The theory of automorphic forms is a dynamically expanding part of number theory with an increasing number of connections and applications to other branches of mathematics as well as physics. Research is driven by long standing conjectures and unexpected breakthroughs.

The purpose of this special semester is to bring together established researchers as well as those just starting their careers or studies. We would like to provide a rich and stimulating environment for interactions. There will be ample opportunity for the participants to present classical results and new developments. It is hoped that visits within this program will lead to further collaborations and progress.

The 4th IMA Conference on The Mathematical Challenges of Big Data is issuing a Call for Papers for both contributed talks and posters. Mathematical foundations of data science and its ongoing challenges are rapidly growing fields, encompassing areas such as: network science, machine learning, modelling, information theory, deep and reinforcement learning, applied probability and random matrix theory. Applying deeper mathematics to data is changing the way we understand the environment, health, technology, quantitative humanities, the natural sciences, and beyond ‐ with increasing roles in society and industry. This conference brings together researchers and practitioners to highlight key developments in the state‐of‐the art and find common ground where theory and practice meet, to shape future directions and maximize impact. We particularly welcome talks aimed to inform on recent developments in theory or methodology that may have applied consequences, as well as reports of diverse applications that have led to interesting successes or uncovered new challenges.

Confirmed Invited Speakers Dr Heather Battey, Imperial College London; Prof. Nando de Freitas, Google Deep Mind; Prof. Lenka Zdebrova, EPFL (Swiss Federal Institute Technology);
Prof. Tiago de Paula Peixoto, Central European University;

This is a one-day conference on Field Arithmetic in honor of Prof. Jarden 80th anniversary.

NTM is a series of meetings in Number Theory cofunded by the Department of Mathematical Sciences "G. L. Lagrange" of Politecnico di Torino and the Department of Mathematics "G. Peano" of Università degli Studi di Torino.

It is a biannual conference intended to highlight recent developments in the work of women in arithmetic geometry, and to connect promising young mathematicians with outstanding senior colleagues. Participation is open to everybody. The event is part of the newly established collaborative research center „Geometry and Arithmetic of Uniformized Structures“ between the universities of Frankfurt, Heidelberg, Darmstadt, Mainz and Münster.

The Preliminary Arizona Winter School (PAWS) is a virtual program on topics related to the upcoming AWS, with an intended audience of advanced undergraduate students and junior graduate students.

PAWS 2022 will consist of two concurrent six-week lecture series.

• Ronnie Nagloo: Introduction to model theory with applications

  Model theory is a branch of mathematical logic dealing with abstract structures, historically with connections to other areas of mathematics. The developments, over the past several decades, have allowed for a strengthening of those connections as well as new striking applications to areas such as diophantine and analytic geometry, algebraic differential equations, and combinatorics. This course will serve as an introduction to the basics of model theory, with a view towards some of the above applications.

• Padmavathi Srinivasan: Heights in Diophantine geometry

  The height of a rational number is a measure of its arithmetic complexity. For example, although the numbers 5 and 500000/100001 are close, the second is arithmetically more complex, and has a much larger height. The height of a rational number is easy to define -- it is simply the maximum of the absolute value of the numerator and the denominator when the number is expressed in lowest form. It is not immediately clear how one can extend this definition to more general algebraic numbers such as the squareroot of 2. In this series of lectures, we will develop the theory of heights of algebraic numbers, and present "Weil's height machine" for defining heights more generally for solutions to systems of polynomial equations in algebraic numbers (i.e., heights of algebraic points on varieties).

  Height functions are a key tool in proving many important finiteness theorems in Diophantine Geometry. The main property of height functions is that there are only finitely many points of bounded height and degree on any given variety. Understanding how quickly the number of points grow as the height grows for various classes of varieties is an active area of research in number theory today! As an application of the theory of heights, we will prove the Mordell--Weil theorem for elliptic curves, namely that the set of rational solutions to cubic equations such as y^2 = x^3 - 2x + 2 is finitely generated.

Registration: You can apply for the program here. The deadline to apply is July 15th, 2022.

Each lecturer will be accompanied by graduate student assistants, who will be in charge of writing weekly problem sets and facilitating weekly, hour-long problem solving and discussion meetings with groups of students. Recommended background for the program is a first course in abstract algebra.

The school will feature an online (Zulip) discussion board where students can ask questions and interact with the speakers and assistants outside of scheduled meeting times.

