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Welcome to MathMeetings.net! This is a list for research mathematics conferences, workshops, summer schools, etc. Anyone at all is welcome to add announcements.

## Know of a meeting not listed here? Add it now!

#### Updates 2017-10

- Secure connections (https) now activated and all traffic is automatically redirected to use https. Thanks to Let's Encrypt for providing the certificate!
- Spam protection now provided by Google reCaptcha.
- New json and xml interfaces for access by other software.

Additional update notes are available in the git repository (GitHub).

# Upcoming Meetings

## December 2018

### On the Langlands Program: Endoscopy and Beyond

Meeting Type: thematic program

Contact: see conference website

### Description

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.

## January 2019

### Explicit Number Theory

Meeting Type: conference

Contact: see conference website

### Description

### Arithmetic of Shimura Varieties

Meeting Type: conference

Contact: see conference website

### Description

### Atelier PARI/GP

Meeting Type: software workshop

Contact: see conference website

### Description

### Trace functions over finite fields and their applications in analytic number theory

Meeting Type: winter school

Contact: see conference website

### Description

### Derived Algebraic Geometry

Meeting Type: research program

Contact: see conference website

### Description

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.

### Connections for Women: Derived Algebraic Geometry, Birational Geometry and Moduli Spaces

Meeting Type: conference

Contact: see conference website

### Description

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.

### Abelian varieties over finite fields

Meeting Type: software development workshop

Contact: see conference website

### Description

This will be a hands-on workshop focused on a specific computational problem: enumerating all isomorphism classes of abelian varieties of of dimension g over a finite field of cardinality q, for a suitable range of integers g and q. Isogeny classes of abelian varieties over finite fields have been previously classified by Weil polynomials and can be found in the L-functions and Modular Form Database. The goal of this workshop is to refine this to the level of isomorphism classes, and, whenever possible, to construct explicit representatives for each isomorphism class. By exploiting recent theoretical and computational advances and assembling an appropriate team of experts, we hope to make rapid and substantial progress during this short, focused workshop and to have results available in advance of the conference on the Arithmetic of Low-dimensional Abelian Varieties that will take place at ICERM in June.

### Introductory Workshop: Derived Algebraic Geometry and Birational Geometry and Moduli Spaces

Meeting Type: conference

Contact: see conference website

### Description

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.

## February 2019

### Francophone Computer Algebra Days (Journées nationales de calcul formel)

Meeting Type: research school

Contact: see conference website

### Description

Computer Algebra refers to the study and design of algorithms for manipulating mathematical expressions and objects. It lies at the interface between Mathematics, Computer Science and various application fields. It covers a wide range of subjects, such as effective linear algebra, algorithmic number theory, integration and summation in closed form, differential and polynomial system solving, or special functions. The French Computer Algebra community is internationally renowned for the excellence of its theoretical work. Several libraries produced by its members are part of mainstream software packages such as Maple or Sage. This success is notably due to the Journées nationales de calcul formel (JNCF), which are a remarkable opportunity for researchers to discuss recent and ongoing work with their peers.

Expected outcomes include:

A better integration of young researchers. The JNCF are an ideal opportunity for young researchers to present their results for the first time and also to get an overview of the various advances in Computer Algebra. This is especially important in the Computer Algebra community, where researchers need to build skills in both Computer Science and Mathematics.

New collaborations and interactions. The JNCF have traditionally been an opportunity to create successful collaborations between researchers from different parts of France. We now would like the JNCF to open to an international community, while remaining primarily French-speaking. The previous editions already included courses by colleagues from other European countries, and we intend to continue this trend. We also plan to better advertise the next JNCF in Mediterranean countries.

### Geometry and Arithmetic of Surfaces

Meeting Type: workshop

Contact: Juliette Bruce, Wanlin Li

### Description

Geometry and Arithmetic of Surfaces is a workshop for early career researchers (undergraduates, graduate students, post docs, etc.) being held at the University of Wisconsin - Madison, February 9-10, 2019. This workshop will focus on providing a diverse group of early career researchers the opportunity to learn about interesting cutting edge topics in the arithmetic and algebraic geometry of surfaces from a diverse set of prominent active researchers: Bianca Viray, Katrina Honigs, Anthony Várilly-Alvarado, and Jesse Kass.

