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Discrete Structures: Harmonic Analysis and Probability

This workshop will focus  on progress in analysis of geometric functional inequalities in particular, but not exclusively, on discrete model: the Hamming cube.

Talks will give an overview of the area, its connections to PDE, combinatorics, martingale methods, stochastic optimization problems, various notions of convexity, and applications to random graphs.  The harmonic analysis on Hamming cube is a tool for studying the Renyi random graph model (e.g. sharp threshold results of the type of Margulis graph connectivity theorem). 

The geometric functional inequalities are often rooted in their discrete versions on Hamming cube, where the harmonic analysis has many interesting specifics and often uses probabilistic ideas. For example, martingale method leads to important geometric inequalities on Hamming cube and  Gauss space, such as e.g. the concentration of measure inequalities. The concentration of measure phenomenon, in its turn, has utmost importance in applied mathematics, in questions of spectral initialization or compressed sensing.

Speakers:  Franck Barthe (Toulouse), Sergey Bobkov (Minnesota), Dario Cordero-Erausquin (Paris 6), Paata Ivanisvili (UC Irvine), Ryan O’Donnell (Carnegie Mellon), Stefanie Petermichl (Toulouse), Mark Rudelson (Michigan), Ramon van Handel (Princeton), Alexander Volberg (Michigan State)

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Venue: Mathematical Institute, University of Oxford

Analysis and Probability

This workshop will focus on the recent progress in Analysis with particular attention to the problems that are either directly motivated by Probability or use probabilistic ideas and techniques. For example, recent progress of Logunov and Malinnikova who proved the Nadirashvili’s conjecture and obtained the lower bound in Yau’s conjecture about the size of nodal sets for Laplace eigenfunctions or analytic proof by Tolsa and Volberg of Tsirelson’s theorem about harmonic measures. Talks will give an overview of the current state of the area and present recent progress in several long-standing open questions.

Participants: Kari Astala (Aalto), Oleg Ivrii (Caltech), Nikolai Makarov (Caltech), Andrei Okounkov (Columbia), Alexei Poltoratski (Texas A&M), Eero Saksman (Helsinki), Misha Sodin (Tel Aviv), Xavier Tolsa (UA Barcelona), Yilin Wang (ETH)

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Venue: Mathematical Institute, University of Oxford

Modular Representation Theory

Modular representation theory is the study of representations of groups and related algebras over fields of positive characteristic. It was initially developed by Brauer, with a view towards the structure of finite groups. Over the last decades it has grown into a deep and beautiful subject with connections to number theory, the Langlands program and algebraic geometry.

The goal of this workshop is to explore the breadth of modern modular reprensentation theory. There will be talks from relevant areas including: mod p Langlands correspondence, deformations of Galois representations, modular representations of algebraic groups, Lie algebras and p-adic groups, modular Deligne-Lusztig theory, and local-global conjectures in finite groups.

Speakers:  Noriyuki Abe (Tokyo), Pramod Achar (LSU), Roman Bezrukavnikov (MIT), Cédric Bonnafé (Montpellier), Ana Caraiani (Imperial), Joe Chuang (City London), Olivier Dudas (Paris 7), Jens Eberhardt (UCLA), Amit Hazi (City London), Bao Le Hung (Northwestern), Brandon Levin (Arizona), Carl Mautner (UC Riverside), Gil Moss (Utah), Gabriel Navarro (Valencia), Simon Riche (Clermont Auvergne), Raphael Rouquier (UCLA), Peng Shan (Tsinghua), Jack Thorne (Cambridge)

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Venue: Mathematical Institute, University of Oxford

Periods, Representations, and Arithmetic: Recent Advances on the Gan-Gross-Prasad Conjectures and their Applications

In the early 2000s, Gan, Gross, and Prasad  formulated a series of conjectures connecting the representation theory of classical groups, periods of automorphic forms, algebraic cycles on Shimura varieties, and special values of L-functions. There has been a lot of progress since on their conjectures and the arithmetic applications. This workshop will explore recent developments as well as possible future directions in the subject. 