We will facilitate additional virtual events, some purely social to build a community, and some more structured sessions on timely and pertinent topics like "What is graduate school in Math like?", "Tips for applying to graduate school this Fall," "How do I navigate the first year of graduate school?" "How do I look for an thesis advisor?", "How to get the most out of the upcoming AWS."

We encourage undergraduate students to take their PAWS course as an independent study with a faculty member at their home institution.

• Organizers: Renee Bell, Isabel Vogt, Hang Xue, with Alina Bucur, Brandon Levin, Anthony Várilly-Alvarado, and David Zureick-Brown.
• Funded by the National Science Foundation.

The goal of the FRNTD is to provide a venue for faculty, graduate students, and undergraduates on the Front Range who are interested in number theory to meet, learn, and collaborate.

For more information and to register, please visit our website: https://sites.google.com/colorado.edu/front-range-number-theory-day/home. If you would like to give a five-minute talk, please include your title on the registration form.

This conference is intended to provide an overview of current research in homotopy theory, which has vastly expanded since its roots in the 19th century from the study of topological spaces by algebraic and combinatorial means to other fields of mathematics, including algebraic geometry, differential topology, and mathematical physics.

We will have morning talks presenting comprehensive viewpoints by a selection of experts, with the afternoon dedicated to more specialized talks - all of which should be accessible to a general audience of homotopy theorists.

This is a one-day regional topology conference focussing on algebraic and low-dimensional topology. It is planned as the third installment of the UNYTS series, which was started in 2017 under the name Topology Day. The goal of the conference is to build connections both within the New York State region and with nearby regions to create a more cohesive topology community.

Weekend regional conference in number theory and arithmetic geometry, featuring two mini-courses and additional research lectures.

A conference in honor of Bruno Kahn's 64th birthday

Mathematics Department at Tulane University is happy to announce the 2022 Clifford Lectures in memory of our late distinguished colleague and friend Professor Slawomir Kwasik. Our plenary speaker is Professor Reinhard Schultz (UC Riverside), who will give the following three lectures: Lecture 1. Rational invariants in algebraic and geometric topology. Lecture 2. Isovariant homotopy and classifications of group actions on manifolds. Lecture 3. High-dimensional topology and low-dimensional group actions. In addition to the above lectures, we will host talks by several distinguished researchers in the field of geometric topology. We invite everyone interested in attending the lectures to fill-out the registration form available at the conference website

Maple in Mathematics Education and Research

This conference is dedicated to exploring different aspects of the math software Maple, including Maple's impact on education, new symbolic computation algorithms and techniques, and the wide range of Maple applications. Attendees will have the opportunity to learn about the latest research, share experiences, and interact with Maple developers.

The conference will take place online, and will consist of two days of live presentations from Maplesoft and the community. Presentations will be recorded and made available after the conference.

The fifteenth Annual Binghamton University Graduate Combinatorics Algebra and Topology Conference (BUGCAT Conference) will meet November 5-6, 2022 at Binghamton university. Graduate students at all levels, as well as faculty, are invited to give a 30-minute talk; talks may be expository or on current research. The deadline for submitting talks is October 18, 2022. This year, we have three distinguished keynote speakers: Yair Minsky(Yale University), Rigoberto Florez (The Citadel) and Kristen Wicklengren (Duke University). The 2022 BUGCAT conference is supported by the NSF grant, the Binghamton University Graduate Student Organization, the Binghamton University Division of Research, and an endowment from the Kappe family in memory of Wolfgang Kappe. For more information visit :http://seminars.math.binghamton.edu/BUGCAT/index.html

The meeting will highlight recent advances in homotopical algebra and geometry, their interactions and applications. Since its emergence in the 1960s, homotopical algebra has steadily gained in importance and applications. The latter range from algebraic topology and geometry to differential geometry and mathematical physics to topological data analysis and computer science. The conference brings together researchers from these different fields to discuss recently developed topics, ongoing work and speculative new ideas. The event offers a unique opportunity to meet scholars who share a common interest in homotopy theory. There will be sufficient time for discussions between the participants; a poster session for junior researchers will be organized.

