for classical groups”. Yamana will discuss his paper Invent. Math. 196 (2014), no. 3, 651–732, beginning with an introduction to theta correspondence.

The discussion meeting will gather together several national and international speakers to get a sense of some of the current directions in this subject without making it overtly technical.

Speakers:

• Michael Hopkins
• Jacob Lurie
• Matthew Morrow
• Kirsten Wickelgren

This workshop will bring together researchers at various frontiers, including arithmetic geometry, representation theory, mathematical physics, and homotopy theory, where derived algebraic geometry has had recent impact. The aim will be to explain the ideas and tools behind recent progress and to advertise appealing questions. A focus will be on moduli spaces, for example of principal bundles with decorations as arise in many settings, and their natural structures.

This week is an introductory school to prepare junior participants for the 3-month thematic program "Reinventing rational points" at the Institut Henri Poincaré (April–July 2019). The study of rational points uses a rich mix of methods ranging from algebraic geometry to analytic number theory and has a constantly growing tool kit. The mini-courses will be an introduction to this broad area with an emphasis on new methods and developments. Among the themes to be covered are: Galois cohomology and arithmetic duality theorems; the Brauer group of an algebraic variety; the Brauer-Manin obstruction and descent obstructions to local-to-global principles for rational and integral points, and for zero-cycles; interactions of analytic number theory and geometry, applications of the circle method, sieve methods and additive combinatorics; rational points in families of varieties over local and global fields; new approaches to counting points and varieties. We plan to discuss the state of the art in the research on the Batyrev–Manin principle on the growth of rational points and the Colliot-Thélène conjecture on the Brauer–Manin obstruction, their versions, interrelations and generalisations.

BrAG is an established series of regular meetings of British algebraic geometers. Our goal is to create a series that further strengthens the British algebraic geometry community, and that integrates postgraduate students and young researchers. The meetings will feature a number of pre-talks for graduate students, a poster session, and will include plenty of time for informal interactions between the participants.

This workshop, sponsored by AIM and the NSF, will be devoted to definability and decidability problems in number theory.

The main topics for the workshop are

• H10 and existential definability of Z for Q and big subrings of Q.
• Decidability of the first-order and the existential theory in finite and infinite algebraic extensions of Q. The workshop will bring together mathematicians working in algebraic geometry, number theory, model theory and computability theory to work on problems in decidability/computability and definability in number theory.
• Definability of valuation rings over infinite algebraic extensions of Q and over function fields.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

This workshop will be focused on presenting the latest developments in moduli theory, including (but not restricted to) recent advances in compactifications of moduli spaces of higher dimensional varieties, the birational geometry of moduli spaces, abstract methods including stacks, stability criteria, and applications in other disciplines.

This conference will gather researchers and students around two important topics in mixed and characteristic p geometry : the computation of stable models of curves, and the geometry of groups and torsors in characteristic p.

In the Summer 2019, there will be a four-week long Thematic Program in Commutative Algebra and its Interaction with Algebraic Geometry, on the occasion of Bernd Ulrich’s 65th birthday. It will be held at the Center for Mathematics of the University of Notre Dame from Tuesday, May 28 until Friday, June 21, 2019. The organizers of the program are Craig Huneke, Sonja Mapes, Juan Migliore, Claudia Polini, and Claudiu Raicu.

1. Undergraduate school Tuesday, May 28–Saturday, June 1, 2019 3 courses with afternoon exercise sessions
2. MSRI graduate school and Macaulay2 workshop Monday June 3– Friday June 14, 2019 4 courses with exercise sessions plus a Macaulay2 workshop
3. International conference in honor of Bernd Ulrich Sunday June 16–Friday June 21, 2019 4 talks per day (2 in the morning, 2 in the afternoon) 30 minutes Q&A each day (only for peridoctoral students) two poster sessions

In this workshop, we will explore a number of themes in the arithmetic of abelian varieties of low dimension (typically dimension 2--4), with a focus on computational aspects. Topics will include the study of torsion points, Galois representations, endomorphism rings, Sato-Tate distributions, Mumford-Tate groups, complex and p-adic analytic aspects, L-functions, rational points, and so on. We also seek to classify and tabulate these objects, in particular to understand explicitly the underlying moduli spaces (with specified polarization, endomorphism, and torsion structure), and to find examples of abelian varieties exhibiting special behavior. Finally, we will pursue connections with related areas, including the theory of modular forms, related algebraic varieties (e.g., K3 surfaces), and special values of L-functions.

