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Analytic Number Theory

This workshop will cover all aspects of analytic number theory.  There have been a number of spectacular advances in recent years, on gaps between primes, mean-values of L-functions, Vinogradov’s mean-value theorem, rational points on algebraic varieties, and additive combinatorics, for example.

The aim of the workshop will be to bring together researchers in these various fields, to learn about the latest developments, facilitate interaction between different areas, promote collaborations, and to suggest future directions.

Invited speakers: Tim Browning (Bristol), Vorrapan Chandee (Burapha), Chantal David (Concordia), John Friedlander (Toronto), Leo Goldmakher (Williams), Andrew Granville (Montreal), Gergely Harcos (Renyi Institute), Adam Harper (Cambridge), Roger Heath-Brown (Oxford), Harald Helfgott (ENS), Emmanuel Kowalski (ETH Zurich), Dimitris Koukoulopoulos (Montreal), James Maynard (Montreal and Oxford), Lillian Pierce (Hausdorff Centre), Maksym Radziwill (IAS), Fernando Shao (Stanford), Kannan Soundararajan (Stanford), Trevor Wooley (Bristol)

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Venue: Mathematical Institute, University of Oxford

Symplectic Topology

The origins of symplectic topology lie in classical dynamics, and the search for periodic orbits of Hamiltonian systems.  It is now understood to arise naturally in algebraic geometry, in low-dimensional topology, in representation theory and in string theory.  Following seminal ideas of Gromov and Floer from the 1980s, several of the most powerful tools in symplectic topology revolve around invariants counting pseudoholomorphic curves.  An important theme in recent years has been that holomorphic curve invariants are not independent, but are bound together and governed by very rich algebraic structures, making connections to integrable systems, and to the theory of A-infinity and L-infinity algebras.  On the one hand, the algebraic structures make the invariants more powerful; on the other, these algebraic structures often underscore the connections to other fields, where similar structures arise with different origins.

The aim of this workshop is to bring together experts in various parts of holomorphic curve theory and to expose the latest developments in:

(i) Floer cohomology theory, the Fukaya category, and aspects of homological mirror symmetry
(ii) contact and Stein topology, and their connections to low-dimensional topology and to dynamics.

Recent areas of progress include flexibility results in Stein topology,  formulations and proofs of homological mirror symmetry, renewed ideas about flux and categorical dynamics, existence theorems for quasi-morphisms, sharp constraints on symplectic or Lagrangian embeddings, and relations to knot theory and finite type invariants.   Whereas much recent activity has been directed towards foundational issues, the workshop is intended to focus more on applications of the theory, particularly those where progress has been and is being made, and to highlight some of the new questions that progress raises.

Speakers: Mohammed Abouzaid (Columbia), Denis Auroux (Berkeley), Paul Biran (ETH Zurich), Vincent Colin (Nantes), Tobias Ekholm (Uppsala), Yasha Eliashberg (Stanford), Kenji Fukaya (Simons Center), Helmut Hofer (IAS), Michael Hutchings (Berkeley), Dusa McDuff (Columbia), Mark McLean (Stony Brook), Emmy Murphy (MIT), Tim Perutz (UT Austin), Leonid Polterovich (Tel Aviv), Paul Seidel (MIT), Nick Sheridan (Princeton)

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Venue: Mathematical Institute, University of Oxford

Functional Transcendence around Ax-Schanuel

Schanuel’s conjecture governs the transcendence properties of the exponential function.  In a differential field it is a theorem of Ax (1971).  Natural analogues for the uniformising maps of Shimura varieties, and related formulations, are of interest in Diophantine geometry, differential Galois theory, model theory, and complex geometry.  This workshop will gather participants from all these disciplines to share methods, results, and conjectures around this topic.

Speakers: Daniel Bertrand (Jussieu), Alexandru Buium (New Mexico), Philipp Habegger (Darmstadt), Jonathan Kirby (East Anglia), Bruno Klingler (Jussieu), Piotr Kowalski (Wroclaw), Angus Macintyre (Queen Mary London), David Masser (Basel), Ngaiming Mok (Hong Kong), Anand Pillay (Notre Dame), Michael Singer (North Carolina State), Emmanuel Ullmo (Paris Sud), Sai Kee Yeung (Purdue), Umberto Zannier (Pisa), Boris Zilber (Oxford)

Partially supported by EPSRC grant EP/J019232/1.