List of speakers:

Nils Baas (Norwegian University of Science and Technology) Wojciech Chacholski (Royal Institute of Technology Stockholm) Benoît Fresse (University of Lille) Julien Grivaux (Sorbonne University) Vladimir Hinich (University of Haifa) Andrey Lazarev (Lancaster University) Ieke Moerdijk (Utrecht University) Joost Nuiten (Institut de Mathématiques de Toulouse) Christian Sämann (Heriot-Watt University Edinburgh) Hisham Sati (New York University Abu Dhabi) Claudia Scheimbauer (Technical University Munich) Alexander Schenkel (University of Nottingham) Martina Scolamiero (Royal Institute of Technology Stockholm) Sergey Shadrin (University of Amsterdam) Bruno Vallette (University Sorbonne Paris Nord)

Scientific Committee:

Vladimir Dotsenko (University of Strasbourg) Kathryn Hess (Ecole Polytechnique Fédérale de Lausanne) Damjan Pistalo (University of Luxembourg) Norbert Poncin (University of Luxembourg) Urs Schreiber (New York University in Abu Dhabi)

The workshop will focus on the interaction between homotopy theory and symplectic topology and low dimensional topology that is mediated by Floer theory. Among the topics covered are foundational questions, applications to concrete geometric questions, and the relationship with finite dimensional approaches.

TARGET AUDIENCE The school is aimed at mathematics, computer science and engineering advanced undergraduate or graduate students interested in algorithms, theory of computing, combinatorics, optimization, game theory, and their applications.

LECTURERS

Marthe Bonamy

Université de Bordeaux, LaBRI

Vincent Cohen-Addad

Google Research

László Végh

London School of Economics and Political Science

APPLICATIONS and FINANCIAL ASSISTANCE:

Accepted students will receive financial assistance for their local expenses (meals and accommodation) in Valparaiso. For more information and access to the application form see:

https://eventos.cmm.uchile.cl/discretas2023/postulaciones/ [https://eventos.cmm.uchile.cl/discretas2023/postulaciones/]

Application deadline: October 15, 2022

VENUE Instituto de Sistemas Complejos de Valparaiso (http://www.iscv.cl/ [http://www.iscv.cl/]), Valparaiso, Chile.

The fundamental conjecture of Birch and Swinnerton-Dyer relating the Mordell–Weil ranks of elliptic curves to their L-functions is one of the most important and motivating problems in number theory. It resides at the heart of a collection of important conjectures (due especially to Deligne, Beilinson, Bloch and Kato) that connect values of L-functions and their leading terms to cycles and Galois cohomology groups.

The study of special algebraic cycles on Shimura varieties has led to progress in our understanding of these conjectures. The arithmetic intersection numbers and the p-adic regulators of special cycles are directly related to the values and derivatives of L-functions, as shown in the pioneering theorem of Gross-Zagier and its p-adic avatars for Heegner points on modular curves. The cohomology classes of special cycles (and related constructions such as Eisenstein classes) form the foundation of the theory of Euler systems, providing one of the most powerful methods known to prove vanishing or finiteness results for Selmer groups of Galois representations.

The goal of this semester is to bring together researchers working on different aspects of this young but fast-developing subject, and to make progress on understanding the mysterious relations between L-functions, Euler systems, and algebraic cycles.

Number Theory concerns the study of properties of the integers, rational numbers, and other structures that share similar features. It is a central branch of mathematics with a well-known feature: it is often the case that easy-to-state problems in number theory turn out to be exceedingly difficult (e.g. Fermat’s Last Theorem), and their study leads to groundbreaking discoveries in other fields of mathematics.

A fundamental theme in number theory concerns the study of integer and rational solutions to Diophantine equations. This topic originated at least 3,700 years ago (as documented in babylonian clay tablets) and it has evolved into the highly sophisticated field of Diophantine Geometry. There are deep and fruitful interactions between Diophantine Geometry and seemingly distant fields such as representation theory, algebraic geometry, topology, complex analysis, and mathematical logic, to mention a few. In recent years, these connections have led to a large number of new results and, specially, to the partial or complete resolution of important conjectures in the field.

While the study of rational solutions of diophantine equations initiated thousands of years ago, our knowledge on this subject has dramatically improved in recent years. Especially, we have witnessed spectacular progress in aspects such as height formulas and height bounds for algebraic points, automorphic methods, unlikely intersection problems, and non-abelian and p-adic approaches to algebraic degeneracy of rational points. All these groundbreaking advances in the study of rational and algebraic points in varieties will be the central theme of the semester program “Diophantine Geometry” at MSRI. The main purpose of this program is to bring together experts as well as enthusiastic young researchers to learn from each other, to initiate and continue collaborations, to update on recent breakthroughs, and to further advance the field by making progress on fundamental open problems and by developing further connections with other branches of mathematics. We trust that younger mathematicians will greatly contribute to the success of the program with their new ideas. It is our hope that this program will provide a unique opportunity for women and underrepresented groups to make outstanding contributions to the field, and we strongly encourage their participation.

