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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 2019-07

- We're now authenticating email; this should decrease the chance that confirmation emails land in Spam or Junk folders.

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

# Upcoming Meetings

## October 2019

### Automorphic p-adic L-functions and regulators

Meeting Type: conference

Contact: Mladen Dimitrov

### Description

The aim of this workshop is to provide an overview of recent developments in theory of p-adic L-functions associated to automorphic representations, covering both the construction of p-adic L-functions, and their relations to Euler systems in Galois cohomology via regulator maps. The workshop will consist of three mini-courses, aimed at younger researchers, and more specialised individual lectures.

There will be three mini-courses, each consisting of four lectures, on the following topics:

Construction of p-adic L-functions for automorphic forms on GL(2n), using the automorphic modular symbols introduced in work of Dimitrov Construction of p-adic L-functions for GSp(4), using Pilloni’s higher Hida theory evalulation of global cohomology classes under the syntomic regulator, using the methods of Darmon—Rotger. The other talks will explore connections of these topics with other related areas of current research, such as Iwasawa theory, the theory of Hecke varieties and the theory of L-invariants.

### 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 (Ï•,GË†)-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.

### Oberwolfach Seminar: Topological Cyclic Homology and Arithmetic

Meeting Type: week-long meeting with talks by organizers and participants

Contact: Dustin Clausen, Lars Hesselholt, Akhil Mathew

### Description

We organize an Oberwolfach Seminar on Topological Cyclic Homology and Arithmetic. The purpose of the seminar is to introduce the higher algebra refinements of determinant and trace, namely, algebraic *K*-theory and topological cyclic homology, along with their budding applications in arithmetic geometry and number theory. In particular, we will use these ingredients to build Clausen's Artin map from *K*-theory of locally compact topological *R*-modules to the dual of his Selmer *K*-theory of *R*, and explain that for *R* a finite, local, or global field, this implies Artin reciprocity. If you wish to participate, please follow the instructions described here
to register at ag@mfo.de by August 11, 2019.

### Illustrating Number Theory and Algebra

Meeting Type: workshop/conference

Contact: Katherine Stange

### Description

The symbiotic relationship between the illustration of mathematics and mathematical research is now flowering in algebra and number theory. This workshop aims to both showcase and develop these connections, including the development of new visualization tools for algebra and number theory. Topics are wide-ranging, and include Apollonian circle packings and the illustration of the arithmetic of hyperbolic manifolds more generally, the visual exploration of the statistics of integer sequences, and the illustrative geometry of such objects as Gaussian periods and Fourier coefficients of modular forms. Other topics may include expander graphs, abelian sandpiles, and Diophantine approximation on varieties. We will also focus on diagrammatic algebras and categories such as Khovanov-Lauda-Rouquier algebras, Soergel bimodule categories, spider categories, and foam categories. The ability to visualize complicated relations diagrammatically has led to important advances in representation theory and knot theory in recent years.

### Number Theory Series in Los Angeles

Meeting Type: conference

Contact: Jim Brown

### Description

NTS-LA is a biannual regional number theory theory conference located in Los Angeles. While each meeting with feature two plenary talks by faculty and one plenary talk from a graduate student from outside of Southern California, the majority of talks will consist of 20 minute contributed talks. The purpose of these meetings is to establish a community of people interested in number theory in Southern California, to allow faculty at institutions that do not have funds for regular seminars to attend high-quality research talks, and to provide a friendly environment for students and faculty to present their research.

## November 2019

### Modular Forms, Arithmetic, and Women in Mathematics (MAAIM)

Meeting Type: conference

Contact: Lea Beneish, Hannah Larson, David Zureick-Brown

### Description

The goals of this conference include facilitating interaction between modular forms and arithmetic geometry researchers, providing graduate students with an opportunity to present their work, and strengthening networks for mathematicians from underrepresented groups including but not limited to women.

