During the 2017-18 academic year, the School will have a special program on Locally Symmetric Spaces: Analytical and Topological Aspects. Akshay Venkatesh of Stanford University will be the Distinguished Visiting Professor.

The topology of locally symmetric spaces interacts richly with number theory via the theory of automorphic forms (Langlands program). Many new phenomena seem to appear in the non-Hermitian case (e.g., torsion cohomology classes, relations with mixed motives and algebraic K-theory, derived nature of deformation rings). One focus of the program will be to try to better understand some of these phenomena.

Much of our understanding of this topology comes through analysis ("Hodge theory"). Indeed harmonic analysis on locally symmetric spaces plays a foundational role in the theory of automorphic forms and is of increasing importance in analytic number theory. A great success of such harmonic analysis is the Arthur-Selberg trace formula; on the other hand, the analytic aspects of the trace formula are not fully developed, and variants such as the relative trace formula are not as well understood. Thus analysis on such spaces, interpreted broadly, will be another focus of the program.

he word "period" is used to designate any number represented by the integral of an algebraic differential form over a cycle in an algebraic variety over \mathbb{Q} (or \overline{\mathbb{Q}}). These include many numbers of interest in number theory and mathematical physics (multiple zeta values, Mahler measures, superstring amplitudes, ...), and also have deep connections with special values of motivic L-functions.

The trimester will be divided into five "activities", each concentrating on one topic and including one or several introductory courses, and also three one-week workshops featuring lectures on current work:

Motives and Periods (Jan 3 - Jan 14)
Workshop: Periods and Regulators (Jan 15 - Jan 19)
Regulators (Jan 20 - Feb 4)
Amplitudes (Feb 5 - Feb 25)
Workshop: Amplitudes and Periods (Feb 26 - Mar 2)
Picard-Fuchs Equations and Geometry (Mar 3 - Mar 25)
Workshop: Picard-Fuchs Equations and Hypergeometric Motives (Mar 26 - Mar 30)
Hypergeometric Motives (Mar 31 - Apr 20)

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.

A primary goal of the Upstate Number Theory Conference is to bring together the specialists from the various branches of Number Theory in the Upstate New York region and surrounding areas, and to expose the younger researchers to new and old problems in the field.

We are happy to announce 1st International Conference on Mathematical and Related Sciences that will be held in 30 April-4 May 2018 in IC Hotels Santai Family Resort, Antalya-Turkey with the sponsorship of Düzce University. The central aim of this conference is to share knowledge and results in theory, methodology and new advances and research results in mathematical and related sciences. It focuses on new trends and approaches for Mathematicians, Scientists, Researchers and Engineers. Current research in computational and applied mathematical sciences require multi-disciplinary knowledge, not only in sciences and engineering but also in technologies of computing. This conference offers academic researchers, developers and practitioners an opportunity to discuss various aspects of computational and applied mathematical sciences and engineering related computational methods and problems solving techniques for science and engineering research. We are very pleased to invite all researchers to join us by sending high quality studies.

The BATMOBYLE is a vehicle for bringing together the algebraic and tropical geometry groups in the Boston area, originally between Brown and Yale.

Overview: This workshop will present recent developments on wave propagation, scattering and diffusion in random medias at the interface of probability theory, mathematical physics and PDEs. Accessible lectures by leading mathematicians will catalyze interactions among both junior and senior researchers in fundamental and applied fields.

Registration: Please consult the website for instructions on how to register.

Funding: Due to support from the NSF, some funding is available primarily for early career participants (preference given towards graduate students, postdocs and early tenure-track). Please consult the website and follow the instruction in order to apply for funding. Funds will be allocated on a rolling basis until depleted, so do not wait to apply.