There will be a limited amount of funding for junior participants. We expect to be able to provide partial travel support for some participants from within the United States. Accommodation for funded participants will be arranged by the organizers for the nights of Friday, February 8 and Saturday, February 9.

Early career researchers from generally underrepresented groups, in particular women and non-binary researchers, are strongly encouraged to apply, and will be given priority when it comes to funding decisions.

### Free Computational Mathematics

Meeting Type: conference

Contact: see conference website

### Description

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

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

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

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

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

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

### Elliptic curves: arithmetic and computation

Meeting Type: CIMPA school

Contact: Valerio Talamanca

### Description

The aim of this is school to introduce the participants to the arithmetic and computational aspects of the theory of elliptic curves. Elliptic curves lie at the crossroads of algebra, analysis, geometry and arithmetic. We will develop the theory of elliptic curves from its very beginning also providing an introductory course on algebraic curves and the Riemann Roch theorem. Topics that will be covered include: basic geometric and arithmetic results for elliptic curves over number fields and over finite fields, the Mordell-Weil theorem for elliptic curves, Galois representations attached to elliptic curves,. and the Birch and Swinnerton Dyer Conjecture. On the computational side we will run six hours of training sessions on PARI/GP and SAGE.

### Higher Dimensional Arithmetic Geometry

Meeting Type: conference

Contact: Sho Tanimoto

### Description

### Braids, Resolvent Degree and Hilbert’s 13th Problem

Meeting Type: conference

Contact: see conference website

### Description

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.

### Non-Archimedean Geometry and Applications

Meeting Type: conference

Contact: see conference website

### Description

### Algebraic and Analytic Aspects of Automorphic Forms

Meeting Type: school/workshop

Contact: see conference website

### Description

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

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

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

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

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

## March 2019

### Arizona Winter School: Topology and Arithmetic

Meeting Type: conference

Contact: see conference website

### Description

Speakers:

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

### Hawaii Number Theory 2019 (HINT)

Meeting Type: conference

Contact: see conference website

### Description

Hawaii Number Theory 2019 (HINT) aims to bring together a wide variety of number theorists to enhance collaboration and community. Right after HINT is the AMS Spring Central and Western Joint Sectional Meeting, also to be held at UH Mānoa. So come join us for a week of number theory in Hawaiʻi nei.

### International Conference on Arithmetic Geometry in honor of Michael Rapoport's 71st birthday

Meeting Type: conference

Contact: see conference website

### Description

### Cohomology of Arithmetic Groups, Lattices and Number Theory: Geometric and Computational Viewpoint

Meeting Type: conference

Contact: see conference website

### Description

The cohomology of arithmetic groups sits squarely at the intersection of several fields of mathematics. For example, it connects to number theory and arithmetic geometry via Galois representations and Hecke operators, and to representation theory, via its relationship to automorphic forms and automorphic representations. It also has deep connections with geometry, topology, and algebra, through its connections with algebraic K-theory, locally symmetric spaces, reduction theory, and lattices. Explicit calculations have played an increasingly important role in the theoretical development of the subject and its applications. For example, explicitly computing the cohomology gives tools to formulate conjectures about automorphic forms and special values of L-functions, and to try to understand the increasing influence in number theory of the torsion in cohomology. As the scale and complexity of the calculations have increased it has become more and more common for such computations to be performed with the aid of computers. This CIRM conference will bring together international experts with diverse skill sets, and expertise in computational techniques relevant to such calculations and their applications to cohomology of groups, algebraic K-theory, arithmetic geometry, and lattices. The main goals are to foster new collaborations, to introduce young researchers to these topics, and to broaden our theoretical knowledge with a view to extending the scope of computer aided calculations in this area.

### Derived algebraic geometry and its applications

Meeting Type: conference

Contact: see conference website

### Description

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.

## April 2019

### Hot Topics: Recent progress in Langlands Program

Meeting Type: conference

Contact: see conference website

### Description

The purpose of the workshop is to explain Vincent Lafforgue's ground breaking work, constructing the automorphic to Galois direction of the Langlands correspondence for function fields. There will also be a number of talks on more recent developments and related results.

### The fifth mini symposium of the Roman Number Theory Association

Meeting Type: conference

Contact: Fabrizio Barroero, Marina Monsurrò, Francesco Pappalardi, Valerio Talamanca, Alessandro Zaccagnini

### Description

This is a meeting organised by the Roman Number Theory Association.

