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## Or choose your own subject tags below

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!
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- New json and xml interfaces for access by other software.

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

# Upcoming Meetings

## July 2018

### AGRA 2018, a two-week school on Arithmetic, Groups and Analysis

Meeting Type: Summer school

Contact: see conference website

### Description

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

### Algorithmic Number Theory Symposium ANTS-XIII

Meeting Type: conference

Contact: see conference website

### Description

### Witt Vectors, Deformations, and Absolute Geometry

Meeting Type: conference

Contact: see conference website

### Description

### Genova-Torino-Milano Seminar: some topics in Commutative Algebra and Algebraic Geometry

Meeting Type: Workshop

Contact: Paolo Lella

### Description

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

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

SPEAKERS

- V. Antonelli (Politecnico di Torino)
- G. Caviglia (Purdue University)
- L. Chiantini (Università degli Studi di Siena)
- A. De Stefani (University of Nebraska-Lincoln)
- A. Hochenegger (Università degli Studi di Milano)
- S. Mongodi (Politecnico di Milano)
- P. Saracco (ULB - Université Libre de Bruxelles)

Schedule, titles and abstracts are available at https://www.eko.polimi.it/index.php/gtm2015/gtm2018.

A social dinner is planned on July 17.

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

### Explicit Methods in Number Theory

Meeting Type: invitational workshop

Contact: see conference website

### Description

Organisers

- Karim Belabas, Bordeaux
- Bjorn Poonen, Cambridge MA
- Fernando Rodriguez Villegas, Trieste

### Young Researchers in Mathematics (YRM) 2018

Meeting Type: conference

Contact: see conference website

### Description

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

### 2018 ICM satellite conference in Number Theory

Meeting Type: conference

Contact: Henri Darmon, Fred Diamond, Kiran Kedlaya, Aftab Pande, Richard Taylor, Marie-France Vigneras

### Description

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

### USC K-theory Summer School 2018 - Computations in stable motivic homotopy theory

Meeting Type: Summer School

Contact: Aravind Asok

### Description

**When**: July 31 - August 3, 2018

**Where**: The University of Southern California

**What**: The theme of the summer school is ``Computations in stable motivic homotopy theory." For more precise information see the webpage of the summer school.

**Who**: The intended audience is graduate students and postdocs interested in the subject.
Limited funding is available to support travel to and housing at the summer school. Please register at the webpage for the summer school to request funding. Priority for funding requests will be given to those applications complete before July 1, 2018.

**Organizers**: Aravind Asok and Oliver Röndigs

## August 2018

### International Congress of Mathematicians

Meeting Type: international congress

Contact: see conference website

### Description

Satellite conferences will appear later with their own entries.

### Arithmetic Geometry and Quantum Field Theory

Meeting Type: conference

Contact: see conference website

### Description

### Arithmetic Geometry and Quantum Field Theory

Meeting Type: conference

Contact: see conference website

### Description

### Arithmetic of curves

Meeting Type: collaboration workshop

Contact: see conference website

### Description

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

### Instructional school in Iwasawa theory

Meeting Type: Summer school

Contact: Guillermo Mantilla-Soler, Otmar Venjakob

### Description

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

### New Trends in Analytic Number Theory

Meeting Type: summer school

Contact: see conference website

### Description

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

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

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

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

### Tropical geometry and moduli spaces

Meeting Type: conference

Contact: see conference website

### Description

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

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

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

### Arithmetic Geometry, Number Theory, and Computation

Meeting Type: conference

Contact: Bjorn Poonen, Andrew Sutherland

### Description

### Arithmetic Algebraic Geometry: A conference in honor of the 60th Birthday of Grzegorz Banaszak

Meeting Type: conference

Contact: see conference website

### Description

### Arithmetic and geometry of cubic hypersurfaces

Meeting Type: conference

Contact: see conference website

### Description

### Masterclass: Mahler measures and special values of L-functions

Meeting Type: Masterclass

Contact: Fabien Pazuki, Riccardo Pengo

### Description

This Masterclass is meant to illustrate to young researchers the latest developments in the study of special values of L-functions and its connection to Mahler’s measure, polylogarithms, hypergeometric functions and K-theory.