Speakers: Scott Armstrong (New York University) Guillame Bal (Chicago University) Liliana Borcea (University of Michigan) Maury Bramson (University of Minnesota) Josselin Garnier (École Polytechnique) Svetlana Jitomirskaya (University of California, Irvine) Jianfeng Lu (Duke University) Jonathan Mattingly (Duke University) James Nolen (Duke University) George Papanicolaou (Stanford University) Lenya Ryzhik (Stanford University) Sylvia Serfaty (New York University) Sasha Sodin (Queen Mary University of London) Thomas Spencer (Institute for Advanced Study) Simone Warzel (Technical University of Munich)

Acknowledgements: We wish to acknowledge support from the NSF, as well as the KI-Net, Packard Foundation and Simons Foundation.

This one-day regional meeting of statisticians, engineers, computer scientists, mathematicians, and practitioners interested in the theory and applications of Statistical Machine Learning is being organized by the ISU Departments of Statistics and Industrial & Manufacturing Systems Engineering. The purpose of the meeting is to bring together a diverse set of researchers and practitioners to discuss current motivating problems and methodological advances in theory-based machine learning. Reduced price registration is offered for students and all poster-presenters. Grants supported by Principal Financial and professional societies are available to partially cover expenses for student and young investigator poster-presenters from outside ISU.

In this workshop, Bas Edixhoven (Leiden), Ilhan Ikeda (Bosphorus) and Winfried Kohnen (Heidelberg) will be the invited speakers. We aim to bring the researchers together who are interested in elliptic curves and modular forms as well as Langlands program. It is possible to give short talks here. We welcome you here in Turkey.

Strength in Numbers is a two-day workshop focusing on number theory and related areas, aimed primarily at graduate students. We especially encourage applications from underrepresented groups.

Each morning session will begin with two lectures by senior speakers, which will be accessible to graduate students in early years. Junior participants are invited to give short talks, on a topic of their choice. A novel feature of this workshop is a presentation by Professor Erin Maloney, a psychologist specializing in math education, math research and surrounding problems. Math anxiety, stereotype threat and impostor syndrome are just a few of these issues. The presentation will be followed by a panel on professional development, led by the invited speakers. The participants are encouraged to ask questions, share their experiences and seek advice during this session. Registered participants will be given the opportunity to submit their questions aimed at the panel members, ahead of time, via an online form with the option of anonymous submissions.

The program will feature a dozen one-hour talks, including two introductory workshop talks which will focus on interactions between representation theory and topology.

The workshop will turn around three series of lectures on topics in motivic homotopy theory, with a particular emphasis on some recent applications in enumerative geometry.

The main goal of this workshop will be to provide a (detailed) introduction to some of the main tools and ingredients which go into the development of this theory.

The lectures are intended for graduate students, postdocs and young researchers. The sequence of lectures will be complemented by several research talks by experts in the field.

The appearance of Probability in Number Theory can be traced back to a famous collaboration of Erdős and Kac. Nowadays, probabilistic techniques are routinely used in the study of integers and L-functions. However, until recently there had not been much room for modern and deep techniques of probability theory. During the past few years this has changed notably. Conversely, number theoretic techniques and heuristics have been proven effective in resolving standing problems in combinatorics and discrete probability theory. The goal of this month-long program is to bring together experts from Number Theory and Probability to highlight and facilitate the interactions between these two fields of mathematics.

During the first week of our program, there will be a workshop (an ISM discovery school) aimed at young researchers (postdocs and advanced graduate students) who work or are interested in the field of Probabilistic Number Theory. The workshop will consist of mini-courses given by Kevin Ford (Illinois), Adam Harper (Warwick), K. Soundararajan (Stanford) and Terence Tao (UCLA).

The remainder of the program will gather at CRM several of the leading experts in the fields of Probability and Number Theory. We also invite applications for five month-long postdoctoral positions (details to follow). Among other things, we will run a frequent research seminar for the participants of our program.

This four day event centered around the topic of Chromatic Homotopy Theory will consist of a graduate student workshop on May 16-17 and research talks by prominent researchers in the field on May 18-20. Participants who are not part of the graduate student workshop are invited arrive on May 17 and take part in the second day of the workshop.

More information will be posted on our website soon.