The venue

The symposium will take place in the Arigiletum situated in Via Madonna dei Monti 40. This historical building host the Department of Architectures of Roma Tre University and is located nearby Via dei Fori Imperiali in the historical center of Rome.

### Geometric and Analytic Methods for Rational Points

Meeting Type: spring school

Contact: see conference website

### Description

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

### Reinventing Rational Points

Meeting Type: thematic program

Contact: see conference website

### Description

Rational points on algebraic varieties represent a modern way of thinking about one of the oldest problems in mathematics: integral and rational solutions of Diophantine equations. In arithmetic geometry one views rational points in the context of geometric properties of underlying algebraic varieties. In analytic number theory many different analytic techniques are used to count the number of rational or integral points, and so understand their “average” behaviour. In logic, rational points feature prominently in the work on Hilbert’s Tenth Problem over Q, which asks for an algorithm to decide the existence of rational solutions to all Diophantine equations. Here one searches for examples of “weird” or “far from average” behaviour of rational points.

There is a large body of conjectures that describe the behaviour of rational points. They include various versions of Mazur’s conjectures on the real topological closure of the set of rational points. A related circle of conjectures deals with the Brauer–Manin obstruction designed to describe the closure of the set of rational points inside the topological space of adelic points. It is conjectured that the Brauer–Manin obstruction should exactly describe this closure for certain classes of algebraic varieties such as rationally connected varieties, K3 surfaces and algebraic curves. Supportive heuristic and theoretical evidence for these difficult conjectures is slowly emerging from the work of many people. The Batyrev–Manin conjectures on the growth of rational points of bounded height have received much attention by analytic number theorists. New techniques that have revolutionised analytic number theory, such as additive combinatorics (Green, Tao, Ziegler) or arithmetic invariant theory (Bhargava, Gross), have made it possible to solve some of the long standing problems in arithmetic geometry. A new feature in recent years has been an increased interaction between the analytic and geometric thinking: questions motivated by various counting problems give rise to novel geometric ideas, whereas conjectures coming from geometry open up new fields of investigation for analytic number theorists.

The aim of this thematic period is to bring together senior and junior mathematicians from the various domains related to rational points to foster new interactions and new research.

### 3rd International Congress: Algebra, Number Theory and Applications April 24-27, 2019 OUJDA MOROCCO

Meeting Type: conference

Contact: Abdelmalek Azizi, Abdelkader Zekhnini, Mohammed Taous

### Description

The purpose of this meeting is to introduce young researchers to some of the most attractive research areas of algebra, number theory and their applications, to present the latest contributions and achievements in these areas. This scientific event will be animated by great specialists as the Bourbaki group's member J. Osterlé and it will be dedicated, among others, to our former teachers : Dr. Ahmed Kerkour from Morocco and Dr. Gunther Frei from Switzerland.

### Simons Symposium: p-adic Hodge theory

Meeting Type: conference

Contact: see conference website

### Description

### Equidistribution: Arithmetic, Computational and Probabilistic Aspects

Meeting Type: conference

Contact: see conference website

### Description

This three-week program on equidistribution considers interactions between probability, number theory, computer science and mathematical logic.

An infinite sequence s(1), s(2), s(3), ... of real numbers is equidistributed (also called uniformly distributed) modulo 1 if the sequence formed by the fractional parts of each term is equidistributed in the unit interval, which means that for each subinterval of the unit interval, asymptotically, the proportion of terms falling within that subinterval is equal to its length. For instance, any sequence of independent identically distributed (i.i.d) random real variables is equidistributed modulo 1. We do not know whether the sequence of integral powers of e (or pi, log 2, root 2, etc.) is equidistributed modulo 1.

To determine whether a given sequence of real numbers is equidistributed and, if this is the case, to estimate the speed of convergence to equidistribution (discrepancy) are usually difficult questions. Lacunary sequences are those where the ratio of successive terms is bounded away from one. A particular family consists of the sequences where the ratio of successive terms is a constant integer greater than 1. These are of the form (b^k x)_k where x is a real and b is an integer greater than 1. These sequences are equidistibuted modulo 1 exactly when every block of digits in the base b expansion of x occurs with the same asymptotic frequency as every other block of equal length. Emile Borel called these real numbers x normal to base b. We know no example of a familiar mathematical constant which is normal to some integer base.