The Connections Workshop features presentations by both leading researchers and promising newcomers whose research has contact with the interrelated topics of algebraic cycles, L-values, and Euler systems. The goal is to present a variety of diverse results, so as to forge new connections, foster collaborative projects, and establish mentoring relationships. While emphasis will be placed on the work of women mathematicians, the workshop is open to all researchers.

The Introductory Workshop aims to provide a coherent overview of current research in algebraic cycles, L-values, Euler systems, and the many connections between them. This includes the study of special cycles on Shimura varieties and moduli spaces of shtukas, integral representations of L-values and the construction of p-adic L-functions, and the construction of Euler systems from special elements in Chow groups or higher Chow groups of Shimura varieties. Workshop lectures will be organized into short lecture series, so as to allow each series to begin with expository lectures on foundational results before moving on to current research.

The purpose of this school is to familiarise advanced Master’s students, Ph.D. students and early career researchers with the central themes of research in arithmetic geometry,and teach them both classical and state-of-the-art tools in the field. The research school is an introduction to the conference SAGA, “Symposium on Arithmetic Geometry and its Applications” and it consists of four 3-day mini-courses, taught by teams of internationally renowned researchers, who are known to be excellent teachers.

Courses:

• Galois representations and modular forms: Alex Ghitza and Anna Medvedovsky
• Modularity and diophantine applications: Samir Siksek and Samuele Anni
• Local-global principles: Rachel Newton and Ekin Ozman
• Jacobians and models: Tim Dokchitser and Elisa Lorenzo García

The thematic month “Arithmetic and Information Theory” focuses on arithmetic geometry, information theory and their interplay.

Arithmetic geometry is a fundamental and growing area of research in modern mathematics, deeply connected to almost all of its branches. As a measure of its importance, about a quarter of the Fields medalists have worked on problems in arithmetic geometry. There is a vast number of different techniques that have been developed in the field. Algebraic Geometry Codes Theory studies (amongst others) varieties, and in particular curves, with many rational points, having in mind that maximal curves are the aspirational target to obtain codes with the best correction rate. This maximality corresponds to extremal points in the distribution of the Frobenius endomorphism acting on the Tate module of the Jacobian of the curves. For elliptic curves, we are confronted with the Sato-Tate conjecture which is concerned with the statistics to modulo p reductions of families of elliptic curves defined over the field of rational numbers. Galois representations on ´etale cohomology play a central role in this field. The most remarkable aspects of this series of conferences is the omnipresent exchange between applied (codes, cryptography) and pure (arithmetic and geometry) mathematics. The first branch supplies the second with new problems. For example, the impulse of Serre, Manin, and Ihara, motivated by the introduction of Goppa codes, led to hundreds of articles devoted to the study of the number of points on curves over finite fields. A more specialized and recent example concerns quantum codes that lead to the study of couples code/sub-code having special properties. As a counterpart, arithmetic and geometry allow remarkable new results to be obtained in their applications. For example, cryptosystems constructed from elliptic curves over finite fields, or the best asymptotically excellent codes constructed using algebraic varieties.

This workshop will highlight talks on various aspects of Diophantine Geometry. The goal of the workshop is to bring together researchers at different career stages and of various backgrounds in order to establish new collaborations and mentoring relationships. Although we will showcase the research of mathematicians who identify as women or gender minorities, this workshop is open to all.

This workshop will feature expository lectures about current developments in Diophantine geometry. This includes the uniform Mordell—Lang for rational points on curves, the Andre—Oort conjecture for special points on Shimura varieties, and effective results via Chabauty method, and related topics in Arakelov theory, unlikely intersections, arithmetic statistics, arithmetic dynamics, and p-adic Hodge theory.

The conference will focus on the recent developments in arithmetic geometry, its applications and explicit and computational methods in approaching the relevant problems. There will be four main axes around which all talks will be organised: Automorphic forms and moduli spaces; Galois representations: geometric and constructive, modularity; Rational points and obstructions; Effective methods in arithmetic geometry. Leading experts in the field have accepted to contribute to the conference, making it a particularly interesting event.