On the evening of the second day of the conference (November 2), we plan to have a reception with a public lecture by Professor Raman Parimala, followed by a showing of the CWM film “Journeys of Women in Mathematics.” Talks and registration will be held at Emory University's Mathematics and Science Center (rooms E208, W201, and the Atrium).

### Analytic Number Theory

Meeting Type: conference

Contact: see conference website

### Description

### Prime Numbers, Determinism and Pseudorandomness

Meeting Type: conference

Contact: see conference website

### Description

The recent results of Green and Tao on the existence of arbitrarily long arithmetic progressions of prime numbers have showed the strength of the interactions between combinatorics, number theory and dynamical systems. Other advances, like the results of Bourgain, Green, Tao, Sarnak and Ziegler on the randomness principle for the Möbius function, the resolution of the Gelfond conjectures concerning the sum of digits of prime and square numbers, as well as those of Golston, Pintz and Yildirim and then Zhang and Maynard on small gaps between primes, the recent results of Pintz on the existence of arbitrarily long arithmetic progressions of generalized twin prime numbers show the vitality of this domain of research. The difficulty of the transition from the representation of an integer in a number system to its multiplicative representation (as a product of prime factors) is at the source of many important open problems in mathematics and computer science. The conference will be devoted to the study of independence between the multiplicative properties of integers and various ”deterministic” functions, i. e. functions produced by a dynamical system of zero entropy or defined using a simple algorithm. This area is developing very fast at international level and the conference will be an opportunity to help to develop techniques that were recently introduced to study of relations between prime numbers, polynomial sequences and finite automata, the study of the pseudorandom properties of certain arithmetic sequences and the search of prime numbers in deterministic sequences. This goal is related to several recent works by Bourgain, Green, Sarnak, Tao and others concerning the orthogonality of the Möbius function with deterministic sequences and obtaining prime number theorems for these sequences.

### p-adic cohomology and arithmetic geometry 2019

Meeting Type: conference

Contact: see conference website

### Description

### Front Range Number Theory Day

Meeting Type: conference

Contact: Ozlem Ejder, Sarah Arpin, Hanson Smith

### Description

This will be the third installment of Front Range Number Theory Day, held in the Fall and Spring semesters of each academic year. This is a day-long conference at Colorado State University, sponsored by the CSU Mathematics Department, CU Center for Number Theory, the CU Research and Innovation Office, and (pending) NSF.

Our speakers this year will be: David Grant (CU Boulder), Padmavathi Srinivasan (UGA), and John Voight (Dartmouth).

Registered participants are invited to give a five minute lightning talk about their research. Register through the link on our website.

### Birational Geometry, Kahler-Einstein Metrics and Degenerations

Meeting Type: conference

Contact: Julius Ross

### Description

### Oregon Number Theory Days

Meeting Type: conference

Contact: Shabnam Akhtari

### Description

## December 2019

### Rational Points on Higher Dimensional Varieties

Meeting Type: conference

Contact: Sho Tanimoto

### Description

Visit the conference website for more info.

### Zeta Functions

Meeting Type: conference

Contact: see conference website

### Description

Zeta functions are ubiquitous objects in Number Theory and Arithmetic Geometry. They are analytic, algebraic, or combinatorial in nature. Families of zeta functions (or more generally of L-functions) naturally appear in a broad variety of active research fields e.g. au- tomorphic forms and Artin representations, Drinfeld modules, arithmetic dynamics, abelian varieties over global fields, inequities in the distribution of sequences indexed by prime num- bers or more generally by places of global fields...

The main purpose of the “Zeta functions” conference is to gather experts of the theoretical and computational branches of number theory and arithmetic geometry together with students and young researchers to have them interact and explore further the richness of the information encoded by zeta and L-functions. Our conference proposal aims at synthesizing complementary points of view coming from distant fields: the analytic approach in the classical theory of zeta and L-functions, the theory of Artin L-functions in connection with the Langlands program, zeta and L-functions coming from arithmetic geometry in the spirit of the Weil conjectures, zeta functions arising in dynamics...

