The programme will focus on the areas of Algebraic K-theory, Algebraic Cycles and Motivic Homotopy Theory. These are fields at the heart of studying algebraic varieties from a cohomological point of view, which have applications to several other fields like Arithmetic Geometry, Hodge theory and Mathematical Physics.

It was in the 1960s that Grothendieck first observed that the various cohomology theories for algebraic varieties shared common properties, which led him to explain the underlying kinship of such cohomology theories in terms of a universal motivic cohomology theory of algebraic varieties. The theory of Algebraic Cycles, Higher Algebraic K-theory, and Motivic Homotopy Theory are modern versions of Grothendieck's legacy. In recent years it has seen some spectacular developments, on which we want to build further.

The programme will also specifically explore the connections between the following areas:

Algebraic K-theory, Motivic Cohomology, and Motivic Homotopy Theory;
Hodge theory, Periods, Regulators, and Arithmetic Geometry;
Mathematical Physics.

For this, we shall bring together mathematicians working on different aspects of this broad area for extended periods of time, promoting exchange of ideas and stimulating further progress.

During the programme there will be four workshops. At the very beginning, there will be a workshop aimed at giving a younger generation of mathematicians an overview of and introduction to this interesting, but broad area. Later there will be a workshop for each of the three areas listed above, aimed at the latest developments and applications of that area.

The study of integer lattices serves as a bridge between number theory and geometry and has for centuries received the attention of illustrious mathematicians including Lagrange, Gauss, Dirichlet, Hermite and Minkowski. In computer science, lattices made a grand appearance in 1982 with the celebrated work of Lenstra, Lenstra and Lovász, who developed the celebrated LLL algorithm to find short vectors in integer lattices. The role of lattices in cryptography has been equally, if not more, revolutionary and dramatic, playing first a destructive role as a potent tool for breaking cryptosystems, and later as a new way to realize powerful and game-changing notions such as fully homomorphic encryption. These exciting developments over the last two decades have taken us on a journey through such diverse areas as quantum computation, learning theory, Fourier analysis and algebraic number theory.

We stand today at a turning point in the study of lattices. The promise of practical lattice-based cryptosystems together with their apparent quantum-resistance is generating a tremendous amount of interest in deploying these schemes at internet scale. However, before lattice cryptography goes live, we need major advances in understanding the hardness of lattice problems that underlie the security of these cryptosystems. Significant, ground-breaking progress on these questions requires a concerted effort by researchers from many different areas: (algebraic) number theory, (quantum) algorithms, optimization, cryptography and coding theory.

The goal of the proposed special semester is to bring together experts in these areas in order to attack some of the main outstanding open questions, and to discover new connections between lattices, computer science, and mathematics. The need to thoroughly understand the computational landscape and cryptographic capabilities of lattice problems is greater now than ever, given the possibility that secure communication on the internet and secure collaboration on the cloud might soon be powered by lattices.

VANTAGe is a new virtual seminar on open conjectures in number theory and arithmetic geometry (NT&AG). The seminar will provide open access to world class mathematics, with a focus on progress on unsolved problems in NT&AG. The purpose of the seminar is to provide a viable way for researchers to be involved with cutting-edge research in NT&AG without the expense and environmental impact of travel. Another aim of the seminar is to advance understanding on the most exciting open problems in this field. As the goal of the seminar is to build communication among researchers developing the fields of NT&AG, we expect speakers and participants to uphold the highest standards for clear exposition and respectful interactions.

This first open conjecture for the seminar is about Class groups of number fields. The lectures for this topic will be on 1/21, 2/4, 2/18, 3/3 at 1 ET = 12 CT = 11 MT = 10 PT.

Invited Speakers: Chao Li (Columbia University), Raphaël Rouquier (UCLA), John Voight (Dartmouth College), Melanie Matchett Wood (UC Berkeley), and Daxin Xu (CalTech).

The Banff International Research Station will host the "Equivariant Stable Homotopy Theory and p-adic Hodge Theory" workshop in Banff from March 1 to March 06, 2020.

Algebraic topology has had a long and fruitful collaboration with algebraic geometry, with each providing techniques and problems to the other. This workshop is aimed at an exciting, evolving incarnation of this story: applications of equivariant stable homotopy to number theory. Recent work on the foundations of equivariant stable homotopy theory (starting with the Hill--Hopkins--Ravenel work on the Kervaire invariant one problem) and Lurie's development of the foundations of ''derived algebraic geometry'' now allows systematic exploration and organization of ''equivariant derived algebraic geometry''. This allows us to do ordinary algebraic geometry in commutative ring spectra.

New foundations in this area have been spectacularly applied to phenomena seen in the trace methods approach to computing algebraic K -theory. For instance, although the theory of equivariant commutative ring spectra was described decades ago, few of the subtleties in the theory were understood or explored. The modern approaches to computing algebraic K-groups step through equivariant commutative ring spectra via the natural S1-action on topological Hochschild homology. Ongoing and transformative work by Bhatt--Morrow-Scholze in p-adic Hodge theory uses cyclotomic spectra and therefore subtle equivariant information. This workshop, at the vanguard of work in this area, seeks to bring together experts in algebraic topology, (derived) algebraic geometry, and number theory to explore these exciting new connections.

