Wed, 29 Jan 2025
15:00
L3

Emergent Phenomena in Critical Models of Statistical Physics: Exploring 2D Percolation

Prof Hugo Duminil-Copin
(IHES)
Abstract

For over 150 years, the study of phase transitions—such as water freezing into ice or magnets losing their magnetism—has been a cornerstone of statistical physics. In this talk, we explore the critical behavior of two-dimensional percolation models, which use random graphs to model the behavior of porous media. At the critical point, remarkable symmetries and emergent properties arise, providing precise insights into the nature of these systems and enriching our understanding of phase transitions. The presentation is designed to be accessible and does not assume any prior background in percolation theory.

 

About the Speaker

Hugo Duminil-Copin is is a French mathematician recognised for his groundbreaking work in probability theory and mathematical physics. He was appointed full professor at the University of Geneva in 2014 and since 2016 has also been a permanent professor at the Institut des Hautes Études Scientifiques (IHES) in France. In 2022 he was awarded the Fields Medal, the highest distinction in mathematics.

Wed, 29 Jan 2025
17:00
Lecture Theatre 1, Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG

Can we truly understand by counting? - Hugo Duminil-Copin

Hugo Duminil-Copin
(IHES)
Further Information

Hugo will illustrate how counting can shed light on the behaviour of complex physical systems, while simultaneously revealing the need to sometimes go beyond what numbers tell us in order to unveil all the mysteries of the world around us.

Hugo Duminil-Copin is is a French mathematician recognised for his groundbreaking work in probability theory and mathematical physics. He was appointed full professor at the University of Geneva in 2014 and since 2016 has also been a permanent professor at the Institut des Hautes Études Scientifiques (IHES) in France. In 2022 he was awarded the Fields Medal, the highest distinction in mathematics. 

Please email @email to register to attend in person.

The lecture will be broadcast on the Oxford Mathematics YouTube Channel on Thursday 20 February at 5-6pm and any time after (no need to register for the online version).

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Tue, 04 Feb 2025
10:00
L4

Twisting Higgs modules and applications to the p-adic Simpson correspondence I (special time!)

Ahmed Abbes
(IHES)
Abstract

In 2005, Faltings initiated a p-adic analogue of the complex Simpson correspondence, a theory that has since been explored by various authors through different approaches. In this two-lecture series (part I in the Algebra Seminar and part II in the Arithmetic Geometry Seminar), I will present a joint work in progress with Michel Gros and Takeshi Tsuji, motivated by the goal of comparing the parallel approaches we have developed and establishing a robust framework to achieve broader functoriality results for the p-adic Simpson correspondence.

The approach I developed with M. Gros relies on the choice of a first-order deformation and involves a torsor of deformations along with its associated Higgs-Tate algebra, ultimately leading to Higgs bundles. In contrast, T. Tsuji's approach is intrinsic, relying on Higgs envelopes and producing Higgs crystals. The evaluations of a Higgs crystal on different deformations differ by a twist involving a line bundle on the spectral variety.  A similar and essentially equivalent twisting phenomenon occurs in the first approach when considering the functoriality of the p-adic Simpson correspondence by pullback by a morphism that may not lift to the chosen deformations.
We introduce a novel approach to twisting Higgs modules using Higgs-Tate algebras, similar to the first approach of the p-adic Simpson correspondence. In fact, the latter can itself be reformulated as a twist. Our theory provides new twisted higher direct images of Higgs modules, that we apply to study the functoriality of the p-adic Simpson correspondence by higher direct images with respect to a proper morphism that may not lift to the chosen deformations. Along the way, we clarify the relation between our twisting and another twisting construction using line bundles on the spectral variety that appeared recently in other works.

Wed, 05 Feb 2025
16:00
Lecture Room 4

Twisting Higgs modules and applications to the p-adic Simpson correspondence II

Ahmed Abbes
(IHES)
Abstract

[This is the second in a series of two talks; the first talk will be in the Algebra Seminar of Tuesday Feb 4th https://www.maths.ox.ac.uk/node/70022]

In 2005, Faltings initiated a p-adic analogue of the complex Simpson correspondence, a theory that has since been explored by various authors through different approaches. In this two-lecture series (part I in the Algebra Seminar and part II in the Arithmetic Geometry Seminar), I will present a joint work in progress with Michel Gros and Takeshi Tsuji, motivated by the goal of comparing the parallel approaches we have developed and establishing a robust framework to achieve broader functoriality results for the p-adic Simpson correspondence.