Our goal is for the workshop to bring together researchers working on abelian varieties in a number of facets to establish collaborations, develop algorithms, and stimulate further research.

The interplay between D-modules and representation theory has became one of the key features in representation theory in the recent years. The aim of the conference is to report on recent progress and to present some new directions concerning this relationship, especially those motivated by applications to the p-adic and to the geometric Langlands program.

A two week program on arithmetic geometry will take place in Carthage in the city of Tunis (Tunisia), from June 17 to 28, 2019. It will focus on a number of areas of important progress over the last three or four years notably: p-adic Hodge theory; p-adic Langlands program; ramification of étale l-adic sheaves; special values of L functions and automorphic and motivic periods and Conjectures of Deligne, Beilinson, Gan-Gross-Prasad.

The first week will be devoted to a summer school and the second week to a conference. The summer school will take place from June 17 to 21, 2019. It will consist of 5 courses of three hours each and few lectures by junior researchers on topics close to the courses. The conference will take place from June 24 to 28, 2019. It will consist of 20 lectures. Registration on the web is mandatory both for the Summer School and the Conference.

The Journées Arithmétiques meetings, held every two years, cover all aspects of number theory. The venues alternate between locations in France and locations elsewhere in Europe.

An international summer school and conference on perfectoid spaces will take place July 8–12, 2019 in Rennes. The first part of the week, until the Thursday morning, will feature courses by international specialists on perfectoid rings, adic spaces and perfectoid spaces. From Thursday afternoon until the end of the conference invited speakers will present their latest results in the field. Participants and lecturers are generally expected to be present the entire week, but they also have the possibility to attend only the research conference on Thursday and Friday.

The Journal of Number Theory will host a number theory conference every two years to publicize recent advances in the field. The JNT is sponsoring the David Goss Prize of 10K USD to be awarded every two years at the JNT Biennial to a young researcher in number theory. Proceedings of the JNT Biennial conferences will appear in a special volume of the JNT.

ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 is hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It will include a Summer School (August 3-4) and a Conference (August 5-8) with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.

In many Latin American countries, political instability, institutional weakness and a lack of government support for scientific research have hindered the development of mathematics. There have been signs of progress in recent years. In 2014, Brazilian mathematics received international recognition when Artur Avila became the first South American to be awarded a Fields Medal. In 2018, the International Congress of Mathematicians will be hosted in a Latin American country for the first time. Within the last five years, several major conferences, such as the Mathematical Congress of the Americas, the AGRA winter schools, and PRIMA 2017, have been organized with the specific aim of increasing mathematical activity in Latin American countries.

In spite of all of this progress, there is still room for improvement. Number theory research in South and Central America continues to be largely confined to geographically isolated pockets of activity, concentrated within a small number of subfields. Many of the strongest math students go abroad for their training, in some cases because they cannot find viable Ph.D. supervisors in the research areas that they hope to pursue in their home countries. In most areas, Latin American mathematicians continue to be poorly represented at major international conferences. The proposed workshop aims to address some of these issues. Our main objectives are as follows:

Facilitate collaboration between North, Central, and South American number theorists.

The primary aim of the proposed workshop is to promote collaboration between number theorists in North, Central, and South America. To do this, we will model our workshop after several other workshops that have been extremely successful at sparking new collaborations: the American Institute of Mathematics workshops, the AMS Mathematics Research Communities workshops, and the BIRS-sponsored Women In Numbers workshops. Participants will be divided into small project groups led by senior researchers. Most of the time during the workshop will be spent working on research in these project groups. The goal is for researchers to leave the workshop with the beginning of a research paper or, at least, with a list of good candidates for future collaborators and a deeper understanding of a timely subject.

Foster research in timely areas of number theory. All of our confirmed participants have impressive research track records, and several are leading mathematicians by world standards. All of our project groups are on areas central to current research in the field, and all of these areas can also be said to lie in the crossroads between number theory and other fields. In several cases, this requires little explanation: the study of the arithmetic of algebraic varieties lies in the intersection of number theory and algebraic geometry; the study of modular forms, which originated in complex analysis, has been essential to number theorists since Ramanujan. The Langlands program is inherently about building connections, particularly with representation theory.