The conference is dedicated to the memory of Bas Edixhoven (Professor, Leiden University)

The conference will be devoted to the study of algebraic varieties over finite fields from an algebraic geometry point of view as a combinatorial one and their applications to coding theory.

Featured topics will include theoretical, effective and algorithmic aspects of point counting as well as algebraic geometry codes constructed from curves and higher dimensional varieties.

This conference is devoted to the algebraic and combinatorial aspects of codes and cryptography and the links between codes and cryptography. It aims at gathering international experts, including researchers from the University of Aix-Marseille, in this essential and very active field of mathematics and theoretical computer science. This event aims also to provide a forum for researchers working on algebraic and combinatorial methods for coding and cryptography, exchange ideas and interests in open problems, and further explore their applications in cryptography error-correcting codes and communications. We plan to issue a ”Call for Paper” for specialized talks and select some presentations to be published in the congress proceedings in the international journal Advances in Mathematics of Communications édited by American Institute of Mathematical Sciences and handled by Jintai Ding and Sihem Mesnager as Editors-in-Chief.

The theme of the conference will be explicit and computational methods in number theory and arithmetic geometry in a broad sense. The format will include scientific talks as well as time for informal collaboration and for coding projects related to (for example) PARI/GP, SageMath, Magma, OSCAR or the L-Functions and Modular Forms Database.

On the one hand, various topics where explicit computations have been the key for proving important results will be presented. These will be found in the context of modular forms, the study of rational points, as well as results towards the Birch and Swinnerton-Dyer conjecture. On the other hand, we will also focus on recently stated conjectures, for example the paramodular conjecture by Brumer and Kramer, and challenge participants to exhibit new examples to support such conjectures (in the case of the paramodular conjecture only one non trivial example is currently known).

We expect that the colloquium will lead to the emergence of new ideas and methods at the interface of these different fields, to new results as well as to new projects and collaborations.

This conference will be organized with the support of the National Research Agency in the framework of the project "MELODIA".

Speakers:

• Laura DeMarco (Chicago)
• Jonathan Pila (Oxford)
• Thomas Scanlon (Berkeley)
• Jacob Tsirmerman (Toronto)

Families of p-adic automorphic forms are well studied objects of arithmetic geometry since the pioneering work of Hida and Coleman. Their study resulted in the definition of geometric objects, called eigenvarieties, that parametrize systems of Hecke eigenvalues of p-adic automorphic forms. Conversely, the rich geometry of these varieties gives insights about p-adic (and thereby also about classical) automorphic forms. Recent techniques from perfectoid geometry, locally analytic representation theory and the point of view of the p-adic Langlands program give new insights and impulses.

The spring school will give an introduction to both p-adic automorphic forms and eigenvarieties as well as the necessary background in p-adic analytic geometry. The courses will be complemented by research talks that will focus on recent developments in the area.

The first week of the spring school will focus on p-adic analytic geometry, the analogue of complex analytic geometry over p-adic base fields. We will study classical rigid analytic spaces from the point of view of adic spaces and introduce perfectoid spaces. The second week will focus on p-adic automorphic forms and eigenvarieties. We will introduce and compare several approaches to p-adic automorphic forms.

Spring School lectures:

• John BERGDALL, University of Arkansas
• Eugen HELLMANN, University of Münster
• Ben HEUER, University Bonn
• Katharina HÜBNER, University of Heidelberg
• Adrian IOVITA, Univ. Concordia / Padua
• Christian JOHANSSON, University of Gothenburg
• Judith LUDWIG, University of Heidelberg
• James NEWTON, University of Oxford

Research Talks:

• Rebecca BELLOVIN, University of Glasgow
• Chris BIRKBECK, University College London
• Christian JOHANSSON, University of Gothenburg
• Lucas MANN, University Bonn
• Joaquín RODRIGUES JACINTO, Univ.Paris Saclay
• Mingjia ZHANG, University Bonn
• Vincent PILLONI, Université Paris-Saclay

A conference dedicated to the memory of Claus-Günther Schmidt.

Committee: Haruzo Hida (UCLA), Fabian Januszewski (Paderborn University), Tadashi Ochiai (Osaka University), A. Raghuram (Fordham University)

For details, please refer to the conference website.