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Venue: Mathematical Institute, University of Oxford

Advances in Probability: Integrability, Universality and Beyond

Since its discovery over two hundred years ago, the Gaussian distribution has come to represent one of mathematics greatest societal contributions – a robust theory explaining and analyzing much of the randomness inherent in the physical world. However, not all systems are well described by Gaussian theory. For example, classical extreme value statistics or Poisson statistics better capture the randomness and severity of events ranging from natural disasters to emergency room visits. 

Recently, significant research efforts have been focused on understanding systems which are not well described in terms of any of the classically developed statistical universality classes. The failure of these systems to conform with classical descriptions is generally due to the non-linear relationship between natural observables and underlying sources of random inputs and noise. The integrability (or exact solvability) of a few key models for these systems allows for detailed descriptions of new scaling limits and statistics, while universality results prove that these limiting behaviors are robust with respect to changing some of the underlying details of the models. 

This workshop will bring together experts at the forefront of recent advances in the probabilistic study of complex random systems and aims to probe the interplay between newly developed methods of integrability and universality in relation to these systems.

The workshop will focus on the following topics:

  • Random interface growth, particle systems, stochastic PDEs, and the Kardar-Parisi-Zhang equation and universality class
  • Random matrix theory
  • Two-dimensional equilibrium statistical mechanics such as the Ising model, percolation, quantum Liouville gravity, and its relation to the Schramm Loewner evolution
  • Logarithmically correlated processes such as Gaussian free field, and branching diffusion processes

Speakers:  Gerard Ben Arous (NYU), Nathanael Berestycki (Cambridge), Alexei Borodin (MIT), Amir Dembo (Stanford), Christina Goldschmidt (Oxford), Geoffrey Grimmett (Cambridge), Alice Guionnet (MIT), Grégory Miermont (ENS Lyon), Jason Miller (MIT), Ashkan Nikeghbali (Zurich), Neil O’Connell (Warwick), Yuval Peres (Microsoft), Jeremy Quastel (Toronto), Fabio Toninelli (Lyon), Craig Tracy (UC Davis), Vincent Vargas (Paris), Ofer Zeitouni (Weizmann, NYU)

Image: Jason Miller

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Venue: Mathematical Institute, University of Oxford

Motives and Automorphic Forms

The inter-related study of motives and automorphic forms comprises some of the most central ideas and problems in number theory of our times.  The arithmetic geometry of Diophantine equations eventually leads, following the well-known philosophy of Grothendieck, to the investigation of their constituent motives which, in turn, should be built from automorphic forms via the Langlands programme.

This workshop will attempt a survey of the latest developmetns in this research programme, especially the interplay between the influence of Archimedean and non-Archimedean geometry.

Invited participants: Massimo Bertolini (Duisburg-Essen), George Boxer (Harvard), Ana Caraiani (Princeton), Laurent Fargues (Jussieu), Michael Harris (Jussieu), Mark Kisin (Harvard), George Pappas (Michigan State), Vytautas Paskunas (Duisburg), Vincent Pilloni (Lyon), Dinakar Ramakrishnan (Caltech), Sug Woo Shin (Berkeley), Jack Thorne (Cambridge), Bertrand Toën (Toulouse), Eric Urban (Columbia), Eva Viehmann (Munich), Jared Weinstein (Boston), Andrew Wiles (Oxford), Sarah Zerbes (UCL)

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Venue: Mathematical Institute, University of Oxford

Geometry and Dynamics on Moduli Spaces

The moduli space of curves is important in both algebraic geometry and geometric topology.  It also supports some interesting dynamical systems.  The dynamics of the SL(2,ℝ)-action and of the Teichmüller geodesic flow on the moduli space of curves, and the asymptotic monodromy of the Hodge bundle along this flow have numerous applications to the dynamics and geometry of measured foliations, to the dynamics of billiards in polygons and of interval exchange transformations, and to the geometry of flat surfaces.

The dynamics on the moduli space and its geometry interact on many levels, some of which are only now emerging.  This workshop aims to survey some of the recent deveopments in the area.