This Spring School will gather together PhD students and junior researchers who use category-theoretic ideas or techniques in their research. It will provide a forum to learn about important themes in contemporary category theory, both from experts and from each other.

Three invited speakers will each present a three-hour mini-course, accessible to non-specialists, introducing an area of active research. There will also be short talks contributed by PhD students and postdocs, and a poster session.

The focus of the Spring School is on aspects of pure category theory as they interact with research in other areas of algebra, geometry, topology and logic. Any "categorical thinker" - that is, any mathematician whose work makes use of categorical ideas - is welcome to participate.

Organizing Committee: Caucher Birkar (University of Cambridge), Christopher Hacon (the University of Utah), Chenyang Xu (M.I.T.) with help from Jingjun Han (Johns Hopkins University).

Principal Japanese Organizers: Keiji Oguiso (University of Tokyo), Shunsuke Takagi (University of Tokyo).

This one year long program at Johns Hopkins University will feature 3 graduate-level courses, one conference, three Kempf lectures, three Monroe H. Martin lectures, several colloquiums and weekly seminars.

Tentative schedule for the conference: March 16--22, 2020.

Conference on moduli spaces.

Periods occur in various branches of mathematics and as the title of our conference indicates, their study intertwines arithmetic, Diophantine analysis, differential equations, and algebraic geometry. Many interesting results have been proved in recent years and many challenging problems on periods are still open. The aim of our conference is to bring together specialists who cover all these different points of view and their ramifications, with special attention towards possible applications to broader areas of the techniques developed in the study of periods and their realizations.

Yves André has contributed in many ways to this ongoing adventure and this conference will not only be the opportunity to listen to a broad range of recent developments in mathematics around the topic of periods, but also to celebrate his 60th birthday.

The Georgia Algebraic Geometry Symposium is a conference series, jointly organized by the University of Georgia, Emory University and Georgia Tech.

The general topic of this conference is number theory with a focus on special values of L-functions. Please consult the web page for further information.

We are organising a three day meeting in honour of Professor Sir Peter-Swinnerton-Dyer who died in December 2018 at the age of 91.

Swinnerton-Dyer was one of the most influential number theorists of his generation worldwide. He is probably best known for the famous conjecture of Birch and Swinnerton-Dyer (one of the Millenium Clay Maths Problems), which relates the arithmetic of elliptic curves to the value of its Hasse-Weil L-function. This conjecture gave rise to a huge field of research relating special values of L-functions to arithmetic data. Swinnerton-Dyer was also one of founding figures in the arithmetic of surfaces and higher-dimensional varieties, such as local-to-global principles for rational points over number fields. He obtained fundamental results for conic bundles and cubic surfaces, and started the research of rational points on surfaces fibred into elliptic curves using completely new methods.

The meeting aims to celebrate the tremendous and wide-reaching contributions to mathematics of the late Sir Peter Swinnerton-Dyer.

In the early years of the Newton Institute, Swinnerton-Dyer served as honorary executive director under Michael Atiyah.

Speakers:

Brian Birch
Martin Bright
John Coates
Lilian Matthiesen
Alexei Skorobogatov
Rodolfo Venerucci
Claire Voisin
Andrew Wiles 
Olivier Wittenberg
Sarah Zerbes
Henri Darmon

This school provides an introduction to some of the main topics of the trimester program. It is mainly directed at PhD students and junior researchers.

The following speakers will give courses on the following topics:

• Arthur-Cesar le Bras, Gabriel Dospinescu: p-adic geometry
• George Boxer, Vincent Pilloni: Higher Hida theory
• Patrick Allen, James Newton: Automorphy lifting
• Eva Viehmann, Cong Xue: Shtukas
• Sophiel Morel, Timo Richarz: Geometric Satake

In recent years a remarkable new link has been unfolding between o-minimality (a branch of model theory), on the one hand, and Diophantine geometry and Hodge theory on the other. It has been discovered that some central objects of study in the latter two areas, such as theta functions and period maps, can be studied within the structure R_{an,exp}, an o-minimal structure that has long been an object of investigation in o-minimal geometry. A combination of ideas from these three areas has led to resolutions of several outstanding conjectures in Diophantine geometry and, more recently, in Hodge theory.

The goal of this workshop will be to introduce these three different areas and their fruitful interactions, and to serve as a bridge between experts already working in some of these fields. The schedule will consist of tutorial minicourses on the three main topics (o-minimality, Diophantine geometry and Hodge theory) with a focus on the areas where they intersect; and more advanced research talks by experts in each of these areas.

This workshop is one of the workshops of a special RIMS year "Expanding Horizons of Inter-universal Teichmüller Theory". The workshop will review fundamental developments in several branches of anabelian geometry, as well as report on recent developments. The list of speakers includes major contributors to anabelian geometry and birational anabelian geometry. Anabelian geometry, together with higher class field theory and the Langlands correspondences, is one of three generalisations of class field theory.