The topical workshop will be dedicated to Shouwu Zhang, to mark the occasion of his 60th birthday, and to honour his numerous beautiful contributions to the theory of Shimura varieties and special values of L-functions. It will highlight cutting edge work on topics such as the construction of Euler systems; relations between special cycles on Shimura varieties and L-functions, such as generalized Gross-Zagier formulas and the Tate conjecture; the construction of Galois representations in cohomology; and related aspects of the theory of automorphic forms.

A conference on Homotopy theory at Northwestern to honor the contributions of Paul Goerss.

ln this workshop, we will consider variants of Artin's primitive root conjecture leading to the study of the Galois groups of various radical extensions. Beyond the case of the multiplicative group studied by Lenstra and others, there are now also interesting results for elliptic radicals, and for division points in more general abelian varieties. ln this context, the elliptic analogue of Artin's conjecture is the Lang-Trotter conjecture, which is still open after more than 40 years.

The Galois representations associated to various division points in abelian varieties are central to understanding the Galois groups of the radical extensions that one tries to explicitly describe in this context, as they control the behaviour of the primes in the underlying problems. Understanding these Galois representations, and the entanglement between the extensions generated by different prime-power radicals, is essential to progress in this area.

ln this circle of problems and questions, one encounters interesting restrictions to local-global principles that will be addressed in this workshop, not only in the context of radical extensions.

This workshop will 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. The workshop will also celebrate the mathematics of Hendrik W Lenstra Jr.

A central topic in Diophantine Geometry is to understand how the geometry of a variety influences the arithmetic of its algebraic points, and conversely. Conjectures of Bombieri, Lang, and Vojta suggest that rational points of algebraic varieties satisfying suitable approximation conditions, are algebraically degenerate. On the other hand, conjectures on unlikely intersections suggest that algebraic points of special type —e.g. torsion points in semi-abelian varieties, special points in Shimura varieties— avoid subvarieties, unless the subvariety itself is also special (in a technical sense).

In recent years, a number of techniques have led to outstanding progress on Lang-Vojta conjectures, such as the Subspace Theorem, p-adic approaches to finiteness, and modular methods. Similarly, spectacular progress has been achieved on unlikely intersection conjectures thanks to new methods and tools, such as height formulas for special points, connections to model theory, refined counting results, and new theorems of Ax-Shanuel type (bi-algebraic geometry).

The goal of this workshop is to create the opportunity for these two groups to interact, to share their techniques, to update on the most recent progress, and to attack the outstanding open questions in the field.

The two directions described above are rather technical and specialized, and it seems necessary to bring together these groups of researchers to explain to each other not only the latest developments in their fields, but also the methods that made possible these breakthroughs. Thus, in this workshop we expect to have lectures explaining the main methods, as well as talks presenting the most recent progress in the subject by the world leading experts.

The field of arithmetic statistics is a fast-moving area of number theory, dealing with distributional results for many basic and important objects: ranks of elliptic curves, central values of L-functions, number fields, modular symbols, orbits of discrete groups etc. The techniques involved come from diverse areas of mathematics e.g. automorphic forms, ergodic theory, spectral theory, dynamical systems, probabilistic number theory, representation theory and random matrix theory. This workshop intends to bring together researchers of all career levels to present their work and exchange ideas and techniques on arithmetic statistics. Motivated by the programme of Mazur and Rubin based on their computational study of elliptic curves over abelian extensions of fixed degree, the meeting will focus on modular symbols, Manin's noncommutative modular symbols, equidistribution of lattice points on the sphere, Heegner points, and closed geodesics, including the error term in the Prime Geodesic Theorem.

There will be invited talks lasting 50 or 25 minutes and other events aimed at facilitating discussions among participants.

The Langlands program aims to relate systems of polynomial equations with integer coefficients to automorphic forms, i.e. functions on symmetric spaces with a large number of discrete symmetries. The focus of the trimester will be on some manifestations of this program, including:

moduli spaces of shtukas
p-adic techniques in local Langlands and the relation to geometric Langlands
Shimura varieties and more general spaces in global Langlands

The spring school serves an introduction to the conference "Arithmetic Statistics", and aims to provide PhD students and early stage researchers with the necessary background to enable them to fruitfully partake in the conference. lt will also be an occasion for Master students to have an introduction towards research in number theory.

Along with lecture courses, there will be exercise sessions and a possibility to work on small research projects under the guidance of the lecturers and the researchers proposing them.