One challenge is to prove the existence of real numbers enjoying simultaneously prescribed digital and Diophantine approximation properties, a more daunting challenge is to give explicit examples of such numbers. In between, one can try to give computable constructions of them.

An analog of equidistribution occurs when using Monte-Carlo simulation with pseudo random numbers, many problems remain to be studied from the point of view of number theory and computer science.

## May 2019

### Definability and decidability problems in number theory

Meeting Type: conference

Contact: see conference website

### Description

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.

### INdAM special research activity in Padova on "p-adic Langlands program"

Meeting Type: Special research bimester

Contact: see conference website

### Description

### Curves and groups in families

Meeting Type: summer school

Contact: see conference website

### Description

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

### The p-adic Langlands programme and related topics

Meeting Type: workshop

Contact: Eran Assaf, Ana Caraiani, Fred Diamond, James Newton

### Description

The aim of the workshop is to bring together researchers working on a range of topics in and around the local and global p-adic Langlands program. The event is funded by the ERC and the EPSRC, and is being organised as a joint activity of the “p-adic Arithmetic Geometry, Torsion Classes, and Modularity” ERC grant, and the "Langlands Programme - p-adic and Geometric Methods" EPSRC grant.

Limited funding is available to support early career researchers with travel and accommodation. Please email the organisers for more information.

### p-adic Modular Forms and p-adic L-functions

Meeting Type: conference

Contact: see conference website

### Description

Modular forms and L-functions have played a central role in Number Theory in the last decades: they have been the source of inspiration for several central conjectures that have been the fuel of the research since they have been formulated up to now. Between them, the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis, two of the six still open problems of the Millennium Prize Problems set by the Clay Mathematics Institute, Hilbert’s Twelfth and Nine Problem, the Stark’s conjectures and other important related conjectures such as the Main Conjectures of Iwasawa theory and Leopoldt’s conjecture. They also provide key insights towards deep elementary conjectures in Number Theory. An emblematic example is Fermat’s Last Theorem, which states that there are no positive integers a, b, and c satisfying the equation aⁿ+bⁿ=cⁿ for n≥3, proved by A. Wiles. It was the result of the efforts of several mathematicians which had the insight to apply the theory of elliptic curves, modular forms and Galois representations to this problem.

As explained, these conjectures are very deep and modular forms and L-functions appear sometime in an amazing way. Another unexpected feature of them is that they present p-adic analogues which can be approached and, sometime, even imply instances of the original conjectures. The aim of the conference is to join together mathematicians working on two fundamental arithmetic aspects of the theory, namely p-adic modular forms and p-adic L-functions, and report on the most recent developments and impact that these results have on the above mentioned conjectures. The conference will take place from May 20 to May 24, 2019.

### Rational points on Fano and similar varieties

Meeting Type: conference

Contact: see conference website

### Description

### p-arithmetic of automorphic forms: a conference in honour of Jacques Tilouine

Meeting Type: conference

Contact: Boyer Pascal, Conti Andrea, Mokrane Farid

### Description

Introduced at the end of the 1960s, the Langlands Program, whose inspiration comes from class field theory in the early 20th century, has produced spectacular and numerous results in the last 20 years.

The most efficient techniques include the intensive use of families of p-adic automorphic forms and the study of deformations of Galois representations. They have led to remarkable arithmetic applications such as the proofs of the Artin conjecture for GL(2) and of the Sato-Tate theorem.

On the occasion of the 60th birthday of J. Tilouine, we gather some of the best experts whose research interacts with those of Jacques.

### Padova school on Serre conjectures and the p-adic Langlands program

Meeting Type: Spring school

Contact: see conference website

### Description

## June 2019

### Arithmetic of low-dimensional abelian varieties

Meeting Type: conference

Contact: Andrew V. Sutherland

### Description

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

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

### Arithmetic, Geometry, Cryptography and Coding Theory

Meeting Type: conference

Contact: see conference website

### Description

Our goal is to organize a conference devoted to interactions between pure mathematics, in particular arithmetic and algebraic geometry, and the information theory, especially cryptography and coding theory. This conference will be the seventeenth edition, with the first one held in 1987, that traditionally reunites some of the best specialists in the domains of arithmetic, geometry and information theory. The corresponding international community is very active with all of the concerned domains changing rapidly over time.