One of the original aspects of the project lies in the interaction between theoretical considerations and numerical and algorithmic features for diverse families of zeta and L-functions. Rather than a meeting meant for experts in a particular topic we will put the emphasis on the exchange of ideas between people coming from related fields in Number Theory and on inviting young researchers and students to further pursue the study of these interactions that have already proven fruitful and that we believe are still very promising.

### A2C: Algebra, Codes and Cryptography. International Conference in honor of Prof. Mamadou SangharÃ©

Meeting Type: conference

Contact: Laila Mesmoudi

### Description

The first Algebra, Codes and Cryptography conference will be held in Dakar, Senegal on Thursday to Saturday, December 5-7, 2019. The conference aims to provide a forum for researchers from all over the world to present results and exchange ideas on topics in Non-Associative Algebra, Non-commutative Algebra, Cryptology, Coding Theory and Information Security.

### 2019 CMS Winter Meeting

Meeting Type: conference

Contact: Patrick Ingram, Jane Heffernan, Sarah Watson

### Description

The Canadian Mathematical Society (CMS) invites the mathematical community to the 2019 CMS Winter Meeting in Toronto, Ontario from December 6-9. All meeting activities are taking place at the Chelsea Hotel. Four days of awards, mini courses, prize lectures, plenary speakers, and scientific sessions. Early Bird Registration ends October 31st. Registration closes November 15. CMS invites all speakers to submit an abstract for their session or contributed paper. Note: You are now required to register for the meeting before you can submit an abstract. Grants are available to partially fund the travel and accommodation costs for bona fide graduate students at a Canadian or other university.

### Symposium in Honor of Julia Robinson’s 100th Birthday

Meeting Type: conference

Contact: see conference website

### Description

A Symposium on the occasion of Julia Robinson’s 100th birthday will be held on Monday December 9, 2019 at MSRI. Julia Robinson (1919-1985) was a leading mathematical logician of the twentieth century, and notably a first in many ways, including the first woman president of the American Mathematical Society and the first woman mathematician elected to membership in the National Academy of Sciences. Her most famous work, together with Martin Davis and Hilary Putnam, led to Yuri Matiyasevich's solution in the negative of Hilbert’s Tenth Problem, showing that there is no general algorithmic solution for Diophantine equations. She contributed in other topics as well. Her 1948 thesis linked the undecidability of the field of rational numbers to Godel’s proof of undecidability of the ring of integers. Confirmed participants in this day-long celebration of her work and of current mathematics insprired by her research include: Lenore Blum, who will give a public lecture, Lou van den Dries, Martin Davis, Kirsten Eisentrager, and Yuri Matiyasevich.

### 17th IMA International Conference on Cryptography and Coding

Meeting Type: conference

Contact: Conferences Department Institute of Mathematics and its Applications

### Description

The mathematical theory and practice of both cryptography and coding underpins the provision of effective security and reliability for data communication, processing and storage. This seventeenth International Conference in an established and successful IMA series on the theme of â€œCryptography and Codingâ€ solicits both original research papers and presentations on all technical aspects of cryptography and coding.

### Supergeometry, supersymmetry and quantization

Meeting Type: conference

Contact: Andrew Bruce

### Description

The conference will bring researchers across both the fields of mathematics and physics together in order to discuss recently developed topics, on-going work and speculative new ideas within supergeometry and its applications in physics. This event offers a unique opportunity to unite physicists and mathematicians who share a common interest in supermathematics. There will be enough time available for discussions between the participants; a poster session for junior researchers will be organized. The main topics include:

- Supermanifolds and their generalizations (e.g., superschemes, color supermanifolds, noncommutative supergeometry...)
- Super Lie groups and Lie superalgebras
- Supergeometric methods in physics
- Geometric aspects of supersymmetric field theories, superstrings, and supergravity
- Mathematical aspects of the BV-BRST formalism