Speakers:  Raphaël Beuzart-Plessis (Marseille), Pierre-Henri Chaudouard (Paris 7), Dimitar Jetchev (EPFL), Rahul Krishna (Northwestern), Chao Li (Columbia), Philippe Michel (EPFL), Andreas Mihatsch (MIT), Dipendra Prasad (TIFR), Kartik Prasanna (Michigan), Michael Rapoport (Bonn), Brian Smithling (Johns Hopkins), Yichao Tian (Strasbourg), Eric Urban (Columbia), Jingwei Xiao (MIT), Hang Xue (Arizona), Shouwu Zhang (Princeton)

Image: Spherical harmonics, inigo.quilez, Wikimedia Commons

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Venue: Mathematical Institute, University of Oxford

Beyond Spectral Gaps

The goal of this workshop is to bring to Oxford leading experts in the various areas of mathematics and theoretical computer science with an interest in the spectral analysis of graphs and its many applications within mathematics. Expander graphs have now become a ubiquitous tool in much of theoretical computer science and attention has focused in the last decade or so, on the one hand, on applications of these methods to other, seemingly unrelated questions both within pure mathematics and within combinatorics and computer science, and on the other hand, to the development of generalizations of the idea of spectral gap. Our goal is to focus on those exciting recent developments where the goal has not been simply to establish a spectral gap in any given situation, but to obtain more refined spectral information with it or from it. Topics will include interlacing polynomials, strong property (T), monotone expanders, local spectral gaps, small-set expansion, affine sieve, non-linear spectral gaps, matrix expanders, high-dimensional expanders, etc.

This workshop is one of the two events composing Pansu’s Fest.  It will be preceded by the conference Geometry and Analysis: Celebrating the Mathematics of Pierre Pansu.  Participants in either of the events are welcome to attend the other event as well.

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Venue: Mathematical Institute, University of Oxford

Physics from the Point of View of Geometry

In the 1980s the symbiosis between mathematics and physics saw a paradigm shift when physical insight led to compelling mathematical conjectures and their proofs in turn brought validation to physical theories that could not be tested through experiments.  One such example is Kontsevich’s proof of Witten’s conjecture on KdV hierarchies.  Segal’s axiomatisation of conformal field theory (CFT) and the adaptation to topological field theories (TQFT) by Atiyah provided a common language.  These basic concepts were taken forward and are at the origin of a number of  new research programs in geometry, topology, algebra, and mathematical physics. 

One strand, in the hand of topologists, saw the complete computation of invertible TQFTs, the resolution of the cobordism hypothesis, and a computation of invertible topological phases.  This also enabled the complete description of semi-simple field theories in dimension two.  Other highlights include the computation of the Verlinde algebra of loop groups in terms of equivariant twisted K-theory, establishing the formal relation of closed string theory and open string theory via Hochschild homology, and the development of a far reaching program to construct  higher chromatic cohomology theories in terms of higher dimensional TQFTs.

This workshop celebrates the influence of Graeme Segal on these developments. It brings together leading experts working on a variety of different aspects of geometry and physics of conformal and topological field theory.

Speakers:  David Ben Zvi (Austin), Dan Freed (Austin), Mike Hopkins (Harvard), Maxim Kontsevich (IHES), Claudia Scheimbauer (TU Munich), Constantin Teleman (Berkeley), Ulrike Tillmann (INI, Oxford), Edward Witten (IAS)

Registration: Participation in the workshop is by invitation but additional places are available. To register for the Clay Research Conference or to register interest in a workshop, email Naomi Kraker. Students please also provide a letter of reference from your supervisor.

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Venue: University of Oxford

p-adic Hodge Theory and Applications

The subject of  p-adic Hodge theory has seen a number of dramatic breakthroughs in recent years, including the introduction of perfectoid geometry, new approaches to p-adic cohomology theories,  and the geometrization of the local Langlands correspondence. These developments have led to the solution of longstanding conjectures both within p-adic Hodge theory and in adjacent fields, and the introduction of new classes of geometric objects for future study.  

The workshop will focus on some of these recent advances, including the growing interactions of p-adic Hodge theory with commutative algebra, mixed characteristic algebraic geometry, and K-theory.