Continuing with our list of topics: additive combinatorics is a relatively new name for an area that encompasses additive number theory, combinatorial arguments and probabilistic and ergodic ideas. The importance of analytical tools to number theory has been clear since Riemann, and the relevance of harmonic analysis and spectral theory has become clearer and clearer since the mid-20th century. Probabilistic arguments in number theory have been fruitful ever since Erd\H{o}s and Tur\'an. The relevance of ergodic theory and dynamical systems to number theory has been known at least since Furstenberg and Ratner. Geometry and number theory often give two different perspectives on arithmetic groups. In particular, spectral gaps and expanders are terrains where number theory, spectral theory and geometry meet.

Train young researchers. Rather than filling the workshop with invited participants, we will reserve some spaces for young researchers who can apply to work in project groups that match their interests. One of our aims is to provide specialized training for young researchers in Latin American countries and introduce them to interesting problems in areas that may not be well-represented in their home countries. In some cases, this will be their first experience with working on a collaborative project. We will take steps to create a supportive environment so that young researchers will feel encouraged by the experience. We will also hold several panel discussions on topics that will be of particular interest to young researchers (see the Overview for more details).

Provide mentoring opportunities for mathematicians who normally do not get to train young researchers. The project groups are designed to provide a vertical mentoring structure, enabling mathematicians at different stages of their careers to mentor one another. Some of our participants may be faculty members at institutions without Ph.D. programs, and some will come from countries where it is typical for the strongest students to go abroad for graduate school. Such participants will have an exceptional chance to mentor promising young researchers in their project groups.

Attract greater visibility for the work of Latin American number theorists. A growing number of Latin Americans are working in number theory. By assembling this group, we will demonstrate that there is, in fact, already a fair number of strong number theorists connected to Latin American countries. Holding our workshop at the CMO, and advertising it on the BIRS website, will lend them additional prominence.

Build a network of Spanish-speaking mathematicians. This workshop will provide the foundation for creating a global network of Spanish-speaking mathematicians. In particular, we plan to start an online community -- including a mailing-list and possibly a more visible database -- of self-identified Spanish-speaking number theorists from around the world, organized by research area, which we hope will be useful to future conference organizers. The defining criterion will be an ability and willingness to lecture and work in Spanish.

The goal of this summer school is to introduce Ph.D. students and Postdocs to exciting recent developments in the geometry of algebraic groups and in modular representation theory.

The school will consist of two morning lecture series and of afternoon lectures on topics directly related to the morning lecture series.

Michel Brion (Grenoble) will give one lecture series on the “Structure of algebraic groups and geometric applications”. Geordie Williamson (Sydney) will give the other lecture series “On the modular representation theory of algebraic groups.”

M. Brion’s abstract. The course will first give an overview of the “classical” structure theory of algebraic groups and of some related geometric developments and problems; for example, on automorphism groups of projective algebraic varieties. It will then address results and questions on algebraic groups over arbitrary fields, including the structure of pseudo-reductive groups (work of Conrad, Gabber and Prasad), and of pseudo-abelian varieties (Totaro). Five lectures, 75 minutes each.

G. Williamson’s abstract. This course will be about the modular (i.e. characteristic p) representation theory of reductive algebraic groups, like the general linear and symplectic groups. I will begin by reviewing the algebraic theory, where there are beautiful connections to classical Lie theory and finite group theory. I will then pass to the geometric theory (perverse and parity sheaves) which is behind recent breakthroughs in the subject. The theory is rich in mysteries and open conjectures, and I will try to outline potentially interesting research directions. Five lectures, 75 minutes each.

This is a workshop that aims to support new collaborations between female mathematicians. Before the workshop, each participant will be assigned to a working group according to her research interests. Prior to the conference, the project leaders will design projects and provide background reading and references for their groups.