**Lecture series:**

- François Brunault (École normale supérieure de Lyon)
- Wadim Zudilin (Radboud University Nijmegen & University of Newcastle)

**Speakers:**

- Marie José Bertin (Institut Mathématiques de Jussieu-Paris Rive Gauche)
- Antonin Guilloux (Institut Mathématiques de Jussieu-Paris Rive Gauche)
- Matilde Lalín (Université de Montréal)
- Michalis Neururer (TU Darmstadt)

### Third Japanese-European symposium on symplectic manifolds and moduli spaces

Meeting Type: conference

Contact: see conference website

### Description

The study of holomorphic symplectic varieties and the related topic of certain moduli spaces are vibrant areas of research. Although they are studied all over the world, particularly many researchers working on these topics are situated in Europe and Japan. The Japanese-European Symposium on Symplectic Varieties and Moduli Spaces" has the goal to intensify the scientific exchange between researchers in these fields despite of the geographical distance. It is mainly aimed at young researchers, starting from last year PhD students.

After the first Japanese-European Symposium on Symplectic Varieties and Moduli Spaces was held in Kyoto in October 2015 and the second edition took place in Levico Terme in September 2017, there will be a third edition in August 2018 in Kagurazaka (Tokyo). The future plan is to run this conference every year for some time, once in Europe and once in Japan.

The conference is meant to encourage interactions between participants, so the schedule will leave time for discussions. There should be roughly 20 speakers, and most participants will also be speakers, presenting their area of expertise. Additionally there will be two senior speakers giving several connected lectures.

### Nairobi Workshop in Algebraic Geometry

Meeting Type: conference

Contact: see conference website

### Description

## September 2018

### Varieties: Arithmetic and Transformations

Meeting Type: conference

Contact: see conference website

### Description

Focus points

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

### Stark's Conjectures, Iwasawa theory and related topics

Meeting Type: Workshop

Contact: Henri Johnston, Andreas Nickel

### Description

See website.

### Arithmetic of Differential Equations

Meeting Type: summer school

Contact: see conference website

### Description

Teachers:

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

### School and Conference "Motives in St. Petersburg"

Meeting Type: conference, summer school

Contact: Alexey Ananyevskiy, Grigory Garkusha, Ivan Panin

### Description

### Women in Automorphic Forms

Meeting Type: conference

Contact: see conference website

### Description

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

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

### Arithmetic Ramsey Theory

Meeting Type: conference

Contact: see conference website

### Description

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

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

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

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

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

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

### Arithmetic Geometry : l-adic and p-adic aspects

Meeting Type: conference

Contact: Ahmed Abbes, Takeshi Saito

### Description

### International Summer School on Arithmetic Geometry

Meeting Type: summer school

Contact: Dino Festi, Ariyan Javanpeykar, Davide Cesare Veniani

### Description

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

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

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

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

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

### Vladimir Voevodsky Memorial Conference

Meeting Type: conference

Contact: Daniel Grayson

### Description

Dear mathematicians,

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

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

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

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

On behalf of the organizing committee,

Thierry Coquand

Dan Grayson

Marc Levine

Charles Weibel

### Joint meeting of the Italian Mathematical Union, the Italian Society of Industrial and Applied Mathematics and the Polish Mathematical Society

Meeting Type: conference

Contact: Karol Szczypkowski

### Description

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

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

### Open questions in number theory and cryptography

Meeting Type: conference

Contact: see conference website

### Description

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

### School in Arithmetic Geometry

Meeting Type: summer school

Contact: see conference website

### Description

### Varieties and Group Actions

Meeting Type: summer school, conference

Contact: see conference website

### Description

Confirmed lecturers:

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

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

The titles of lecture series are provisional.

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

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

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

### Arbeitsgemeinchaft "Motives, foliations and the Conservativity conjecture"

Meeting Type: Arbeitsgemeinschaft

Contact: see conference website

### Description

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

### A two-day journey in Computational Algebra and Algebraic Geometry

Meeting Type: conference

Contact: Paolo Lella

### Description

Confirmed speakers are

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

### Special Values of Automorphic L-functions and Associated p-adic L-Functions

Meeting Type: conference

Contact: see conference website

### Description

## October 2018

### Conference on Arithmetic Algebraic Geometry, On the occasion of Michael Rapoport’s 70th birthday

Meeting Type: conference

Contact: see conference website

### Description

### Complex Algebraic Geometry

Meeting Type: conference

Contact: see conference website

### Description

The discussion meeting on Complex Algebraic Geometry will be centered around the "Infosys-ICTS Ramanujan lectures" of Professor Claire Voisin. Complex algebraic geometry is a rich, classical area of mathematics. We shall focus on some specific themes including, but not restricted to vector bundles over complex varieties, topology of varieties, Chow groups, etc.