Speakers: Yves Benoist (Paris Sud), Simion Filip (Chicago), Pascal Hubert (Marseille), Igor Krichever (Columbia), François Labourie (Paris Sud), Erwan Lanneau (Fourier), Elon Lindenstrauss (Hebrew), Howard Masur (Chicago), Carlos Matheus (Paris 13), Maryam Mirzakhani (Stanford), Martin Möller (Goethe), Andrei Okounkov (Columbia), Kasra Rafi (Toronto), Caroline Series (Warwick), Corinna Ulcigrai (Bristol), Amie Wilkinson (Chicago), Scott Wolpert (Maryland), Alex Wright (Stanford), Jean-Christophe Yoccoz (College de France), Peter Zograf (Steklov), Dmitri Zvonkine (Jussieu)

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Venue: Mathematical Institute, University of Oxford

Algebraic Topology: manifolds unlocking higher structures

From one point of view algebraic topology is the study of higher structures, which are ubiquitous in mathematics and nature.  Much development of effective machinery in algebraic topology has been motivated by the study of manifolds.  While these powerful tools continue to solve deep problems in manifold theory, such as Mumford conjecture and the Kervaire invariant problem, the tables have also been turned around with manifolds, via the cobordism hypothesis, defining the higher structures of interest and entirely new fields have emerged exploiting these higher structures in topology, geometry and algebra.

This workshop will expose and explore recent developments and advances in the study of topological moduli spaces, topological field theories, infinity categories, derived algebraic geometry and related topics.

Invited speakers: Clark Barwick (MIT), Julie Bergner (UC Riverside), Andrew Blumberg (Austin), John Francis (Northwestern), Dan Freed (Austin), Soren Galatius (Stanford), Kathryn Hess (EPFL), Lars Hesselholt (Nagoya, Copenhagen), Nick Kuhn (Virginia), Wolfgang Lueck (Bonn), Ib Madsen (Copenhagen), Oscar Randal-Williams (Cambridge), Marco Schlichting (Warwick), Chris Schommer-Pries (MPIM), Neil Strickland (Sheffield), Peter Teichner (MPIM, Berkeley), Constantin Teleman (Berkeley), Nathalie Wahl (Copenhagen)

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Venue: Mathematical Institute, University of Oxford

Algebraic Geometry: Old and New

The workshop focuses on related old topics in algebraic geometry that are transformed by a recent new point of view, and new topics that have applications to old problems. Some of the themes that will feature are: Fano manifolds, special metrics and moduli; stable objects in derived categories; applications of mirror symmetry.

The workshop will pause on Wednesday, 28 September for the Clay Research Conference, at which János Kollár will give a plenary talk on Celestial surfaces and quadratic forms.

Speakers: Christian Boehning (Warwick), Tom Coates (Imperial), Kento Fujita (Kyoto), Mark Gross (Cambridge), Dominic Joyce (Oxford), Anne-Sophie Kaloghiros (Brunel), Conan Leung (CUHK), Yuchen Liu (Princeton), Eduard Looijenga (Utrecht and Tsinghua), Emanuele Macri (Northeastern), Yuji Odaka (Kyoto), Bernd Siebert (Hamburg), Cristiano Spotti (Aarhus), Jacopo Stoppa (SISSA Trieste), Song Sun (Stony Brook), Alessandro Verra (Roma 3), Xiaowei Wang (Rutgers), Jaroslaw Wisniewski (Warsaw) 

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Venue: Mathematical Institute, University of Oxford

Geometric Representation Theory and Beyond

There are powerful new tools and ideas at the forefront of geometric representation theory, particularly categorical incarnations of ideas from mathematical physics, algebraic, arithmetic and symplectic geometry, and topology: perhaps geometry is no longer the future of geometric representation theory. Speakers at this workshop have been invited to present their own vision for the future of the field.