List of confirmed speakers: · Glenn Barnich (Université Libre de Bruxelles) · José Miguel Figueroa-O'Farrill (University of Edinburgh) · Rita Fioresi (University of Bologna) · Janusz Grabowski (Polish Academy of Sciences, Warsaw) · Richard Kerner (Sorbonne University) · Hovhannes Khudaverdyan (University of Manchester) · Alexei Kotov (University of Hradec Kralove) · Dimitry Leites (Stockholm University) · Yuri Manin (Max Planck Institute for Mathematics, Bonn) · Sergei Merkulov (University of Luxembourg) · Ruben Mkrtchyan (Yerevan Physics Institute) · Valentin Ovsienko (CNRS, Reims) · Ivan Penkov (Jacobs University Bremen) · Jian Qiu (University of Uppsala) · Vladimir Salnikov (CNRS, La Rochelle) · Urs Schreiber (NYU Abu Dhabi and Czech Acedemy, Prague) · Albert Schwarz (University of California, Davis) · Ekaterina Shemyakova (University of Minneapolis) · Francesco Toppan (Brazilian Center for Physics Research, Rio de Janeiro) · Luca Vitagliano (University of Salerno) · Alexander Voronov (University of Minnesota) · Ted Voronov (University of Manchester) Scientific Committee:

Andrew Bruce (University of Luxembourg), Steven Duplij (University of Münster), Janusz Grabowski (Polish Academy of Sciences), Norbert Poncin (University of Luxembourg), Ted Voronov (University of Manchester) Organizing Committee:

Andrew Bruce (University of Luxembourg), Eduardo Ibargüengoytia (University of Luxembourg) .

### West Coast Number Theory Conference

Meeting Type: conference

Contact: see conference website

### Description

Hello Number Theorists,

Please join us for the 50th anniversary edition of the West Coast Number Theory Conference, to be held Dec 16-20, 2019, where it all began back in 1969:

```
Asilomar Conference Grounds
Pacific Grove, CA
http://westcoastnumbertheory.org
```

The early registration rate for students, postdocs, and retirees is $75; for all others it is $100. Early registration ends November 1, 2019, after which registration will increase by $25. Financial assistance is available for those who need it.

Full details on registration and accommodations can be found here on the website.

We hope you can make a special effort to come celebrate with us in December!

## 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.

### Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting

Meeting Type: conference

Contact: see conference website

### Description

The 2020 Simons Collaboration on Arithmetic Geometry, Number Theory & Computation Annual Meeting will focus on three main themes:

- Development and organization of software and databases supporting research in number theory and arithmetic geometry
- Fundamental research in arithmetic geometry inspired by computation and leading to new algorithms
- Explorations of L-functions, modular forms, and Galois representations with elegant and unusual properties

### 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 LovÃ¡sz, 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.

### Atelier PARI/GP 2020

Meeting Type: software development workshop

Contact: see conference website

### Description

### Combinatorial Algebra meets Algebraic Combinatorics 2020

Meeting Type: Conference

Contact: Susan Cooper, Sara Faridi, Mayada Shahada, Daniele Turchetti

### Description

This workshop is the 17th in a series of weekend workshops that bring together the mathematical community sharing interests in algebraic combinatorics, commutative algebra and combinatorial algebraic geometry.

Invited Speakers

Chris Francisco (Oklahoma State University) Ezra Miller (Duke University) Stephanie van Willigenburg (University of British Columbia) Josephine Yu (Georgia Institute of Technology) (tentative)

Call for papers:

If you would like to give a talk at this conference, please send titles and abstracts to Sara Faridi (faridi@dal.ca) by November 15, 2019.

## February 2020

### Workshop on Galois Representations

Meeting Type: workshop

Contact: aftab pande

### Description

Summer workshop on Galois Representations

As part of the 2020 summer program at UFRJ, we are organizing a one week workshop on Galois Representations at UFRJ.