Confirmed group leaders:

Irene Bouw (Ulm)
Rachel Newton (Reading) and Ekin Ozman (Bogazici)
Damaris Schindler (Utrecht) and Lilian Matthiessen (KTH)
Ramla Abdellatif (Picardie Jules Verne)
Cecilia Salgado (Rio de Janeiro)
Elisa Gorla (EPFL)
Eimear Byrne (Dublin) and Relinde Jurrius (Neuchâtel)
Kristin Lauter (Microsoft)
Marcela Hanzer (Zagrev)
Lejla Smajlovic (Sarajevo)

The purpose of the workshop is to support and expand research efforts by female mathematicians in the field of algebraic topology. The workshop will bring senior and junior researchers together with advanced graduate students to cooperate on research projects on topics of common interest. The focus of the workshop will be on active collaboration to advance the projects, rather than on knowledge communication through lectures.

The aim of arithmetic geometry is to solve equations on integers by geometric methods. One of the most prominent achievements of this approach is certainly the Langlands program, which makes a connection between representations of the absolute Galois group of $\mathbb Q$ and certain adelic representations of reductive algebraic groups. In the early 2000's, Christophe Breuil suggested the existence of a purely $p$-adic version of the Langlands correspondence and supported his vision by numerous examples. Almost twenty years after, the $p$-adic Langlands correspondence has become a major topic in number theory.

Besides, following the rapid development of computer science throughout the 20th century, a large panel of algorithmical tools has been deployed and are now quite performant, in particular for attacking questions in Number Theory. A computational approach to the (classical) Langlands correspondence has been already investigated in recent times as well. We believe that the time has come to begin to extend it to the $p$-adic Langlands correspondence.

This conference is a first step towards this perspective. It will bring together the most internationally recognized experts in $p$-adic Langlands correspondence on the one hand and effective aspects of the Langlands correspondence on the other hand. Young researchers, and more generally researchers who are familiar with one side (either the abstract one or the effective one) and are willing to learn the other side, are particularly encouraged to attend our event: a enthousiastic program with 2 mini-courses, a bunch of short lectures and an introduction to the mathematical software SageMath is specially designed for them.

In this school, we intend to understand connections between the arithmetic theory of modular forms and new developments in p-adic Hodge theory that grew from the breakthrough work of Peter Scholze on perfectoid spaces (see P. Scholze "Perfectoid spaces" Publ. Math. de l’IHES 116 (2012)).

p-adic methods play a key role in the study of arithmetic properties of modular forms. This theme takes its origins in Ramanujan congruences between the Fourier coeffcients of the unique eigenform of weight 12 and the Eisenstein series of the same weight modulo the numerator of the Bernoulli number B12. After the work of Deligne on Ramanujan's conjecture it became clear that congruences between modular forms reflect deep properties of corresponding p-adic representations. The general framework for the study of congruences between modular forms is provided by the theory of p-adic modular forms developed in fundamental papers of Serre, Katz, Hida and Coleman (1970's-1990's).

p-adic Hodge theory was developed in pioneering papers of Fontaine in 80’s as a theory classifying p-adic representations arising from algebraic varieties over local fields. It culminated with the proofs of Fontaine's de Rham, crystalline and semistable conjectures (Faltings, Fontaine-Messing, Kato, Tsuji, Niziol,...). In order to classify all p-adic representations of Galois groups of local fields, Fontaine (1990) initiated the theory of (φ, Г)-modules. This gave an alternative approach to classical constructions of the p-adic Hodge theory (Cherbonnier, Colmez, Berger). The theory of (φ, Г)-modules plays a fundamental role in Colmez's construction of the p-adic local Langlands correspondence for GL2. On the other hand, in their famous paper on L-functions and Tamagawa numbers, Bloch and Kato (1990) discovered a conjectural relation between p-adic Hodge theory and special values of L-functions. Later Kato discovered that p-adic Hodge theory is a bridge relating Beilinson-Kato Euler systems to special values of L-functions of modular forms and u sed it in his work on Iwasawa-Greenberg Main Conjecture. One expects that Kato’s result is a particular case of a very general phenomenon.

The mentioned above work of Scholze represents the main conceptual progress in p-adic Hodge theory after Fontaine and Faltings. Roughly speaking it can be seen as a wide generalization, in the geometrical context, of the relationship between p-adic representations in characteristic 0 and characteristic p provided by the theory of (φ,Г)-modules. As an application of his theory, Scholze proved the monodromy weight conjecture for toric varieties in the mixed characteristic case. On the other hand, in a series of papers, Scholze applied his theory to the study of the cohomology of Shimura varieties. In particular to the construction of mod p Galois representations predicted by the conjectures of Ash (see P. Scholze “On torsion in the cohomology of locally symmetric space" (Ann. Of Math. 182 (2015)). Another striking application of this theory is the geometrization of the local Langlands correspondence in the mixed characteristic case. Here the theory of Fontaine—Fargues plays a fundamental role.