The Langlands program and the theory of automorphic forms are fundamental subjects of modern number theory. Langlands’ principle of functoriality, and the notion of automorphicL-functions, are central pillars of this area. After more than forty years of development, andmany celebrated achievements, large parts of this program are still open, and retain their mystery. The theory of endoscopy, a particular but very important special case of functoriality, has attracted much effort in the past thirty years. This has met with great success, leading to the proof of the Fundamental Lemma by Ngô, and the endoscopic classification of automorphic representations of classical groups by Arthur. Going beyond these remarkable achievements requires new techniques and ideas ; in the past few years, exciting directions have started to emerge, which may renew our vision of the whole subject.

This brings us to the three main topics of this conference : (1) The "relative Langlands program" is a very appealing generalization of the classical Langlands program to certain homogeneous spaces (mainly spherical ones). It relates period integrals of automorphic forms to Langlands functoriality or special values of L-functions.Remarkable progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures has been made(by Waldspurger, W. Zhang, and others). With the work of Sakellaridis and Venkatesh, this subject has reached a new stage ; we now have a rigorous notion of "relative functoriality",with very promising perspectives. A central tool in all these questions is Jacquet’s relative trace formula, whose reach and theoretical context remain to be fully investigated. (2) Relations between special values of (higher) derivatives of L-functions and height pairings (between special cycles on Shimura varieties and Drinfeld’s Shtuka stacks) will also be part of the program. This includes arithmetic versions of the Gan-Gross-Prasad conjectures, which generalize the celebrated Gross-Zagier formula. In the function field case, striking recent results of Yun and W. Zhang give geometric meaning to higher central derivatives of certain L-functions. (3) Ways of going beyond endoscopy, and proving new cases of functoriality. This includes Langlands’ original idea of using the stable trace formula to study poles of L-functions ; but also other related proposals that have attracted lot of recent attention, such as the Braverman-Kazhdan approach through non-standard Poisson summation formulas, or new methods to go "beyond endoscopy in a relative sense”, as developed by Sakellaridis.

This conference aims to gather leading experts in this vital area of mathematics (including several researchers from Aix-Marseille University) ; to attain a state-of-the-art overview of the different directions that are being actively pursued ; and to promote collaboration and the exchange of ideas between those approaches.

Speakers

In the recent years, new connections have emerged between complex algebraic varieties and non-archimedean spaces. They could be made precise by relying on the theory of Berkovich spaces and tropical geometry. The aim of the workshop is to present the latest techniques as well as several applications.

The Casa Matemática Oaxaca (CMO) will host the "Advances in Mixed Characteristic Commutative Algebra and Geometric Connections" workshop in Oaxaca, from June 7 to June 12, 2020.

One of the big ideas in modern mathematics is that integers (like 1, 2, 3, 4, 5, ...) in many formal ways behave similarly to polynomial equations (like y = x^2, which defines the parabola). Frequently, and perhaps surprisingly, many questions in mathematics are easier to study for polynomials than for integers. Hence intuition and results for polynomials can tell us about the integers. Commutative algebra lives at the intersection of both perspectives, and one fundamental object of study is polynomials with integer coefficients, this is called the mixed characteristic case. Recently, Yves Andre proved a long standing open conjecture in commutative algebra in this mixed characteristic setting, relying on constructions of Scholze (and then Bhatt gave a simplified proof of the same conjecture).

This workshop aims to foster and discuss these and other recent tools, to study some remaining open problems in mixed characteristic. The workshop will bring together a diverse group of researchers from different fields, such as commutative algebra, algebraic geometry, and number theory.

This conference is part of the celebration of the recent creation of the new Number Theory group at the University of Bath.

It will revolve around topics in analytic number theory, algebraic number theory, and algebraic geometry, related to the study of Diophantine equations.

With this workshop we would like to promote the interaction between the following five fields:

• Berkovich spaces,
• Tropical geometry,
• p-adic differential equations,
• Arithmetic D-modules and representations of p-adic Lie groups,
• Arithmetic applications of p-adic local systems.

While the first two are already tightly linked, the role of Berkovich spaces in the last 3 topics is only emerging and within this, the role of tropical geometry has not yet been explored. More generally, we consider this conference to be a good opportunity to study new techniques recently introduced into the field. We are convinced that each of these areas has plenty of potential and that a fruitful interaction between them might nourish their development. The aim of the conference is precisely to give leading experts in these each of these domains the opportunity to meet, present their last results and open challenges, and encourage an exchange that will drive forward these exciting and rapidly developing subjects.

A poster session is planned. Students are welcome to submit posters.

A week-long course designed for PhD students in nearby fields.

Speakers: Johannes Nicaise and Tony Yue Yu.

Last day to register: April 15.

Combinatorial anabelian geometry concerns the reconstruction of scheme- or ring-theoretic objects from more primitive combinatorial constituent data. In this sense, it is closely philosophically related to inter-universal Teichmüller theory.

The purpose of the present workshop is to expose fundamental, introductory aspects of combinatorial anabelian geometry, as well as more recent developments related to the Grothendieck-Teichmüller group and the absolute Galois groups of number fields and mixed-characteristic local fields.