Dates: 10 - 14 February 2020.

Location: Instituto de Matemática, Universidade Federal do Rio de Janeiro (UFRJ).

Program: 1 mini course in the morning and 2 research talks in the afternoons by junior researchers (preferably from South America).

### A CIMPA research school on Group Actions in Arithmetic and Geometry

Meeting Type: Cimpa research school

Contact: Valerio Talamanca

### Description

The concept of a group is central to essentially all of modern mathematics. In Number theory and geometry, where groups take central stage in various shapes such as symmetry groups, Galois groups, fundamental groups, reflection groups and permutation groups, the conceptual unification that it provides is most strikingly illustrated. The School will help the students acquiring a good background on the Langlands program, which, after all, is about relations between symmetries in geometry, analysis and number theory. In this school, we present groups and the natural objects they act on in a variety of arithmetic and geometric contexts. Special emphasis will be given to concrete examples, and practical and computational aspects of groups and their actions will be stressed. The topics to be treated include finite fields, coding theory, covering spaces, representation theory, modular forms and Galois theory.

## March 2020

### Equivariant Stable Homotopy Theory and p-adic Hodge Theory

Meeting Type: conference

Contact: see conference website

### Description

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.

### Arizona Winter School 2020: Nonabelian Chabauty

Meeting Type: winter school

Contact: see conference website

### Description

### Topics in Category Theory: A Spring School

Meeting Type: Spring School

Contact: Guy Boyde (Southampton), Aryan Ghobadi (QMUL), Emily Roff (Edinburgh)

### Description

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.

### Arithmetic Algebraic Geometry

Meeting Type: conference

Contact: see conference website

### Description

### Interactions between group theory, number theory, combinatorics and geometry

Meeting Type: conference

Contact: see conference website

### Description

Finite group theory has very close connections with many areas of mathematics and other sciences. It has been very useful in solving major problems in these areas; in turn, many problems originated outside of group theory have impacted the field. In this workshop, the focus will be on problems that have influenced group theory significantly, and also where group theory has led to fundamental advances. Most especially we will focus on number theory, combinatorics, and geometry.

### Higher Dimensional Algebraic Geometry--An event in honor of Prof. Shokurov's 70th Birthday

Meeting Type: conference

Contact: Jingjun Han

### Description

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.

### Arithmetic groups and 3-manifolds

Meeting Type: conference

Contact: Haluk Sengun

### Description

Arithmetic groups provide a fruitful link between various areas, such as geometry, topology, representation theory and number theory. Methods from geometry and topology hinge on the fact that arithmetic groups are lattices in Lie groups, whereas the theory of automorphic forms establishes a connection to representation theory and number theory. This interplay is especially intriguing in the setting of hyperbolic 3-manifolds. Indeed many conjectures in 3-manifold theory tend to be much more accessible for hyperbolic 3-manifolds whose fundamental groups are arithmetic, and conversely such manifolds provide the simplest set-up in which some of the most exciting new phenomena in the Langlands program can be studied. This conference will bring together researchers with various backgrounds around links between number theory and 3-manifolds. Central topics of the conference are the cohomology of arithmetic groups, the relation between torsion and L²-torsion, profinite invariants of 3-manifolds, and number theoretic ramifications.

### Madison Moduli Weekend

Meeting Type: Conference

Contact: Brandon Boggess, Soumya Sankar

### Description

Conference on moduli spaces.

### Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups

Meeting Type: learning conference

Contact: see conference website

### Description

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

Meeting Type: conference

Contact: see conference website

### Description

## April 2020

### Periods, Motives and Differential equations: between Arithmetic and Geometry

Meeting Type: conference

Contact: https://periodes.sciencesconf.org/resource/page/id/1

### Description

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.