Organizers: Faculty of Sciences and Mathematics University of Niš, Serbia and Faculty of Mathematics, University of Belgrade, Belgrade, Serbia

Co-organizer: Faculty of Science, University of Kragujevac, Kragujevac Serbia,

Conference Topics: Differential Geometry, Topology, Lie Groups, Mathematical Physics, Discrete Geometry, Integrable Systems, Visualization, as well as other subjects related to the main themes are welcome.

Governmental sponsors: Ministry of Education, Science and Technological Development of the Republic of Serbia

International Advisory Commitee: David Blair (East Lansing, USA), Victor Buchstaber (Moscow, Russia), Bang-Yen Chen, (East Lansing, USA), Ryszard Deszcz (Wroclaw, Poland), Branko Dragovich (Belgrade, Serbia), Anatoly T. Fomenko (Moscow, Russia), Graham Hall (Aberdeen, UK), Louis Kauffman (Chicago, USA), Oldrich Kowalski (Prague, Czech Republic) Miodrag Mateljević (Belgrade, Serbia), Josef Mikes (Olomouc, Czech Republic), Svetislav Minčić (Niš, Serbia), Emil Molnar (Budapest, Hungary), Masafumi Okumura (Japan), Leopold Verstraelen ( Belgium), Andrei Vesnin (Novosibirsk, Russia), Kostadin Trencevski (Macedonia), Luc Vrancken (Vallenciennes, France).

Program Committee: Miroslava Antic, Mirjana Đorić, Stana Nikčević, Miroslava Petrović-Torgašev, Ljiljana Radović, Zoran Rakić, Mića Stanković, Ljubica Velimirović, Srdjan Vukmirovic.

Additive combinatorics is a very active area of mathematics. It is the crossing point of number theory, harmonic analysis, ergodic theory, and combinatorics. Beyond the numerous conferences and workshops, special semester-long programs have been organized on the subject at Princeton (IAS), Cambridge (Newton Institute), UCLA (IPAM), and UC Berkeley (MSRI). In this series of lectures, Dr. Jozsef Solymosi (University of British Columbia) provides an elementary introduction to additive combinatorics using discrete geometry, algebra, extremal combinatorics, and a bit of algebraic geometry. He also shows how to apply these techniques in order to attack Erdos type problems in discrete geometry. The lectures and the monograph to be written are intended for graduate students and researchers working on related areas, and to help those who intend to enter the field.

Dr. Solymosi will give 10 lectures on the topic of the conference, which will be complemented by 4 one-hour lectures (by Gyula Karolyi, Giorgis Petridis, Orit Raz, and Joshua Zahl). Problems to be solved will be distributed and discussed. Open problem sessions will provide opportunity for collaboration among the participants. This program is designed for graduate students and early-career researchers. As seating is limited, prospective participants not affiliated with the University of South Carolina must submit an application before January 31, 2018 at: http://mathprograms.org/db/programs/622

The application must include a CV with date or expected date of Ph.D., an email address, a letter of reference, a statement of research interests, a publication list, and a statement on how participation is expected to enhance the applicants career.

Selection of participants will take place in the first two weeks of February. Acceptance letters will be sent out as soon as final selections have been made. Selection will focus on the background of the participants in relevant areas and the expected effect of the program on their career. In particular, we encourage young mathematicians from the Southeast and members of underrepresented groups in the mathematical sciences to apply. Approved applicants will be provided dorm accommodation and partial reimbursement of other expenses.

Arrival to the conference should be scheduled for May 20, 2018. The scientific program is scheduled for May 21-25, 2018.

The 1st International Conference on Computational Methods and Applications in Engineering will take place from May 23 to May 26, 2018 at Politehnica University of Timisoara, Timisoara, Romania. On behalf of the organizing committee, we would like to invite you, your colleagues, postdocs, and graduate students to participate in this conference.This international conference will provide a forum for engineers, mathematicians, and scientists in the area of applied mathematics and their computational methods in engineering applications. The primary objective of this conference is to enhance interdisciplinary work between scientists and engineers in the USA and Romania. In addition to plenary lectures, there will be sessions for mini-symposia and contributed talks.