Speakers:  Benjamin Antieau (Northwestern), Ana Caraiani (Imperial), Kestutis Česnavičius (Paris-Saclay), Dustin Clausen (Copenhagen), Pierre Colmez (Jussieu), Matthew Emerton (Chicago), Laurent Fargues (Jussieu), Ofer Gabber (IHES), Lars Hesselholt (Copenhagen), Arthur-César Le Bras (Paris 13), Linquan Ma (Purdue), Matthew Morrow (Jussieu), Wieslawa Nizioł (Jussieu), Alexander Petrov (Harvard), Peter Scholze (Bonn)

Participation in the workshop is by invitation, but a limited number of additional places is available. To register for the Clay Research Conference or to register interest in a workshop, email Naomi Kraker. Students please also provide a letter of reference from your supervisor.

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Venue: University of Oxford

O-minimality and Diophantine Geometry

The last ten years have seen a flourishing interaction between o-minimality, a part of model theory, and diophantine geometry. The catalyst for this interaction was Pila’s proof in 2011 of the André-Oort conjecture for products of modular curves. This applied the Pila-Wilkie counting theorem, using a method introduced by Pila and Zannier, in combination with results in function transcendence. Pila introduced further ideas to prove these transcendence results using o-minimality.

Since then there have been many further developments. Many of these have taken place in the framework of the Zilber-Pink conjecture, a conjecture on unlikely intersections in mixed Shimura varieties, which implies the André-Oort conjecture and a great deal more. Work by various researchers has shown that the Pila-Zannier method can also apply to these unlikely intersection problems, and now both o-minimality and functional transcendence are important tools in several areas of diophantine geometry.

More recently, Bakker, Brunebarbe, Klingler and Tsimerman have applied ideas from o-minimality, in the study of Hodge theory. Their work further develops applications of o-minimality to functional transcendence. This in turn has applications back to diophantine geometry. For instance, the general Ax-Schanuel result established by Bakker and Tsimerman for period maps plays a key role in the work of Lawrence and Venkatesh.

This workshop, organised in honour of Jonathan Pila’s 60th birthday, brings together leading experts in these various developments. The speakers include a blend of established experts and promising junior researchers, as well as people from nearby areas in which o-minimality or functional transcendence are applied.

Speakers:  Enrico Bombieri (IAS) tbc, Laura Capuano (Roma Tre), Gabriel Dill (Hannover), Ziyang Gao (Leibniz Hannover),  Philipp Hagbegger (Basel) tbc, Annette Huber (Freiburg), David Masser (Basel), Kobi Peterzil (Haifa), Marta Pieropan (Utrecht), Jonathan Pila (Oxford), Peter Sarnak (Princeton and IAS), Tom Scanlon (Berkeley), Harry Schmidt (Basel), Margaret Thomas (Purdue), Jacob Tsimerman (Toronto), Umberto Zanier (Pisa)

Participation in the workshop is by invitation, but a limited number of additional places is available. To register for the Clay Research Conference or to register interest in a workshop, email Naomi Kraker. Students please also provide a letter of reference from your supervisor.

Graduate students interested in these topics may also be interested in the CMI-HIMR Summer School on Unlikely Intersections in Diophantine Geometry.

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Venue: University of Oxford

Stability and Instability in General Relativity

Speakers:  Piotr Bizon (Jagiellonian), Grigorios Fournodavios (Princeton), Dejan Gajic (Radboud), Carsten Gundlach (Southampton), Gustav Holzegel (Münster), Anna Iljas (NYU), Christoph Kehle (IAS), Jonathan Luk (Stanford), Georgios Moschidis (Princeton), Malcolm Perry (Cambridge), Hans Ringstrom (KTH), Jorge Santos (Cambridge), Yakov Shlapentokh-Rothman (Toronto) tbc, Jacques Smulevici (Sorbonne), Jeremie Szeftel (Sorbonne), Rita Teixeira Da Costa (Princeton), Helvi Witek (Illinois Urbana-Champaign), Zoe Wyatt (Cambridge)

Participation in the workshop is by invitation, but a limited number of additional places is available. To register for the Clay Research Conference or to register interest in a workshop, email Naomi Kraker. Students please also provide a letter of reference from your supervisor.

Getting to Oxford

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Venue: University of Oxford