The conference is thus an important occasion for young mathematicians (graduate students and post-docs) to interact with established researchers in order to exchange new ideas.

The conference talks will be devoted to recent advances in arithmetic and algebraic geometry, and number theory, with a special accent on algorithmic and effective results and applications of these fields to the information theory.

The conference will last one week and will be organized as follows :

- One or two plenary talks per day at the beginning of each session. They will be given by established researchers, some of them new to the established AGC2T community, so that that new emerging topics can be introduced, that may give rise to new applications of arithmetic or algebraic geometry to the information theory.
- Shorter specialized talks, often delivered by young mathematicians.

As with the previous editions of the AGC2T, we would like to publish the acts of the conference as a special volume of the Contemporary Mathematics collection of the AMS.

Conference Topics

- Number theory, asymptotic properties of families of global fields, arithmetic statistics, L-functions.
- Arithmetic geometry, algebraic curves over finite fields or number fields, abelian varieties : point counting, the invariant theory and classification of curves.
- Coding theory, algebraic-geometric codes constructed from curves and higher dimensional varieties, decoding algorithms. - Cryptography, elliptic curves and abelian varieties : the discrete logarithm problem, pairings, explicit computation of isogenies, multiplication over finite fields, APN functions, bent and hyper-bent funtions.

### Ideals, Varieties, Applications: Celebrating the Influence of David Cox

Meeting Type: conference

Contact: see conference website

### Description

### PIMS Workshop on Arithmetic Topology

Meeting Type: conference

Contact: see conference website

### Description

The last 10 years have brought a burst of activity at the intersection of algebraic topology, number theory and algebraic geometry. This has led to a wealth of: New theorems, such as Ellenberg–Venkatesh–Westerland’s breakthrough results on the Cohen–Lenstra heuristics for function fields; New sources of heuristics in topology, such as Vakil–Wood’s predictions from the Grothendieck ring, or the notion and coincidences of homological densities as in Farb– Wolfson–Wood; Refinements of classical enumerative theorems using modern topological tools, such as Kass–Wickelgren’s arithmetic count of the 27 lines on a cubic surface; and A renewed focus on unstable homology, as in Galatius–Kupers–Randal-Williams and Miller–Wilson. We believe that these results are just the beginning of the emerging area of arithmetic topology.

This 5 day workshop will bring together junior and senior researchers from across these areas with the goal of:

```
Giving participants a global view of a fast emerging and multidisciplinary area,
Giving participants a detailed awareness on the range of methods available, and
Emerging with a robust problem list which can help guide activity in the area for the next 5-10 years.
```

### Geometry and Arithmetic of Algebraic Varieties

Meeting Type: conference

Contact: see conference website

### Description

### Explicit methods in arithmetic geometry in characteristic p

Meeting Type: conference

Contact: see conference website

### Description

The AMS’s Mathematics Research Communities (MRC) are a professional development program offering early-career mathematicians a rich array of opportunities to develop collaboration skills, build a network focused in an active research domain, and receive mentoring from leaders in that area. Funded through a generous three-year grant from the National Science Foundation, MRC is a year-long experience that includes:

```
Intensive one-week, hands-on research conferences in the summer;
Special Sessions at the AMS-MAA Joint Mathematics Meetings in the January following the summer conferences;
Guidance in career building;
Follow-up small-group collaborations;
Longer-term opportunities for collaboration and community building among the participants;
```

Over time, each participant is expected to provide feedback regarding career development and the impact of the MRC program.

Women, underrepresented minorities, and individuals from various types of institutions across the country are all encouraged to apply.

The focus of this MRC will be on problems in arithmetic geometry over fields of positive characteristic p that are amenable to an explicit approach, including the construction of examples, as well as computational exploration. Compared to algebraic geometry in characteristic 0, studying varieties over fields of characteristic p comes with new challenges (such as the failure of generic smoothness and classical vanishing theorems), but also with additional structure (such as the Frobenius morphism and point-counts over finite fields) that can be exploited. This often leads to interesting arithmetic considerations: possible topics for the workshop include isogeny classes of abelian varieties over finite fields, Galois covers of curves and lifting problems, and arithmetic dynamics.