This goal of the proposed summer school is twofold:

1. Give an advanced introduction to Scholze's theory.
2. To understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, and lifting of modular forms, completed cohomology, local Langlands program and special values of L-functions.

We wish to bring together experts in the area of arithmetic geometry that will felicitate future research in the direction. We strongly encourage participation of young researchers.

Modularity. Until relatively recently, the celebrated Taylor--Wiles method for establishing the automorphy of Galois representations carried several significant limitations. First, the method applied only to Galois representations expected to come from cohomological automorphic forms of regular weight. For classical modular forms this excludes the case of weight 1 forms. Second, the locally symmetric space in whose cohomology the automorphic form is expected to arise was required to be an algebraic variety (a Shimura variety). This excludes for instance the case of elliptic curves over imaginary quadratic fields, where the locally symmetric space is 3-dimensional, and so cannot even admit a complex structure. Finally, in the absence of results towards Serre's conjecture on the modularity of mod p Galois representations, the Taylor--Wiles method generally only establishes the potential automorphy of Galois representations, i.e., automorphy after a finite base change.

In a major breakthrough, Calegari--Geraghty have introduced a derived version of the Taylor--Wiles method which has the potential to remove the first two of these restrictions. To realize the potential of the Calegari--Geraghty method requires overcoming a number of significant challenges in the theory of automorphic forms and the arithmetic of Shimura varieties. For instance one needs to know the existence of Galois representations attached to torsion classes in the cohomology of locally symmetric spaces, as well as strong forms of local-global compatibility for those representations. Scholze [SchTorsion] (and independently Boxer [Boxer] in some special cases) has addressed the former, and work of Cariani--Scholze [CS] on the vanishing of torsion in the cohomology of non-compact Shimura varieties has made progress towards the latter. These advances already have remarkable applications, such as the proof of potential modularity of elliptic curves over imaginary quadratic fields, as well as the Sato--Tate conjecture for such curves [tenauthor].

In addition to examining these many important developments, the workshop will contemplate possible future improvements to the Calegari--Geraghty method, such as may come from incorporating the derived deformation theory of Galatius--Venkatesh [GV]. We will also explore the prospects for proving actual (rather than potential) modularity of elliptic curves over some CM fields. Another expected topic is work in progress by Boxer--Calegari--Gee--Pilloni on the potential automorphy of abelian surfaces, using the Calegari--Geraghty method, as well as Pilloni's ``higher Hida theory'' for coherent cohomology of Shimura varieties [Pilloni].

Moduli of Galois representations. In ongoing work, Emerton and Gee are constructing moduli stacks which parameterize p-adic Galois representations arising from p-adic local fields. In the classical deformation theory of Galois representations, one considers formal families of deformations of a fixed mod p Galois representation; in contrast, the Emerton--Gee stacks admit non-constant families of mod p Galois representations, raising the possibility of arguing by interpolating between them. Furthermore, thanks to the global geometry of these spaces one has more algebro-geometric tools at one's disposal to study them.

The Emerton--Gee moduli stacks are built out of moduli spaces of integral p-adic Hodge theory data. Several incarnations of p-adic Hodge theory play a role in constructing and understanding these spaces, including Breuil-Kisin modules, Wach modules, and Tong Liu's (ϕ,Gˆ)-modules. Understanding how these different theories interact should a play an important role in the further development of this field. There remains many open questions about these stacks. What are the components of the special fiber? Are they normal? Cohen--Macaulay? What kind of singularities do they have? What is the structure of the line bundles/coherent sheaves on these spaces? Answers to these questions would have broad implications for modularity and the p-adic Langlands program.

The geometry of the Emerton--Gee stacks is closely linked to the Breuil--M\'ezard conjecture, which first arose in the context of attempt to generalize the Taylor--Wiles method. This conjecture measures the complexity of local Galois deformation rings (i.e., the versal deformation rings at closed points of Emerton--Gee stacks) in terms of the modular representation theory of GLn;\ understanding the geometry of local deformation spaces is essential for proving modularity lifting theorems. The Breuil--M\'ezard conjecture is in turn closely connected to the so-called weight part of Serre's conjecture, which can be viewed as a step towards the conjectural p-adic local Langlands correspondence.