Midwest algebraic geometry graduate conference will be held at UIC May 25-27. As this is a conference for graduate students, we welcome the submission of any abstracts for talks and/or posters on original graduate research in algebraic geometry. Limited funds are available for housing for registered participants. The deadline to register to be considered for funding or to submit an abstract is March 15, 2018.

Some financial support will be available for travel and local expenses for both junior and senior participants. Since the venue has a limited capacity, we ask all interested participants to register/apply through the conference website.

Conference on modular forms and related topics has two objectives. The First is to introduce the local community to the recent significant developments in the theory of modular forms and related topics. The second objective is to encourage young Lebanese mathematicians to pursue their graduate studies in this area that has different applications in different mathematical sciences.

In recent years, there have been several important advances in the theory of differential and other functional equations. This field of research lies at the interface between several areas of mathematics, e.g., number theory, model theory, combinatorics and automata theory, theoretical physics, theoretical informatics, etc. This conference aims to gather researchers from a variety of backgrounds, whose research is related to the theory of functional equations. We will address theoretical as well as effective aspects. This is an international conference, every interested researcher can apply.

La théorie des équations différentielles et autres équations fonctionnelles a connu d’importantes avancées ces dernières années. Elle se situe à l’interface de nombreux domaines des mathématiques, dont la théorie des nombres, la théorie des modèles, la combinatoire et la théorie des automates, la physique théorique, l’informatique théorique, etc. L’objectif de cette rencontre est de favoriser les interactions entre mathématiciens issus d’horizons variés et dont les recherches touchent aux équations fonctionnelles. Nous aborderons aussi bien les aspects théoriques qu’effectifs. Il s’agit d’une rencontre internationale, ouverte à toute personne souhaitant y participer.

The purpose is to gather experts from two mathematical communities, arithmetic geometry and complex geometry, who study from different perspectives Shimura varieties and questions related to hyperbolicity of moduli spaces.

The geometry of quotients of bounded symmetric domains $\Omega/\Gamma$ has been the subject of many works. Classical results (Mumford, Tai, Tsuyumine, etc.) and more recent ones (Bakker-Tsimerman, Brunebarbe, Cadorel, etc.) state that in most cases compactifications of such quotients have many differential forms. In particular, they are of general type. The geometry of entire curves has also been investigated (Nadel, Noguchi, Rousseau, etc.) showing that such quotients are in general hyperbolic modulo the boundary. Some recent works (Brunebarbe, Cadorel, etc.) have also established that all subvarieties of $\Omega/\Gamma$ are of general type. These results can be seen as illustrations in this context of the Green-Griffiths-Lang's conjectures which make a natural bridge with arithmetic problems.

Many conjectures on the arithmetic side indeed involve the study of subvarieties of Shimura varieties. There has been a lot of work around the so-called Andr\'e-Oort conjecture on special subvarieties of Shimura varieties (Klingler-Yafaev, Ullmo, Pila, Tsimerman, etc.) which was recently proved for $\mathcal{A}_g$, the moduli space of abelian varieties of dimension $g$, by J. Tsimerman by relying in particular on the so-called Colmez conjecture in average proved by Andreatta-Goren-Howard-Pera. There is on-going work using a similar line of attack that was successful to prove Andr\'e-Oort to investigate its generalization, labelled the Zilber-Pink conjecture for brevity, for example by Daw and Ren in the direction of the hyperbolic Ax-Schanuel conjecture. The interplay between complex geometry and arithmetic geometry relies on direct applications to Shimura varieties of general theorems from complex geometry; on the suggestive analogy between the complex category and the arithmetic setting; but also on more intertwined interactions.

The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. The summer school will expose undergraduate and graduate students to important ideas in number theory. The research conference following the school is open to participants of all levels, including senior and junior faculty, and interested students. In particular, summer school students’ participation in the research conference will allow them hear about recent developments in the areas of elliptic curves, modular forms, and related topics.

This is a seven-day event, consisting of a four-and-a-half-day summer school followed by a two-and-a-half-day conference, from Monday, May 28th to Sunday, June 3rd, 2018, at the University of Connecticut at Storrs. The conference will feature research talks on number theory.