### Quebec-Maine Number Theory Conference 2018

Meeting Type: conference

Contact: see conference website

### Description

In 1998, number theorists at Université Laval and the University of Maine founded the Maine/Québec Conference on Number Theory and Related Topics. Since then it has been held annually on a weekend in early Fall (except 2001, when the conference was cancelled due to the attacks that September). We invite number theorists and mathematicians in related areas from New England, eastern Canada, and beyond to speak about their research and discuss ideas for future work. Each year there is a change in venue: odd years at UMaine, even years at U. Laval.

### Automorphic Techniques in Arithmetic Geometry

Meeting Type: workshop

Contact: Elmar Große-Klönne

### Description

This is the concluding workshop of the first period of the subproject "Automorphic Techniques in Arithmetic Geometry" of the Central European Network for Teaching and Research in Academic Liaison. The aim is to present some of the current research topics using automorphic techniques in arithmetic geometry pursued at the various CENTRAL partner universities.

### Cohomology of Algebraic Varieties

Meeting Type: conference

Contact: Anna Cadoret, François Charles, Cyril Demarche, Bruno Klingler, Ben Moonen

### Description

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

## November 2018

### New Developments in the Theory of Modular Forms over Function Fields

Meeting Type: conference

Contact: Mihran Papikian

### Description

### Young Researchers in Algebraic Number Theory

Meeting Type: conference

Contact: Ariel Weiss, Ciaran Schembri

### Description

Y-RANT is a conference for early career researchers in algebraic number theory and arithmetic geometry to present their research in a network of their peers. We are pleased to invite contributed talks. Funding is available for UK-based participants thanks to support from an LMS Scheme 8 Postgraduate Conference grant.

### Sixth Abel Conference: A Mathematical Celebration of Robert P. Langlands

Meeting Type: conference

Contact: Ben Brubaker

### Description

### 22nd Workshop on Elliptic Curve Cryptography

Meeting Type: summer school, conference

Contact: see conference website

### Description

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

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

### Syntomic cohomology and p-adic Hodge theory

Meeting Type: school

Contact: see conference website

### Description

## December 2018

### D-modules, quantum geometry and related topics

Meeting Type: conference

Contact: Takuro Mochizuki, Masa-Hiko Saito, Claude Sabbah

### Description

D-modules have recently played an important role in various fields of mathematics, including algebraic geometry, mathematical physics, arithmetic geometry, etc. The aim of this conference is to present several important developments related with D-modules. A series of survey talks on four active topics will be given by leading experts, together with research talks on related topics.

### School on cluster algebras

Meeting Type: school

Contact: see conference website

### Description

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

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

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

### 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

### Arithmetic of Shimura Varieties

Meeting Type: conference

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

### Birational Geometry and Moduli Spaces

Meeting Type: research program

Contact: see conference website

### Description

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

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

### Derived Algebraic Geometry

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.

### Thematic month: Complex geometry

Meeting Type: conference

Contact: Erwan Rousseau, Andreas Höring

### Description

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

### 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.

### 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

### 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

### 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.

### British Algebraic Geometry Meeting

Meeting Type: conference

Contact: Julius Ross

### Description

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

### Tropical Geometry: New Directions

Meeting Type: conference

Contact: see conference website

### Description

## 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.

### Recent Progress in Moduli Theory

Meeting Type: conference

Contact: see conference website

### Description

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

### 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.

### Rational points on Fano and similar varieties

Meeting Type: conference

Contact: see conference website

### Description

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

Meeting Type: Spring school

Contact: see conference website

### Description

### Thematic Program in Commutative Algebra and its Interaction with Algebraic Geometry

Meeting Type: undergraduate school, graduate school, conference

Contact: see conference website

### Description

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

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

## 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.

### Representation Theory and D-modules

Meeting Type: conference

Contact: Michel Gros, Thierry Levasseur, Tobias Schmidt

### Description

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

### 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

### 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.

### 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.

### 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

### 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.

### Graduate Summer School on the Geometry and Modular Representation Theory of Algebraic Groups

Meeting Type: Graduate Summer School

Contact: Mark Andrea de Cataldo, Francois Greer, Christian Schnell

### Description

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

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

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

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

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

### 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.

### 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

## May 2020

### The Arithmetic of the Langlands Program

Meeting Type: conference

Contact: see conference website

### Description

## June 2020

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

Meeting Type: conference

Contact: Andrew V. Sutherland

### Description

## August 2020

### 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.