For example, Caraiani--Emerton--Gee--Savitt [CEGS] are able to use known results about the geometric Breuil--M\'ezard conjecture and the weight part of Serre's conjecture for GL2 to analyze the irreducible components of certain Emerton--Gee stacks and relate them to the modular representation theory of GL2. The moduli stack perspective has also already played a role in the proof of the weight part of Serre's conjecture in generic situations in higher dimensions [LLLM1, LLLM2] and in on-going work of Emerton--Gee on the existence of crystalline lifts of mod p representations.

Despite considerable progress (e.g.\ [Herzig, GHS]), there still is no unconditional statement of the weight part of Serre's conjecture beyond the case of GL2. The Emerton-Gee moduli stack may be helpful for understanding this conjecture, as illustrated by the work of [CEGS]. One objective of the workshop will be to formulate an unconditional weight part of Serre's conjecture in terms of the Emerton-Gee stack, and to understand how such a conjecture relates to modular representation theory and to the Breuil-M\'ezard conjecture.

Finally, there are already tantalizing hints, for instance the work of [EGS] proving Breuil's local-global compatibility conjecture for types in the p-adic Langlands program, that the Emerton--Gee moduli stacks will play an important role in future developments on the modularity of Galois representations. However, this avenue is as yet largely unexplored. Another goal of this workshop is to bring together leading experts involved in these two strands of research in order to explore the possible synergies between them.

Local models for Galois deformation spaces. Although the two flavors of moduli spaces (Shimura varieties, Galois deformation spaces) that we have contemplated in this proposal are rather different, Kisin [Kis09a] observed that there is a surprising and fundamental relation between them:\ namely, their singularities are both modeled by relatively simpler moduli spaces called local models of Shimura varieties. These local models have been studied extensively in the context arithmetic of Shimura varieties, so that much is known about their geometry. Kisin's observation led to improved modularity lifting theorems, which in turn played a key role in the eventual proof of Serre's original conjecture for GL2/Q.

Beyond dimension two, in order to study regular weight Galois deformation spaces, there is an additional condition which comes from a subtle analogue of Griffiths transverality in p-adic Hodge theory. In [LLLM1,LLLM2], Le--Le Hung--Levin--Morra give explicit presentations for certain potentially crystalline deformation rings with Hodge--Tate weights (0,1,2) by studying this Griffiths transversality condition, and as an application prove cases of the weight part of Serre's conjecture and other related conjectures in dimension three. In higher dimension, the connection with local models is weaker and does not capture the Griffiths transversality condition. Ongoing work of Le--Le Hung--Levin--Morra constructs local models for Galois deformation spaces in generic situations and will shed light on the structure of generic parts of the Emerton-Gee moduli stack. Further, there are mysterious connections between these local models and objects in geometric representation theory which have not yet been explored.

There are a number of parallels between the mod p and p-adic stories. A striking example of this is Breuil--Hellmann--Schraen's recent proof of a Breuil--M\'ezard type conjecture for locally analytic representations, which furthermore leads to a proof of the locally analytic socle conjecture of Breuil [BHS]. They study the geometry of a p-adic family of Galois representations called the trianguline variety. In another parallel to the mod p picture, they create a link between the geometry of these p-adic families to objects in geometric representation theory.

By sharing these new developments broadly with other experts in the field, the workshop aims to spur further development of connections between moduli of Galois representations and the geometry of (generalized) local models, and of parallels between the p-adic and mod p settings; and to contemplate what the implications might be for the geometry of Emerton--Gee stacks.

This program is focused on the two-way interaction of logical ideas and techniques, such as definability from model theory and decidability from computability theory, with fundamental problems in number theory. These include analogues of Hilbert's tenth problem, isolating properties of fields of algebraic numbers which relate to undecidability, decision problems around linear recurrence and algebraic differential equations, the relation of transcendence results and conjectures to decidability and decision problems, and some problems in anabelian geometry and field arithmetic. We are interested in this specific interface across a range of problems and so intend to build a semester which is both more topically focused and more mathematically broad than a typical MSRI program.