During the day, the summer school will have lectures aimed at undergraduate and graduate students. In the evenings, students will work in groups on problem sets and projects, both computational and theoretical in nature. The conference to follow the workshop will feature international experts on elliptic curves, modular forms, and related topics.

Funding from the National Science Foundation, the National Security Agency, and the Number Theory Foundation will make it possible for undergraduate and graduate students to attend the summer school and conference.

Dear Colleague,

We are pleased to invite you to attend the conference 'Algebraic and geometric methods of analysis' which will take place in Odesa city, Ukraine, from 30 May till 4 June 2018.

The conference continues the traditional annual conference 'Geometry in Odesa' holding from 2004.

The purpose of this conference is to bring together researchers in geometry, topology, algebra, analysis and dynamical systems and to provide for them a forum to present their recent work to colleagues from different nationalities. This way we aim to stimulate discussion about the latest findings in geometrical and topological methods in analysis and to increase international collaboration.

The conference will cover the following topics: - Algebraic methods in geometry - Differential geometry in the large - Geometry and topology of differentiable manifolds - General and algebraic topology - Dynamical systesm and their applications - Geometric problems in mathematical analysis - Geometric and topological methods in natural sciences - History and methodology of teaching in mathematics

The British and Irish Geometry Meeting 2018 (BI-GeM 2018) will bring together experts in geometry and related areas from Ireland, the UK and all over the world to highlight current research trends in geometry and explore interactions of geometry with other subjects such as topology and group theory. The list of speakers includes:

• Barbara Baumeister, Universität Bielefeld (Germany)
• Jürgen Berndt, University College London (UK)
• Tom Brady, Dublin City University (Ireland)
• John Burns, National University of Ireland in Galway (Ireland)
• Lynn Heller, Universität Hannover (Germany)
• Jarek Kedra, University of Aberdeen (UK)
• Mary Rees, University of Liverpool (UK)

The conference is supported by the London Mathematical Society and the Irish Mathematical Society.

Organisers: Thomas Hüttemann and Brian McMaster.

Angelo Vistoli's work in Algebraic Geometry has been extremely influential. From his very first works about intersection theory on algebraic stacks to the more recent ones on the essential dimension, through important results in equivariant algebraic K-theory, Brauer groups and twisted stable maps, Angelo's view and enthusiasm have been a driving force to all the algebraic geometers that had the occasion of talking and collaborating with him.

We celebrate his 60th birthday in June 2018 in Pisa.

The conference will be 4-days long and it will be housed at the mathematics department of the University of Pisa.

A meeting on the Riemann Hypothesis, and on the theory of the zeta-function and other L-functions.

The scope of the conference is geometric measure theory in Euclidean and non-Euclidean spaces, and the connections of GMT with adjacent fields, such as harmonic analysis, fractals, dynamical systems, and sub-Riemannian geometry. The scientific programme of the conference consists of invited talks, contributed talks, and a poster session.

The conference will feature a series of ten lectures by Professor David Cox on topics including elimination theory, polynomial systems in the real world, geometric modeling, geometric constraint theory, and chemical reaction networks. Follow up lectures will be given by other leading experts including Carlos D’Andrea, Jonathan Hauenstein, Hal Schenck, Jessica Sidman, and Alicia Dickenstein. The conference will also feature a problem session, software demonstration, and poster session.

The CBMS and NSF have generously provided funding for the conference, including funds for travel and lodging of attendees (NSF grant DMS-1741730) . To apply for funding, please visit the conference website. Graduate students, recent PhDs, and members of underrepresented groups in mathematics are especially encouraged to apply.

Sponsored by the Conference Board of the Mathematical Sciences, each five-day conference in the Regional Conference series features a distinguished lecturer who delivers ten lectures on a topic of important current research in one sharply focused area of the mathematical sciences; the lecturer subsequently prepares an expository monograph based upon these lectures. The principal lecturer for this conference will be David A. Cox, the William J. Walker Professor of Mathematics at Amherst College. Professor Cox is a world-renowned master expositor and award-winning author of several popular and highly-cited books in the mathematical area of applied algebraic geometry. Professor Cox’s lectures will discuss historical developments in this field in light of modern perspectives, leading right up to current research and applications to such diverse fields as computer aided design, rigidity of mechanical linkages, and chemical reaction networks. Each pair of lectures by Professor Cox will develop a chosen topic and be followed by a further lecture by a specialist he has hand-picked to provide a deeper look at the forefront of current work on that topic.