The lecture courses will take place in the mornings, while exercise sessions will take place in the afternoon, with extra time allotted for round tables and work in groups.

Lecture courses thematics are the following:

• Galois representations and Diophantine equations,
• Complex Multiplication,
• Class field theory,
• Zeta functions and L-functions,
• Abelian varieties.

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 theme of the conference is a type of number theory that has become very popular over the last decades, and that is influenced by the possibility of « experimentally » studying arithmetic objects with the help of a computer. Thanks to the wide availability of computer algebra systems, essentially any number theorist nowadays has this possibility at his fingertips. There are concrete objects that are not so easily determined « by hand », such as fundamental units in number fields of higher degree, or Mordell-Weil generators of point groups of elliptic curves, and a first impression of « what these typically look like » is often obtained by numerical experimentation.

Over the years, substantial datasets relating to number fields, elliptic curves and L-series have become available, enhancing our understanding of the arithmetic world somewhat beyond only the smallest examples, which may fail to show the true asymptotic behaviour.

Conference dedicated to the memory of Bas Edixhoven

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.

In the early 1920s, Emmy Noether introduced the fundamental concept of a Noetherian ring, a notion that has had a remarkably broad impact on mathematics over the last century. In this conference, we celebrate Noether's legacy with research talks from many areas of algebra, broadly construed, including algebraic geometry, commutative algebra, number theory, and representation theory.

Interest in specialization of polynomials and Galois groups goes back at least to the work of Hilbert on the inverse Galois problem. This theory has found an abundance of applications in Algebra, Number Theory, Group Theory, and Arithmetic Geometry. In recent years, the area is blooming and we see striking results that open completely new horizons: The discovery of Hilbert irreducibility properties of algebraic groups, its connection with expanders and random walks, the interrelation with arithmetic-geometric properties of parametrizing varieties, and the exciting progress on the Cohen-Lenstra heuristics. The conference aims to bring together leading experts and young researchers interested in the area. We plan to leave an abundance of free time, dedicated to informal discussions. We believe that this will encourage the transfer of ideas, techniques, and will foster new collaborations and new research directions.

We are going to organize a week long conference during June 26-30, 2023, hosted by the Alfred Renyi Institute of Mathematics in Budapest, Hungary. The topic of the conference will emphasize the rich interactions between complex analysis and complex geometry, within the context of geometric analysis.

In addition, the conference will serve as an opportunity to celebrate the 70+1th birthday of Laszlo Lempert.

This will be a one week conference broadly focused on the topics of the LMFDB, mathematical databases, computation, number theory, and arithmetic geometry. The conference will include invited talks, presentations by authors of papers submitted to the conference and selected by the scientific committee following peer-review, as well as time set aside for research and collaboration. We plan to publish a proceedings volume that will include all of the accepted papers.

This is the ninth Iwasawa conference following conferences in Besancon, Limoges, Irsee, Toronto, Heidelberg, London, Tokyo and Bordeaux. The conference is dedicated to the memory of John Henry Coates.

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

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

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.

During the 2023-24 academic year the School will have a special program on the p -adic arithmetic geometry, organized by Jacob Lurie and Bhargav Bhatt, who will be the Distinguished Visiting Professor.

The last decade has witnessed some remarkable foundational advances in p-adic arithmetic geometry (e.g., the creation of perfectoid geometry and the ensuing reorganization of p-adic Hodge theory). These advances have already led to breakthroughs in multiple different areas of mathematics (e.g., significant progress in the Langlands program and the resolution of multiple long-standing conjectures in commutative algebra), have uncovered new phenomena that merit further investigation (e.g., the discovery of new structures on algebraic K-theory, new period spaces in p-adic analytic geometry, and new bounds on torsion in singular cohomology), and have made hitherto inaccessible terrains more habitable (e.g., birational geometry in mixed characteristic). This special year intends to bring together a mix of people interested in various facets of the subject, with an eye towards sharing ideas and questions across fields.

The conference is about recent developments in Arithmetic Algebraic Geometry. Its central theme is the geometry of Shimura varieties and related spaces in all its facets. Topics to be covered include: Integral models of Shimura varieties and the geometry of their reductions, p-adic and perfectoid geometry, special cycles on Shimura varieties, moduli spaces of Galois representations and (φ, Γ)-modules.

The website will be available soon.