Speakers: David Ben-Zvi (Austin), Alexander Braverman (Toronto and Perimeter Institute for Theoretical Physics), Laurent Fargues (Jussieu), Dennis Gaitsgory (Harvard), Victor Ginzburg (Chicago), Michael Harris (Columbia), Tamas Hausel (IST Austria), David Nadler (Berkeley), Andrei Okounkov (Columbia), Vera Serganova (Berkeley), Wolfgang Soergel (Freiburg), Catharina Stroppel (Bonn), Constantin Teleman (Oxford), Geordie Williamson (Bonn)

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Venue: Mathematical Institute, University of Oxford

Mean Curvature Flow

Mean curvature flow (MCF) is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time.  It has been studied in material science to model things such as cell, grain, and bubble growth.  It also plays an important role in image processing and has been widely studied numerically.   The flow is well-defined classically until the first singularity.  The level set method was invented to study the flow, even past singularities, and it is now widely used in a variety of settings.

Before the flow becomes extinct, changes occur as it goes through singularities.   To understand the flow, it is crucial to understand these.  This can be done completely for embedded curves, but there are infinitely many possible types of singularities in all higher dimensions.  Recent years have seen major progress on some of the most natural questions:  Is it possible to bring the hypersurface into general position so that there is a smaller set of “generic” singularities?  What does the flow look like near these singularities?  What is the structure of the singular set itself? 

This workshop will explore recent developments in MCF, focusing on which singularities can occur, which of these are generic, what the flow looks like near these singularities, and the structure of the singular set.

Speakers: Sigurd Angenent (Wisconsin), Jacob Bernstein (Johns Hopkins), Yoshikazu Giga (Tokyo), Gerhard Huisken (Tübingen), Bruce Kleiner (NYU), Bob Kohn (NYU), Felix Otto (MPIM Leipzig), Felix Schulze (UCL), James Sethian (Berkeley), Peter Topping (Warwick), Lu Wang (Wisconsin), Mu-Tao Wang (Columbia), Brian White (Stanford)

Some of the topics that will be covered in the workshop are described in the paper Level set flow  by Colding and Minicozzi.

Image:  R. I. Saye and J. A. Sethian, Lawrence Berkeley National Laboratory and University of California, Berkeley

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Venue: Mathematical Institute, University of Oxford

Recent Developments on Elliptic Curves

The last few years have witnessed a number of developments in the arithmetic of elliptic curves, notably the proof that there are positive proportions of elliptic curves of rank zero and rank one for which the Birch—Swinnerton-Dyer conjecture is true.  The proof of this landmark result relies on an appealing mix of diverse techniques arising from the newly resurgent field of arithmetic invariant theory, Iwasara theory, congruences between modular forms, and the theory of Heegner points and related Euler systems.  The purpose of this workshop is to survey the proof of this theorem and to describe the new perspectives on the Birch—Swinnterton-Dyer conjecture which it opens up.

Speakers: Mirela Ciperiani (Texas, Austin), Ellen Eischen (Oregon), Benedict Gross (Harvard), Wei Ho (Michigan), Antonio Lei (Laval), Chao Li (Columbia), Kartik Prasanna (Michigan), Victor Rotger (Catalunya), Arul Shankar (Harvard), Ye Tian ( Chinese Academy of Sciences), Eric Urban (Columbia), Rodolfo Venerucci (Duisberg-Essen), Xin Wan (Columbia), Xiaoheng Jerry Wang (Princeton), Andrew Wiles (Oxford), Shou-Wu Zhang (Princeton)

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Venue: Mathematical Institute, University of Oxford

Modern Moduli Theory

The workshop will focus on modern approaches to moduli problems in algebraic geometry, including derived structures on moduli spaces such as shifted symplectic and shifted Poisson structures, novel quotient constructions, and relationships to geometric representation theory.

Speakers: Dario Beraldo (Oxford), Chris Brav (HSE, Moscow), Yalong Cao (Oxford), Ben Davison (IST Vienna), Chris Dodd (Illinois), Barbara Fantechi (SISSA Trieste), Elham Izadi (San Diego), Frances Kirwan (Oxford), Kobi Kremnitzer (Oxford), Sven Meinhardt (Sheffield), Tom Nevins (Illinois), Georg Oberdieck (MIT), Andrei Okounkov (Columbia), Tony Pantev (Penn), Jørgen Rennemo (Oslo and Oxford), Nick Rozenblyum (Chicago), Yukinobu Toda (IPMU, Tokyo), Bertrand Toën (Toulouse)

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Venue: Mathematical Institute, University of Oxford