Summer school on effective geometry and arithmetic of curves (Istanbul June 4-8 2018).

This mathematical school will focus on effective and explicit methods in the theory of algebraic curves and applications to coding theory and cryptography. An important aspect of the school will be tutorials accompanying each of the lecture series. In these tutorials participants will have the opportunity to put into practice the theory from the lectures and experiment with the introduced object using various computer algebra systems. Depending on the background of the students, they might be given small research problems on which they can work together during the week. Active participation and interaction will be a key aspect of the mathematical school. This school also serves the purpose of preparing students for a sequence of conferences and workshops to be organized in the near future.

• Practical information

The school will take place on Boğaziçi university campus in Istanbul from 4th to 8th of June 2018.

We have financial support available for young participants to partially/totally cover accommodation and travel expenses. Applications from developing countries are especially encouraged.

In order to attend the summer school, please send to us a short CV (containing mathematical background and name of one/two reference(s)). Indicate as well if you request financial support for travel and/or accommodation. Deadline for applications will be 15 of May. Female mathematicians are particularly encouraged to apply.

• Lecturers and courses

Nurdagül Anbar RICAM, Linz Algebraic function fields and applications to cryptography and coding theory

Anne Frühbis-Krüger Leibniz Universität Hannover Algorithmic Aspects of Desingularization of 2-dimensional schemes

Christophe Ritzenthaler University Rennes 1 Models of curves of low genus and number of points over finite fields

• Description of each course

Nurdagül Anbar

-Algebraic function fields and their extensions -Congruence function fields, rational places, The Hasse-Weil bound, maximal curves, Serre-bound, Ihara bound, Drinfeld-Vladut bound -Towers of function fields -Examples of exceptional function fields (Hermitian, Ree, Suzuki, GK, etc.) -Goppa codes, almost perfect sequences, low-discrepancy sequences

Anne Frühbis-Krüger

Resolution of singularities is a well-known theoretical and practical tool in algebraic geometry over fields of characteristic zero and for surfaces also in arbitrary characteristic. For surfaces in mixed characteristic, it is by far less used, as there are a number of additional algorithmic issues to consider. In this mini-course we shall first explore the tasks and challenges of algorithmic desingularization in general and explicitly provide basic algorithmic tools like blow-ups. In the second lecture we will focus on the desingulariziation invariant which controls the whole resolution process and on algorithmically determining the locus of maximal order. The last of the three lectures will be devoted to a variant of the algorithm of Cossart-Jannsen-Saito for resolving 2-dimensional schemes in arbitrary and mixed characteristic. The mini-course is designed for PhD-students and advanced Master-students, so there will be an emphasis on instructive examples and explicit algorithmic techniques.

Christophe Ritzenthaler

• Review of some basic properties: divisors, Riemann-Roch, Riemann-Hurwitz
• The canonical map and hyperelliptic curves/non hyperelliptic curves
• Equation of curves of genus up to 5
• Weil conjectures for curves with the proof of Stepanov/Bombieri
• Jacobians: application to cryptography and construction of curves with many points

The organization committee, Alp Bassa and Christophe Ritzenthaler

See the school website.

The International Workshop on Algebraic Topology is a four day event which will take place at the Southern University of Science and Technology in Shenzhen, China from June 6 to June 9, 2018. This is a satellite event for the joint AMS-CMS Special Session on Algebraic and Geometric Topology which will take place at Fudan University in Shanghai, China from June 11 to June 14, 2018.

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 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.

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.

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.

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.

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

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.

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.

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

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.

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).

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.

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.

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

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 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.

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.

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.

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.

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

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.

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.

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 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.

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.