The workshop will also treat results concerning the "resolution of nonsingularities" of hyperbolic curves over mixed-characteristic local fields, such results are closely related to combinatorial anabelian geometry over mixed-characteristic local fields.

The subject of this meeting covers valuation theory and resolution of singularities, along with some topics that are closely related like the theory of singularities of vector fields, problems concerning arc spaces or the Pierce-Birkhoff Conjecture.

It is our aim to gather together researchers on these transversal subjects in order to strengthen interdisciplinarity between different thematics, to contribute to build a community of researchers working on these problems, to develop new research projects, and to support new collaborations.

The ANTS meetings, held biannually since 1994, are the premier international forum for the presentation of new research in computational number theory and its applications. They are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, algebraic geometry, finite fields, and cryptography.

The lecture series and research talks at the IHES Summer School will focus on presenting the latest developments in topics related to categories of motives, calculational and foundational aspects of motivic and equivariant homotopy theory, and the generalisations of these tools and techniques in the setting of non-commutative geometry.

This conference will be on various aspects of the local Langlands correspondence over p-adic fields and methods from p-adic Hodge theory. Topics will include the usual local Langlands correspondence, the p-adic local Langlands correspondence and the relation to coherent sheaves on spaces of Galois representations, and the geometry and cohomology of local Shimura varieties.

The International Symposium on Symbolic and Algebraic Computation (ISSAC) is the premier conference for research in symbolic computation and computer algebra. ISSAC 2020 will be the 45th meeting in the series, which started in 1966 and has been held annually since 1981. The conference presents a range of invited speakers, tutorials, poster sessions, software demonstrations and vendor exhibits with a center-piece of contributed research papers.

The principal aim is to bring together leading researchers interested in the quantitative arithmetic of higher-dimensional varieties on the one hand, and those working on the statistics of algebraic number fields and elliptic curves on the other hand. These two areas have operated more or less independently over the last twenty years, but their borders share an increasingly rich seam of mathematics.

Higher Dimensional Geometry in NYC is a series of six conferences in Higher Dimensional Geometry which will be held at the Simons Foundation in New York City and at the Simons Centre in Stony Brook over the period 2020-2022.

The Women in Algebraic Geometry Collaborative Research Workshop will bring together researchers in algebraic geometry to work in groups of 4-6, each led by one or two senior mathematicians. The goals of this workshop are: to advance the frontiers of modern algebraic geometry, including through explicit computations and experimentation, and to strengthen the community of women and non-binary mathematicians working in algebraic geometry. This workshop capitalizes on momentum from a series of recent events for women in algebraic geometry, starting in 2015 with the IAS Program for Women in Mathematics on algebraic geometry.

Successful applicants will be assigned to a group based on their research interests. The groups will work on open-ended projects in diverse areas of current interest, including moduli spaces and combinatorics, degenerations, and birational geometry. Several of the proposed projects extensively involve experimentation and computation, which will increase the likelihood that concrete progress is made over the course of five days and provide useful training in computational mathematics.

Description:

Non-archimedean geometry is the analogue of complex geometry, where the field of complex numbers is replaced by a field which is complete with respect to a p-adic metric. A fundamental complication is that p-adic spaces are totally disconnected, and therefore basic notions such as analytic con- tinuation must be entirely recast in different language. Nevertheless, the particular properties of the p-adic topologies, while perverse in some sense, provide the key to a rich, fulfilling, and ultimately productive theory. There are various manifestations of non-archimedean geometry, e.g. rigid analytic spaces a la Tate, Berkovich spaces or adic spaces a la Huber. For this summer school we take the point of view of adic spaces with emphasis on rigid analytic spaces which form special examples. Indeed, non-archimedean geometry and the associated area of p-adic Hodge theory for Galois representations play a central role in modern algebraic number theory. It has become increasingly clear that active researchers in algebraic number theory would be greatly benefitted by having a working knowledge of p-adic geometric methods.

Eigencurves - and more generally, eigenvarieties - are rigid analytic versions of modular curves, which parametrize p-adic families of modular forms. The study of such families may be said to have started with Serre in the 1970s, and was extensively developed by Hida in the 1980s; Hida's work on the so-called ordinary modular forms, in particular, was deeply influential in the eventual proof of modularity of elliptic curves by Wiles and others in 1994. However, it was clear even from looking at Serre's results, that a fully satisfactory theory of p-adic families would require consideration of non-ordinary forms, and that such a theory would necessarily require fundamental new ideas. These results were eventually supplied by Coleman in the mid-1990s, and the eigencurve parametrizing was introduced as a parameter space by Coleman and Mazur shortly thereafter. The subject has exploded in the last decades, with generalizations of the eigencurve to higher rank groups, and with the use of increasingly sophisticated technology from p-adic geometry. Furthermore, p-adic families of automorphic forms have taken on an increasingly important role in modern number theory.

The subject of eigencurves lies somewhere between classical arithmetic geometry represented in Canada, and p-adic geometry which is well-represented in Germany, and this we propose to take advantage of the complementary expertise and the broad outlines of a PIMS/Germany collaboration to organize an event where both sides can benefit. Thus, the proposed workshop on eigenvarieties will be an instructional school for students, postdocs, and researchers in other fields. The goal is to provide beginners with a working knowledge of this immensely active and important field, and to encourage collaborations between German researchers and those at PIMS Institutes around the general workshop themes.