The approach I developed with M. Gros relies on the choice of a first-order deformation and involves a torsor of deformations along with its associated Higgs-Tate algebra, ultimately leading to Higgs bundles. In contrast, T. Tsuji's approach is intrinsic, relying on Higgs envelopes and producing Higgs crystals. The evaluations of a Higgs crystal on different deformations differ by a twist involving a line bundle on the spectral variety.  A similar and essentially equivalent twisting phenomenon occurs in the first approach when considering the functoriality of the p-adic Simpson correspondence by pullback by a morphism that may not lift to the chosen deformations.
We introduce a novel approach to twisting Higgs modules using Higgs-Tate algebras, similar to the first approach of the p-adic Simpson correspondence. In fact, the latter can itself be reformulated as a twist. Our theory provides new twisted higher direct images of Higgs modules, that we apply to study the functoriality of the p-adic Simpson correspondence by higher direct images with respect to a proper morphism that may not lift to the chosen deformations. Along the way, we clarify the relation between our twisting and another twisting construction using line bundles on the spectral variety that appeared recently in other works.

Tue, 19 Nov 2024
13:00
L2

Symmetry topological field theory and generalised Kramers–Wannier dualities

Clement Delcamp
(IHES)
Abstract

A modern perspective on symmetry in quantum theories identifies the topological invariance of a symmetry operator within correlation functions as its defining property. Within this paradigm, a framework has emerged enabling a calculus of topological defects in terms of a higher-dimensional topological quantum field theory. In this seminar, I will discuss aspects of this construction for Euclidean lattice field theories. Exploiting this framework, I will present generalisations of the celebrated Kramers-Wannier duality of the Ising model, as combinations of gauging procedures and generalised Fourier transforms of the local weights encoding the dynamics. If time permits, I will discuss implications of this framework for the real-space renormalisation group flow of these theories.

Mon, 04 Nov 2024
15:30
L5

Zariski closures of linear reflection groups

Sami Douba
(IHES)
Abstract

We show that linear reflection groups in the sense of Vinberg are often Zariski dense in PGL(n). Among the applications are examples of low-dimensional closed hyperbolic manifolds whose fundamental groups virtually embed as Zariski-dense subgroups of SL(n,Z), as well as some one-ended Zariski-dense subgroups of SL(n,Z) that are finitely generated but infinitely presented, for all sufficiently large n. This is joint work with Jacques Audibert, Gye-Seon Lee, and Ludovic Marquis.

Mon, 17 Jun 2024
15:30
L3

The Brownian loop measure on Riemann surfaces and applications to length spectra

Professor Yilin Wang
(IHES)
Abstract
Lawler and Werner introduced the Brownian loop measure on the Riemann sphere in studying Schramm-Loewner evolution. It is a sigma-finite measure on Brownian-type loops, which satisfies conformal invariance and restriction property. We study its generalization on a Riemannian surface $(X,g)$. In particular, we express its total mass in every free homotopy class of closed loops on $X$ as a simple function of the length of the geodesic in the homotopy class for the constant curvature metric conformal to $g$. This identity provides a new tool for studying Riemann surfaces' length spectrum. One of the applications is a surprising identity between the length spectra of a compact surface and that of the same surface with an arbitrary number of cusps. This is a joint work with Yuhao Xue (IHES). 


 

Thu, 23 May 2024
16:00
L5

Square roots for symplectic L-functions and Reidemeister torsion

Amina Abdurrahman
(IHES)
Abstract

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

Tue, 02 Mar 2021
12:00
Virtual

Some mathematical problems posed by the conformal bootstrap program

Slava Rychkov
(IHES)
Abstract

The conformal bootstrap program for CFTs in d>2 dimensions is
based on well-defined rules and in principle it could be easily included
into rigorous mathematical physics. I will explain some interesting
conjectures which emerged from the program, but which so far lack rigorous
proof. No prior knowledge of CFTs or conformal bootstrap will be assumed.

Thu, 18 May 2017
17:30
L6

Theories of presheaf type as a basic setting for topos-theoretic model theory

Olivia Caramello
(IHES)
Abstract

I will review the notion of classifying topos of a first-order (geometric) theory and explain the central role enjoyed by theories of presheaf type (i.e. classified by a presheaf topos) in the context of the topos-theoretic investigation of the model theory of geometric theories. After presenting a few main results and characterizations for theories of presheaf type, I will illustrate the generality of the point of view provided by this class of theories by discussing a topos-theoretic framework unifying and generalizing Fraissé’s construction in model theory and topological Galois theory and leading to an approach to the problem of the independence from l of l-adic cohomology.

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