To reflect the inherent interdisciplinary nature of arithmetic geometry, we invite early-career mathematicians with a wide range of backgrounds in number theory, algebraic geometry, and other subjects that intersect these fruitfully, such as dynamics and commutative algebra. During the workshop, the participants will formulate and investigate open problems around areas of current interest in small collaborative groups. They will benefit from the mentorship of a diverse group of senior arithmetic geometers, and from activities tailored to junior researchers. For instance, there will be two problem brainstorming sessions (at the beginning and the end), expository talks on key techniques (such as recent computational advances), and career-related group discussions.

### p-adic dynamics of Hecke operators

Meeting Type: conference

Contact: see conference website

### Description

The study of symmetries, starting with Klein and Hilbert in the late 19th century is of central importance to number theory. One is interested in behaviour of points in a space under its group of "generalized symmetries"; this idea is central to theories such as ergodic theory and dynamical systems that originated with problems in physics, but also to number theory where the symmetries are very much related to Diophantine equations. Mostly this study is done on manifolds in the usual Euclidean space that takes its structure from the usual space in which we live; however, number theory provides us with new exotic spaces and new exotic geometries -- p-adic numbers, p-adic analysis, p-adic manifolds -- that in a certain perspective serve as tools to study local behaviour of complicated global symmetries originating in Euclidean spaces. Our interest is precisely in developing such a theory in the context of very particular spaces and symmetries; to be precise, p-adic Shimura varieties and p-adic Hecke operators.

### Arithmetic, geometry, and modular forms: a conference in honour of Bill Duke

Meeting Type: conference

Contact: see conference website

### Description

### Arithmetic Geometry in Carthage

Meeting Type: Summer School (June 17-21, 2019) & Conference (June 24-28, 2019)

Contact: Ahmed Abbes

### Description

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.

### CMI-HIMR Summer School in Computational Number Theory

Meeting Type: summer school

Contact: Jennifer Balakrishnan, Tim Dokchitser

### Description

### Iwasawa 2019

Meeting Type: Summer school + conference

Contact: Denis Benois, Pierre Parent

### Description

In 2019, the international Iwasawa conference takes place in Bordeaux, from June 24 to 28. It will be preceded by four mini-courses (of four hours each) on the topic.

Mini-courses (June 19 to 22):

- Euler systems (Victor Rotger, Universitat Politècnica de Catalunya)
- p-adic L-functions (Ellen Eischen, University of Oregon)
- Chern classes and Iwasawa theory (Frauke Bleher, University of Iowa)
- Computing arithmetic invariants from overconvergent modular forms (Jan Vonk, Oxford)

Conference speakers (June 24 to 28):

Daniel Barrera Salazar (Universitat Politècnica de Catalunya) Ted Chinburg (University of Pennsylvania) Mladen Dimitrov (Université de Lille) Adrian Iovita (Concordia University and Università degli studi di Padova) Joaquin Rodrigues Jacinto (University College London) Yukako Kezuka (Universität Regensburg) Guido Kings (Universität Regensburg) Antonio Lei (Université Laval) Zheng Liu (Princeton) David Loeffler(University of Warwick) Jan Nekovář (Sorbonne Université) Jishnu Ray (University of British Columbia) Giovanni Rosso (Concordia University and Cambridge) Romyar Sharifi (UCLA) Florian Sprung (Arizona State University) Eric Urban (Columbia University and CNRS) Shunsuke Yamana (Kyoto University).

Scientific committee: Denis Benois, Henri Darmon, Ming-Lun Hsieh, Masato Kurihara, Otmar Venjakob, Sarah Zerbes.

Organizers: Denis Benois, Pierre Parent.

### Rational points on irrational varieties

Meeting Type: conference

Contact: see conference website

### Description

### Boston University-Keio University Workshop in Number Theory

Meeting Type: conference

Contact: Jennifer Balakrishnan, Masato Kurihara, Steve Rosenberg

### Description

## July 2019

### Journées Arithmétiques

Meeting Type: conference

Contact: see conference website

### Description

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.

### Algebraic Geometry, Number Theory and Applications in Cryptography and Robot kinematics

Meeting Type: conference

Contact: see conference website

### Description

The CIMPA School offers an intensive teaching session to graduate students and young researchers in the fields of Algebraic Geometry, Number Theory, Applications in Cryptography and Robot kinematics. This course will provide elements needed for the applications in cryptography and robot kinematics which will be developed at the end of the school. The goal of this course is for every participant to be able to select a suitable hyperelliptic curve $C$ for constructing some cryptosystems based on the discrete logarithm problem in its Jacobian $J_C$.