Topics of Instruction:

The goal of the course will be to understand the foundational work of Coleman, and Coleman-Mazur, and eventually to study the paper on the generalization to the higher dimensional case given by Buzzard. A good overview of the subject is given in the survey article of Kassei.

Invited Speakers:

• John Bergdall, Bryn Mawr College, USA
• George Boxer, University of Chicago, USA
• David Hansen, Max Planck Institute, Germany
• Eugen Hellman, University of Münster
• Christian Johansson, Chalmers Institute of Technology, Sweden
• Judith Ludwig, University of Heidelberg, Germany
• James Newton, King’s College, England
• Vincent Pilloni, Ecole Normale Supérieure de Lyon, France

This will be a workshop in arithmetic and algebraic geometry, similar to the previous iteration (https://stacks.github.io/2017/). The intended participant is a graduate student, or a postdoc, or even a senior researcher. You will work on a single topic in a small group together with a mentor for a week with the aim of producing a text that will be considered for inclusion in the Stacks Project. Part of this process will be seeing how one builds new theory from the foundations. There will also be one or two talks per day covering advanced topics in arithmetic or algebraic geometry.

The Stacks project workshop will have some optional activities you won't see at other workshops. Adding references to and finding mistakes in the Stacks Project (and fixing them) as well as activities related to LaTeX use, Git, and GitHub. Overall these will be aimed at helping you contribute efficiently to the Stacks Project.

The Eighth Pacific Rim Conference on Mathematics will be held at the University of California at Berkeley from Monday 3th August to Friday 7th August 2020. The PRCM will be a high profile mathematical event that will cover a wide range of exciting research in contemporary mathematics. Its objectives are to offer a venue for the presentation to and discussion among a wide audience of the latest trends in mathematical research, and to strength ties between mathematicians working in the Pacific Rim region. The conference will provide junior researchers with opportunities to engage and collaborate with established colleagues within and between the many represented mathematical disciplines.

Mini-courses by Matt Kerr, Andreas Mihatsch, Colleen Robles, plus a couple of research talks.

This conference will be on various aspects of the global Langlands correspondence. Topics will include in particular the geometry and cohomology of Shimura varieties and more general locally symmetric spaces, or moduli spaces of shtukas.

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.

The aim of the workshop is to discover how the problems in number theory and algebraic geometry arising from the Hilbert’s tenth problem for rationals interact with the ideas and techniques in mathematical logic, such as definability from model theory and decidability and degree-theoretic complexity from computability theory. This interaction includes various analogues of Hilbert’s tenth problem and related questions, focusing on the connections of algebraic, number-theoretic, model-theoretic, and computability-theoretic properties of structures and objects in algebraic number theory, anabelian geometry, field arithmetic, and differential algebra.

This conference is dedicated to studying recent advancements concerning Serre weights conjectures and the geometry of Shimura varieties and, in particular, the interaction between these two areas.

Homological tools and ideas are pervasive in number theory. To defend this assertion, it suffices to evoke the role of étale cohomology in the study of the zeta functions of varieties over finite fields through the Weil conjectures, or the cohomological approach to class field theory formulated by Artin and Tate in the 1950's. The theory of motives, a manifestation of a universal cohomology theory attached to algebraic varieties, and the attendant motivic cohomology plays a central role in describing the special values of L-functions of varieties over number fields, via the conjectures of Deligne, Beilinson-Bloch, and Bloch-Kato. Much progress in the Langlands program exploits the fruitful connection between automorphic representations and the cohomology of associated Shimura varieties and more general arithmetic quotients of locally symmetric spaces. The study of special values of L-functions and the Langlands program, widely perceived as two fundamental yet seperate strands of the subject in the early 1990's, were beautifully unified in Wiles' epoch-making proof of the Shimura-Taniyama conjecture, in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic curve. Recent years have seen great strides in our understanding of the cohomology of the arithmetic quotients arising in the study of automorphic representations, spurred in part by the desire to extend the range of applicability of the celebrated Taylor-Wiles method. This has led to new automorphy and potential automorphy results: most spectacularly, perhaps, for abelian surfaces, as well as elliptic curves over general CM fields.

The Banff International Research Station will host the "Arithmetic Aspects of Algebraic Groups" workshop in Banff from September 6 to September 11, 2020.

The investigation of arithmetic groups has been an active and important area of mathematical research ever since it arose in the work of Gauss, Klein, Poincare, and other famous mathematicians of the 18th and 19th centuries. New points of view have recently led to progress on classical problems, opened new directions of inquiry, and revealed unexpected connections with other areas of mathematics. The workshop will bring together experts in the area, researchers in related fields, and young mathematicians who wish to learn about the most recent advances and the most promising directions for the future of the field.

The conference Arakelov Geometry is organized by

José Ignacio Burgos Gil, Walter Gubler, and Klaus Künnemann.

This conference constitutes the eleventh session of the Intercity Seminar on Arakelov Theory organized by Jose Ignacio Burgos Gil, Vincent Maillot and Atsushi Moriwaki with previous sessions in Barcelona, Beijing, Copenhagen, Kyoto, Paris, Regensburg, and Rome.