### Arithmetic, algebra and algorithms — celebrating the mathematics of Hendrik Lenstra

Meeting Type: conference

Contact: Alex Bartel, Alice Silverberg, John Voight

### Description

The purpose of the workshop is to bring together leading, as well as early career researchers on arithmetic statistics and on algorithmic aspects of algebra and number theory, with the aim of fostering collaborations within and between these communities, and to offer early career researchers the opportunity to get a broad overview of the most recent achievements and of the most pressing problems in these fields. Another purpose is to celebrate the mathematics of Hendrik W. Lenstra Jr. on the occasion of his 71st birthday.

### Number theory days in Regensburg – Special values of L-functions

Meeting Type: conference

Contact: see conference website

### Description

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.

### Lattices: From Theory to Practice

Meeting Type: conference

Contact: see conference website

### Description

Recent trends, such as the NIST initiative to standardize post-quantum cryptography, point to large-scale adoption of lattice-based cryptography in the near future. There has consequently been a great deal of attention devoted to making various aspects of lattice-based cryptography practical.

This workshop will focus on questions related to the transition of lattice-based cryptography from theory to practice including the hardness of lattice problems arising from algebraic number theory, and algorithmic solutions to practical issues such as time and space-efficiency, side-channel resistance, and ease of hardware implementations.

The workshop will bring together theoretical and applied cryptographers and computational number-theorists, and will also encourage interaction amongst different communities within and outside cryptography.

## May 2020

### The Arithmetic of the Langlands Program

Meeting Type: conference

Contact: see conference website

### Description

### Swinnerton-Dyer Memorial

Meeting Type: conference

Contact: see conference website

### Description

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
```

### The Sixth Mini Symposium of the Roman Number Theory Association

Meeting Type: conference

Contact: Valerio Talamanca

### Description

### 34th Automorphic Forms Workshop

Meeting Type: conference

Contact: see conference website

### Description

Over the last 33 years, the Annual Workshop on Automorphic Forms and Related Topics has remained a small and friendly conference. Those attending range from students to new PhD's to established researchers. For young researchers, the conference has provided support and encouragement. For accomplished researchers, it has provided the opportunity to mentor as well as a forum for exchanging ideas.

The workshop has become internationally recognized for both its high-quality research talks and its supportive atmosphere for junior researchers. Participants present cutting-edge research in all areas related to automorphic forms. These include mock modular forms, Maass wave forms, elliptic curves, Siegel and Jacobi modular forms, special values of L-functions, random matrices, quadratic forms, applications of modular forms, and many other topics.

In addition to research talks, the workshop has, in the past years, featured panel discussion sessions on the topics of grant writing, mentoring and research partnerships, REUs and outreach, and opportunities for international collaborations. Based on the success of these sessions, we plan to have similar panel sessions this year as well.

This year, the 2020 Automorphic Forms Workshop will be held in Moab, Utah at the Moab Arts and Recreation Center. Moab, in southern Utah, is near Arches and Canyonlands National Parks and other scenic landmarks. The Workshop will be organized and hosted by Brigham Young University.