### Perfectoids

Meeting Type: summer school, conference

Contact: see conference website

### Description

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

### Rational Points 2019

Meeting Type: conference

Contact: see conference website

### Description

### The First Journal of Number Theory Biennial Conference

Meeting Type: conference

Contact: See conference website

### Description

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.

### Geometric methods in p-adic representation theory

Meeting Type: conference

Contact: Konstantin Ardakov, Peter Schneider

### Description

The goal of this workshop is to bring together researchers who work in number theory or representation theory or non-archimedean analysis, with an eye towards recent developments in the p-adic representation theory of p-adic groups.

Among others, the themes of the workshop include:

- applications to the p-adic local Langlands program,
- constructing representations through the cohomology of Drinfeld coverings,
- p-adic analogues of Beilinson-Bernstein localisation,
- techniques from differential-graded categories.

## August 2019

### Arithmetic and Algebraic Geometry

Meeting Type: conference

Contact: Bhargav Bhatt, Evangelia Gazaki

### Description

### Number Theory in the Americas

Meeting Type: collaboration conference

Contact: see conference website

### Description

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.

### Women in Numbers Europe 3

Meeting Type: conference

Contact: see conference website

### Description

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

Confirmed group leaders:

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

## September 2019

### p-adic Langlands correspondence: a constructive and algorithmic approach

Meeting Type: conference

Contact: see conference website

### Description

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

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

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

### Perfectoid spaces

Meeting Type: summer school

Contact: see conference website

### Description

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:

- Give an advanced introduction to Scholze's theory.
- 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.

### Wild Ramification and Irregular Singularities

Meeting Type: conference

Contact: see conference website

### Description

The planned topics include recent advances in ramification of ℓ-adic sheaves, study of irregular holonomic D-modules in higher dimensions, irregular Hodge theory, exponential motives, companions, finiteness results for local systems, etc. There are well-known analogies between wild ramification in characteristic p and irregular singularities of meromorphic differential equations, and one of our aims is to bring experts in these and related areas together.

### New Developments in Representation Theory of p-adic Groups

Meeting Type: conference

Contact: see conference website

### Description

## October 2019

### Modularity and Moduli Spaces

Meeting Type: conference

Contact: see conference website

### Description

**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.

## November 2019

### Analytic Number Theory

Meeting Type: conference

Contact: see conference website

### Description

## January 2020

### K-Theory, Algebraic Cycles and Motivic Homotopy Theory

Meeting Type: thematic research program

Contact: see conference website

### Description

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.

### Lattices: Algorithms, Complexity and Cryptography

Meeting Type: thematic program

Contact: see conference website

### Description

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.

## March 2020

### Arithmetic geometry, cycles, Hodge theory, regulators, periods and heights

Meeting Type: conference

Contact: see conference website

### Description

## May 2020

### The Arithmetic of the Langlands Program

Meeting Type: conference

Contact: see conference website

### Description

### Summer School: The Arithmetic of the Langlands Program

Meeting Type: summer school

Contact: see conference website

### Description

## June 2020

### Arithmetic Geometry, Number Theory, and Computation III

Meeting Type: conference

Contact: Andrew V. Sutherland

### Description

### Foundations of Computational Mathematics (FoCM) 2020

Meeting Type: conference

Contact: see conference website

### Description

## July 2020

### Park City Mathematics Institute: Number theory informed by computation

Meeting Type: conference and summer school

Contact: Bjorn Poonen

### Description

### Workshop on Local Langlands and p-adic methods

Meeting Type: conference

Contact: see conference website

### Description

### Arithmetic Geometry

Meeting Type: conference

Contact: see conference website

### Description

## August 2020

### Workshop on Global Langlands, Shimura varieties, and shtukas

Meeting Type: conference

Contact: see conference website

### Description

### Decidability, definability and computability in number theory

Meeting Type: research program

Contact: see conference website

### Description

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.

### Low-Dimensional Topology and Number Theory

Meeting Type: conference

Contact: see conference website

### Description

### Automorphic Forms and Arithmetic

Meeting Type: conference

Contact: see conference website