The GTA 2020 conference will bring together world class researchers in mathematics. Its main objectives are to discuss recent advances in the fields of algebraic geometry, number theory and their applications, as well as to foster international collaborations on connected topics.

Although contributions from all related areas of mathematics are welcome, particular emphasis will be placed on research interests of our late colleague Alexey Zykin, namely: zeta-functions and L-functions, algebraic geometry over finite fields, families of fields and varieties, abelian varieties and elliptic curves.

In Iwasawa Theory, one of the central questions is the study of the Iwasawa main conjecture, which relates the characteristic ideal of the Selmer group of a motive to its p-adic L-function (when it exists). This in turn leads to information on the Bloch-Kato conjecture, a generalization of the Birch and Swinnerton-Dyer conjecture. Cases of the Iwasawa main conjecture have been established using the machinery of Euler systems, which are collections of cohomology classes satisfying certain norm relations and are related to the L-function of a motive and were first introduced and exploited in the late 80s and early 90s in the works of Thaine, Kolyvagin, Rubin, and Kato.

Bernadette Perrin-Riou, one of the influential, pioneering figures in Iwasawa Theory in the 1990s, is widely acclaimed for the influential ideas she has brought to the subject. Her deep study of the Euler system originally constructed by Kato led to the introduction of her fundamental big logarithm map" (often refereed as thePerrin-Riou map" nowadays), which is a far reaching generalisation of the Coleman power series and is one of the key ingredients in establishing links between Euler systems and p-adic L-functions. Her work also initiated the study of higher rank Euler systems and has been a source of inspiration for many further developments in this direction. Likewise, her p-adic analogue of the Gross-Zagier formula has opened up an area of enquiry that remains active and fertile to the present day. All these, as well as many other important contributions of Perrin-Riou, continue to serve as a model and a guide for today's research in Iwasawa Theory. This workshop is therefore dedicated to the celebration of her 65th birthday.

In the first decade of this century, further progress in the theory of Euler systems was stymied by the fact that few instances were known beyond the basic examples (circular units, elliptic units, Heegner points, and Beilinson elements) introduced and exploited by Thaine, Rubin, Kolyvagin and Kato respectively. Around 2010, the scope of Kato's construction was extended to encompass p-adic families of cohomology classes arising from Beilinson-Flach elements, and diagonal cycles in triple products of Kuga-Sato varieties, with application to the Birch and Swinnerton conjecture in analytic rank zero, in the spirit of the early work of Coates and Wiles. Important progress was then made in establishing the Euler system norm compatibilities of Beilinson-Flach elements. This has opened the floodgates for a wide variety of new Euler system constructions, applying notably to the Rankin-Selberg convolution of two modular forms, Siegel modular forms on GSp(4) and GSp(6), as well as Hilbert modular surfaces. At around the same time, and quite independently, a markedly different strategy has been proposed for studying diagonal on triple products based on congruences between modular forms instead of $p$-adic deformations, leading to remarkable constructions whose scope has the potential to surpass the more traditional approach based on norm-compatible elements. Finally, important progress arising from the method of Eisenstein congruences offer a powerful complementary approach, greatly contributing to the power, usefulness, and widening appeal of Euler system techniques.

The workshop will precede the annual Quebec-Maine conference which will take place at Laval University on Saturday and Sunday (September 26-27, 2020). The workshop will end on Friday at noon so that those who wish to attend can travel to Quebec City in the afternoon. (A roughly 3 hour trip by train or by bus.)

Witt vectors are well-known for their ability to pop up in unexpected places, and can serve as a good starting point for a conversation on many interesting and diverse mathematical subjects. The goal of the conference is to have such a conversation on several recent subjects where Witt vectors appeared yet again (including but not limited to the usual and topological Hochschild Homology, both commutative and non-commutative, prismatic cohomology and p-adic Hodge theory, polynomial functors, geometric representation theory in char p).

If G is a reductive algebraic group over Z, the group G(Z) of its integral points (or any congruence subgroup thereof) acts discretely on the locally symmetric space X:= G(R)/K, where K is a maximal compact subgroup of G(R). The quotients G(Z) X play a fundamental role in the theory of automorphic forms and in number theory. Notably, their cohomology is a rich source of invariants attached to automorphic representations of G, and thus plays a central role in the Langlands program. A fundamental trichotomy governing the topological behaviour of such arithmetic quotients was proposed around 2010 by Bergeron and Venkatesh. A single positive integer d, depending only on the overlying symmetric space X, dictates the expected behaviour of the homology of the arithmetic quotient. When d=0, the cohomology is expect to have very little torsion but lots of characteristic 0 homology, which can be studied via analytic and transcendental methods (de Rham cohomology, Hodge theory). Shimura varieties and even-dimensional real hyperbolic spaces fall into this class. When d=1, one expects to find a lot of torsion but very little characteristic 0 homology. Odd dimensional hyperbolic manifolds, such as the Bianchi three-fold SL2(Z[i]) SL2(C)/U(2), fall into this case. When d is greater than 1, one expects little torsion and little characteristic zero homology.