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

Meeting Type: summer school

Contact: see conference website

### Description

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

### Foundations and Perspectives of Anabelian Geometry

Meeting Type: conference

Contact: see conference website

### Description

This workshop is one of the workshops of a special RIMS year "Expanding Horizons of Inter-universal TeichmÃ¼ller 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.

### BRIDGES: Building Relationships for an Inclusive and Diverse Group of Emerging Students

Meeting Type: conference

Contact: see conference website

### Description

This conference is aimed towards early graduate students and advanced undergraduate students interested in representation theory, number theory, and commutative algebra.

The goal of this conference is to:

- Foster a sense of community amongst underrepresented groups in mathematics,
- Introduce possible research areas,
- Expose the participants to role models and possible mentors.

### Relative Aspects of the Langlands Program, L-Functions and Beyond Endoscopy

Meeting Type: conference

Contact: see conference website

### Description

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.

## June 2020

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

Meeting Type: conference

Contact: Andrew V. Sutherland

### Description

### Advances in Mixed Characteristic Commutative Algebra and Geometric Connections

Meeting Type: conference

Contact: see conference website

### Description

The Casa MatemÃ¡tica 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.

### Connecticut Summer School in Number Theory

Meeting Type: summer school and conference

Contact: Jennifer Balakrishnan, Keith Conrad, Alvaro Lozano-Robledo, Christelle Vincent, Liang Xiao

### Description

CTNT 2020 will take place during the week of June 8th-14th, 2020 (summer school June 8-12, and research conference 12-14), at University of Connecticut.

### Foundations of Computational Mathematics (FoCM) 2020

Meeting Type: conference

Contact: see conference website

### Description

### Canadian Number Theory Association (CNTA XVI)

Meeting Type: conference

Contact: Patrick Ingram

### Description

### On the non-triviality of arithmetic invariants and its applications

Meeting Type:

Contact: see conference website

### Description

### Combinatorial Anabelian Geometry and Related Topics

Meeting Type: conference

Contact: see conference website

### Description

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.

### Resolution of singularities, valuation theory and related topics

Meeting Type: conference

Contact: see conference website

### Description

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 Circle Method” Entering its Second Century

Meeting Type: summer school

Contact: see conference website

### Description

The Circle Method emerged one hundred years ago from ideas of Ramanujan, Hardy and Littlewood, and quickly became the most powerful analytic method for counting solutions to Diophantine equations. As the Circle Method enters its second century, new work is making significant advances both in strengthening results in classical Diophantine settings, and in demonstrating applications in novel settings. This includes function field, number field, adelic, geometric, and harmonic analytic applications, with striking consequences in areas such as ergodic theory, subconvexity for L-functions, and the Langlands program.

This summer school for graduate students and postdocs will present accessible lecture series that demonstrate how to apply the Circle Method in a wide variety of settings. Participants will gain both a foundational understanding of the core principles of the Circle Method, and an overview of cutting-edge applications of the method.

Key Speakers: The following speakers will give a lecture series:

```
Timothy Browning (IST Austria)
Jayce Getz (Duke University)
Ritabrata Munshi (Tata Institute)
Simon Myerson (Universität Göttingen)
Lillian Pierce (Duke University)
```

Additional Speakers:

```
Julia Brandes (University of Gothenburg)
Damaris Schindler (University of Utrecht)
Pankaj Vishe (Durham University)
```

### Fourteenth Algorithmic Number Theory Symposium, ANTS-XIV

Meeting Type: conference

Contact: see conference website

### Description

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.

## July 2020

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

Meeting Type: conference and summer school

Contact: Bjorn Poonen

### Description

### Building Bridges 5th EU/US Summer School and Workshop on Automorphic Forms and Related Topics (BB5)

Meeting Type: Summer School and Workshop

Contact: Samuele Anni, Jim Brown, Jay Jorgenson, Almasa Odžak, Lejla Smajlovic, Lynne Walling

### Description

Automorphic forms are present in almost every area of modern number theory. In recent decades there has been a starburst of activity and progress in this broad area, leading to many new and exciting directions, applications, and connections with other areas. This is a rapidly expanding area, with numerous approaches, tools, and interconnections, as well as connections to other areas of mathematics. The Building Bridges research school offers training to graduate students and early career researchers in the areas of automorphic forms and related topics that continue to be the foci of exciting and influential research activity. The five-day workshop immediately following the summer school aims to foster and strengthen a long-lasting, friendly and supportive exchange between automorphic forms researchers in the EU and the US, and to integrate the summer school students into this community. The organizers have obtained funding to subsidize the costs of successful applicants to the summer school. By popular demand from previous research school and workshop participants, BB5 will take place in Sarajevo, a beautiful city with a friendly, multicultural atmosphere, and low costs of accommodation, food, and transportation.

### Local Langlands and p-adic methods (in honour of Jean-Marc Fontaine)

Meeting Type: conference

Contact: see conference website

### Description

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.

### Arithmetic Geometry

Meeting Type: conference

Contact: see conference website

### Description

### ISSAC: International Symposium on Symbolic and Algebraic Computation

Meeting Type: conference

Contact: see conference website

### Description

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.

### Women in Algebraic Geometry

Meeting Type: research collaboration workshop

Contact: see conference website

### Description

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.

### PIMS - Germany Summer School on Eigenvarieties

Meeting Type: summer school

Contact: see conference website

### Description

**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

## August 2020

### Stacks Project Workshop 2020

Meeting Type: workshop (appropriate for graduate students)

Contact: Pieter Belmans, Aise Johan de Jong, Wei Ho

### Description

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.

### Global Langlands, Shimura varieties, and shtukas

Meeting Type: conference

Contact: see conference website

### Description

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.

### 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.

### Connections for Women: Decidability, definability and computability in number theory

Meeting Type: conference

Contact: see conference website

### Description

The aim of the workshop is to discover how the problems in number theory and algebraic geometry arising from the Hilbertâ€™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 Hilbertâ€™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.

### Low-Dimensional Topology and Number Theory

Meeting Type: conference

Contact: see conference website

### Description

### Introductory Workshop: Decidability, definability and computability in number theory

Meeting Type: conference

Contact: see conference website

### Description

Our workshop will focus research efforts on the interaction of number-theoretic questions with questions of decidability, definability, and computability, bringing together researchers approaching these questions from various sides to work on the core issues. This Introductory Workshop will serve as the introductory event of the MSRI semester program and is designed to introduce the basic structures and ideas of the different communities, and to highlight problems of active current interest.

### Modern Breakthroughs in Diophantine Problems

Meeting Type: conference

Contact: see conference website

### Description

### Automorphic Forms and Arithmetic

Meeting Type: conference

Contact: see conference website

### Description

### Serre weights conjectures and geometry of Shimura varieties

Meeting Type: conference

Contact: see conference website

### Description

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.

### Number Theory-Cohomology in Arithmetic

Meeting Type: thematic research program

Contact: see conference website

### Description

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.

## September 2020

### Arithmetic Aspects of Algebraic Groups

Meeting Type: conference

Contact: see conference website

### Description

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.

### Géométrie algébrique, Théorie des nombres et Applications (GTA)

Meeting Type: conference

Contact: Gaetan Bisson

### Description

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.

### p-adic L-functions and Euler systems in honor of Bernadette Perrin-Riou

Meeting Type: conference

Contact: see conference website

### Description

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 the`

Perrin-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.)

## October 2020

### Arithmetic quotients of locally symmetric spaces and their cohomology

Meeting Type: conference

Contact: see conference website

### Description

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.

## November 2020

### WIN5: Women in Numbers 5

Meeting Type: conference

Contact: see conference website

### Description

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.

### Langlands Program: Number Theory and Representation Theory

Meeting Type: conference

Contact: see conference website

### Description

The Casa MatemÃ¡tica 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.

## December 2020

### p-adic cohomology, p-adic families of modular forms, and p-adic L-functions

Meeting Type: conference

Contact: see conference website

### Description

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 Andreattaâ€“Iovitaâ€“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 Barreraâ€“Dimitrovâ€“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 Blochâ€“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 Bellaicheâ€“Chenevier for unitary groups and Skinnerâ€“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 Breuilâ€“Hellmannâ€“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 Gaussâ€“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 Chojeckiâ€“Hansenâ€“Johansson using functions on the perfectoid tower of modular curves and the construction by Kriz of a new p-adic Maassâ€“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 Bhattâ€“Morrowâ€“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.

## February 2021

### Combinatorial Algebraic Geometry

Meeting Type: thematic research program

Contact: see conference website

### Description

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.