There has been remarkable progress towards understanding how this trichotomy interacts with arithmetic: When d = 0, several interesting recent torsion-freeness results have been obtained by researchers like Caraiani, Emerton, Gee, and Scholze. When d=1, one can ask whether torsion always arises when it's expected to, and with the expected abundance. Torsion can be probed analytically using the Cheeger-Muller theorem. But there are obstructions ("tiny eigenvalues" and "very complex cycles"), which are very interesting in their own right, and need to be overcome in order to prove that there's as much torsion as expected. This torsion growth problem, especially for hyperbolic three-manifolds, has a life of its own even outside number theory, notably in the community of geometric groups theorists. Among the most striking developments arising in the relatively less well explored setting where d is larger than 1, let us mention Peter Scholze's construction of Galois representations attached to (possibly torsion) eigenclasses in the cohomology of arithmetic quotients, which is especially deep in this case. Another highly promising, fundamental breakthrough is manifested in Akshay Venkatesh's conjecture on derived Hecke algebras, which is expected to play an important role in extending the scope of the Taylor-Wiles method beyond the setting of d=0 to which it had been confined until relatively recently. The deep study of torsion in homology and analytic torsion carried out earlier by Bergeron, Venkatesh and others played a very important part in the nascent theory of derived Hecke operators and the attendant motivic action on the cohomology of arithmetic groups. In some very special instances, where G=GL(2) and one focusses on the coherent cohomology of an arithmetic quotient with values in certain automorphic sheaves, Venkatesh's conjectures exhibit a tantalising connection with certain ``tame refinements", in the spirit of conjectures of Mazur and Tate, of conjectures on the values of triple product p-adic L-functions.

The field is still in a very exploratory stage in which precise expectations (conjectural or otherwise) have not yet fully cristallised. For instance, there does not yet seem to be a reasonable conjecture about "how much cohomology", torsion or characteristic zero, to expect when d is greater than 1. Among other reasons, this makes computing in this setting very interesting. The workshop is expected to have a significant computational and experimental component, in which various experts will report on experimental data that might prove valuable in solidifying our expectations.

The Banff International Research Station will host the "WIN5: Women in Numbers 5" workshop in Banff from November 15 to November 20, 2020.

Despite recent progress in gender equality in STEM fields, women continue to be underrepresented in the research landscape of many areas of mathematics, including number theory. The Women in Numbers (WIN) network was created in 2008 for the purpose of increasing the number of active female researchers in number theory. For this purpose, WIN sponsors regular conferences, taking place approximately every three years, where female scholars gather to collaborate on cutting-edge research in the field and produce publishable scientific results. The WIN workshops provide an ongoing forum for involving each new generation of junior faculty and graduate students in state-of-the-art research in number theory. They have to come be highly regarded among the broader number theory community due to the quality of research produced by these collaborations.

WIN5 is the fifth in this series of events, bringing together female number theorists at various career stages for research collaboration and mentorship. As always, the scientific program will centre on onsite collaboration on open research problems in number theory, conducted in small groups comprised of senior and junior scholars as well as graduate students. Groups will publish their initial finding in a peer-reviewed conference proceedings volume, and research partnerships formed at the WIN5 workshop are expected to last well beyond the duration of the event. WIN projects have the potential to grow into fruitful long-term research alliances that have a transforming influence on participants' careers and a significant positive impact on the research landscape in number theory. Past WIN workshop project groups have matured into highly effective research teams producing ongoing scholarly work of exceptional scientific quality.

The Casa Matemática Oaxaca (CMO) will host the "Langlands Program: Number Theory and Representation Theory" workshop in Oaxaca, from November 29 to December 04, 2020.

Langlands functoriality conjectures predict a vast generalization of the classical reciprocity laws of Class Field Theory, providing crossroads between Number Theory and Representation Theory. The conjectures are both local and global and pertain a connected reductive group and its Langlands dual group.

We aim to introduce young mathematicians in M\'exico and Latin-America to topics of current research in the Langlands Program. We will also promote the participation women and of graduate students from a diverse background in a workshop where experts in the field from across the world will gather to expand upon the frontiers of current research. In addition to research talks, there will be three courses that will also be accessible to mathematicians working in closely related fields.

The workshop will be devoted to the varied and fruitful interactions between p-adic cohomology theories, the theory of p-adic deformations of modular forms and Galois representations, and the construction of p-adic L-functions arising from the latter using techniques drawn from the former, with special emphasis on their rich array of arithmetic applications.

The field of p-adic automorphic forms has seen a huge development in the last decades with the construction of p-adic families in many new context. Among this one can cite Hansen's construction of eigenvarieties using overconvergent cohomology, and the coherent approach using (partial) Igusa towers of Andreatta–Iovita–Pilloni. There have been immediate applications to the construction of p-adic L-functions in families and to the proof of several instances of the conjectures by Greenberg and Benois on trivial zeroes, such as the work of Barrera–Dimitrov–Jorza.

But the existence of these eigenvarieties have proved to be useful also for the study of many other interesting arithmetic problems.

The first example is given by the applications to the Bloch–Kato conjecture. Bloch and Kato conjectured that the most interesting arithmetic information concerning varieties (and more generally, geometric Galois representations) are contained in two objects: the Selmer group and the L-function. They also conjecture that all the information coming from the Selmer group can be recovered from the L-function. Some special cases of this conjecture have been proven: we cite for example the work of Bellaiche–Chenevier for unitary groups and Skinner–Urban for elliptic curves in rank less or equal than 2. The key ingredients in these works is the use of deformations of automorphic forms and their Galois representations in p-adic families to construction elements in the Selmer group.

Another example is the study of local properties of Galois representations and the corresponding p-adic Hodge theory. We cite the work of Kedlaya, Pottharst, and Xiao concerning the existence of triangulations in families for p-adic representations of p-adic fields arising from finite slope automorphic forms. Other related results are the works on the smoothness of eigenvarieties at critical points by Bergdall and Breuil–Hellmann–Schraen; it has implication on the existence of companion forms, which are different p-adic automorphic forms sharing the same Galois representations (such as a CM form and its Serre antiderivative).

Very recently two new geometric approaches have been developed in the study of p-adic families and their L-functions.

Andreatta and Iovita introduced the idea of vector bundles with marked sections, which not only allows one to recover their previous constructions of eigenvarieties but let them p-adic interpolate in families the de Rham cohomology of the modular curve and the Gauss–Manin connection. They can then construct triple product p-adic L-functions for finite slope families and anticyclotomic p-adic L-functions when p is inert in the CM field.

At the same time, the introduction of perfectoid spaces and adic geometric has brought new and fresh ideas in the field: one can cite the new construction of classical eigenvarieties by Chojecki–Hansen–Johansson using functions on the perfectoid tower of modular curves and the construction by Kriz of a new p-adic Maass–Shimura operator and anticyclotomic p-adic L-functions in the inert case. His strategy relies on Scholze's Hodge to de Rham comparison isomorphism, which has been recently upgraded to an integral comparison map by Bhatt–Morrow–Scholze.

These new geometric tools have already allowed the construction of new p-adic L-functions; the aim of the workshop is to bring together arithmetic people with the experts in these two innovative approaches to find new exciting applications, both to global (Galois representations and their L-functions) and local (integral p-adic Hodge theory) problems.

Combinatorial algebraic geometry comprises the parts of algebraic geometry where basic geometric phenomena can be described with combinatorial data, and where combinatorial methods are essential for further progress.

Research in combinatorial algebraic geometry utilizes combinatorial techniques to answer questions about geometry. Typical examples include predictions about singularities, construction of degenerations, and computation of geometric invariants such as Gromov-Witten invariants, Euler characteristics, the number of points in intersections, multiplicities, genera, and many more. The study of positivity properties of geometric invariants is one of the driving forces behind the interplay between geometry and combinatorics. Flag manifolds and Schubert calculus are particularly rich sources of invariants with positivity properties.

In the opposite direction, geometric methods provide powerful tools for studying combinatorial objects. For example, many deep properties of polytopes are consequences of geometric theorems applied to associated toric varieties. In other cases geometry is a source of inspiration. For instance, long-standing conjectures about matroids have recently been resolved by proving that associated algebraic structures behave as if they are cohomology rings of smooth algebraic varieties.

Much research in combinatorial algebraic geometry relies on mathematical software to explore and enumerate combinatorial structures and compute geometric invariants. Writing the required programs is a considerable part of many research projects. The development of new mathematics software is therefore prioritized in the program.

The program will bring together experts in both pure and applied parts of mathematics as well mathematical programmers, all working at the confluence of discrete mathematics and algebraic geometry, with the aim of creating an environment conducive to interdisciplinary collaboration. The semester will include four week-long workshops, briefly described as follows.

• A 'boot-camp' aimed at introducing graduate students and early-career researchers to the main directions of research in the program.

• A research workshop dedicated to geometry arising from flag manifolds, classical and quantum Schubert calculus, combinatorial Hodge theory, and geometric representation theory.

• A research workshop dedicated to polyhedral spaces and tropical geometry, toric varieties, Newton-Okounkov bodies, cluster algebras and varieties, and moduli spaces and their tropicalizations.

• A Sage/Oscar Days workshop dedicated to development of programs and software libraries useful for research in combinatorial algebraic geometry. This workshop will also feature a series of lectures by experts in polynomial computations.

This workshop aims at bringing together the leading experts in the field, covering a broad spectrum reaching from the more theoretically-oriented over the explicit to the algorithmic aspects. The fundamental problem motivating the workshop asks for a description of the set of rational points X(Q) for a given algebraic variety X defined over Q. When X is a curve, the structure of this set is known, and the most interesting question is how to determine it explicitly for a given curve. When X is higher-dimensional, much less is known about the structure of X(Q), even when X is a surface. So here the open questions are much more basic for our understanding of the situation, and on the algorithmic side, the focus is on trying to decide if a given variety does have any rational point at all.

This is a workshop with about 50 participants. Participation is by invitation. Every participant is expected to contribute actively to the success of the event, by giving talks and/or by taking part in the discussions.