Since 2014 I have been organising a math-physics colloquium twice per year. Here is an archive of the current and past events.
Symposium Day 08 November, 2019
(special edition: learning/math/physics symposium, co-organised with Prof. Max Welling and his team members Dr. Roberto Bondesan, Dr. Daniel Worall, and Pim de Haan)
Danilo Rezende (DeepMind) : Generative Models and Symmetries
Abstract: In this talk I will discuss how some ideas from Physics such as phase-transitions and Gauge symmetries provide powerful tools to analyse and build generative models. In particular, the study of symmetries in Physics has revolutionised our understanding of the world and permeates every fundamental physical model. Inspired by this, I will focus on our recent work of incorporating Gauge symmetries into normalizing flows generative models and its potential applications in the sciences and ML.
Clément Hongler (EPFL Lausanne): Neural Tangent Kernel and Applications
Abstract: In this talk I will discuss how some ideas from Physics such as phase-transitions and Gauge symmetries provide powerful tools to analyse and build generative models. In particular, the study of symmetries in Physics has revolutionised our understanding of the world and permeates every fundamental physical model. Inspired by this, I will focus on our recent work of incorporating Gauge symmetries into normalizing flows generative models and its potential applications in the sciences and ML.
Stéphane Mallat (Collège de France, ENS Paris): Multiscale Interactions in Physics and Machine Learning with Deep Nets
Abstract: Is there a generic notion of regularity allowing to generate low-dimensional models for high-dimensional problems in physics and machine learning ? There is no mathematical answer to this outstanding question but the success of deep neural networks architectures indicates that such a regularity exists. This talk outlines the role of scale separation, symmetries and sparsity, to define models implemented by convolutional neural networks. I will discuss open problems to regress quantum molecular energies and to define statistical physics models of turbulence, face to face with image classification and generation problems, which appear to have similar regularities.
Math/Physics Colloquium 7 June, 2019
Don Zagier (MPI, Bonn): Knots, q-series, and Modularity
Boris Pioline (UPMC, Paris): Schwarzschild meets Ramanujan: From quantum black holes to mock modular forms
While black holes are by now firmly established as astrophysical objects, they continue to raise conceptual puzzles for theorists -- one of them being the microscopic origin of the Bekenstein-Hawking entropy. An important breakthrough was achieved almost 25 years ago for supersymmetric black holes in string theory, whose micro-states can be described by D-brane bound states and
counted (with signs) exactly. In the simplest case with maximal supersymmetry, the generating functions of exact degeneracies turns out to be a weak modular form, familiar from 19th century mathematics. In cases with reduced supersymmetry, the counting must take into account wall-crossing phenomena, whereby black hole bound states may form or decay depending on the
values of the scalar fields at spatial infinity. Recent work has shown that generating functions of exact degeneracies of single-centered black holes are no longer exactly modular. Rather, they belong to the class of `mock modular forms', which generalizes the 'mock theta functions' introduced by Ramanujan just before his premature death in 1920. I will attempt to review these developments for non-experts, and convey some of the excitement which has happened over the last few years at this juncture between physics and mathematics.
Math/Physics Colloquium 22 May, 2018
Jesper Jacobsen (ENS Paris): Four-point functions in the Fortuin-Kasteleyn cluster model
Abstract: The determination of four-point correlation functions of two-dimensional lattice models is of fundamental importance in statistical physics. In the limit of an infinite lattice, this question can be formulated in terms of conformal field theory (CFT). For the so-called minimal models the problem was solved more than 30 years ago, by using that the existence of singular states implies that the correlation functions must satisfy certain differential equations. This settles the issue for models defined in terms of local degrees of freedom, such as the Ising and 3-state Potts models. However, for geometrical observables in the Fortuin-Kasteleyn cluster formulation of the Q-state Potts model, for generic values of Q, there is in general no locality and no singular states, and so the question remains open. As a warm-up to solving this problem, we discuss which states propagate in the s-channel of such correlation functions, when the four points are brought together two by two. To this end we combine CFT methods with algebraic and numerical approaches to the lattice model.
Hugo Duminil (IHES): Counting self-avoiding paths using discrete holomorphic functions
Abstract: In the early eighties, physicists Belavin, Polyakov and Zamolodchikov postulated conformal invariance of critical planar statistical models. This prediction enabled physicists to harness Conformal Field Theory in order to formulate many conjectures on these models. From a mathematical perspective, proving rigorously the conformal invariance of a model (and properties following from it) constitutes a formidable challenge. In recent years, the connection between discrete holomorphicity and planar statistical physics led to spectacular progress in this direction. Kenyon, Chelkak and Smirnov exhibited discrete holomorphic observables in the dimer and Ising models and proved their convergence to conformal maps in the scaling limit. These results paved the way for the rigorous proof of conformal invariance for these two models.
Other discrete observables have been proposed for a number of critical models, including self-avoiding walks and Potts models. While these observables are not exactly discrete holomorphic, their discrete contour integrals vanish, a property shared by discrete holomorphic functions. This property sheds a new light on the critical models, and we propose to discuss some of its applications. In particular, we will sketch the proof (joint work with Smirnov) of a conjecture made by Nienhuis regarding the number of self-avoiding walks of length n on the hexagonal lattice starting at the origin.
Math/Physics Colloquium 20 Nov, 2017 (co-organised with M. Ozols and M. Walter)
Aram Harrow (MIT): Small quantum computers and big classical data
David Gross (U. of Cologne): Compressed sensing, phase retrieval, and the Clifford group
Abstract: Every time the release button of a digital camera is pressed, several megabytes of raw data are recorded. But the size of a typical jpeg output file is only 10% of that. What a waste! Can't we design a process
which records only the relevant 10% of the data to begin with? The theory of compressed sensing achieves this trick for certain signals. I will give an introduction to this theory and show how repeatedly, there
has been a fruitful interchange of ideas between compressed sensing and quantum physics. I will focus on the most recent example: The Clifford group. In quantum information, it describes a group of unitaries that is large enough to exhibit a rich set of quantum phenomena, while being structured enough to allow for efficient classical simulations. New results on the representation theory of the Clifford group have allowed us show that its orbits - including the important set of stabilizer states - have near-optimal properties both for distinguishing quantum
states and for the compressed-sensing related task of reconstructing signals from phase-insensitive measurements.
Math/Physics Colloquium 24 May, 2017:
Prof. Minhyong Kim (U. Oxford) : Quantum number theory? Some speculative remarks
Abstract: The area of arithmetical algebraic geometry was created to a large extent through the interaction of two ideas: Weil's conjecture that the regular behaviour of zeta functions of varieties over finite fields should have a topological explanation, and Grothendieck's theory of schemes, whereby geometries can be associated to abstract rings, and maps between spaces to solutions of equations. Their combination makes it possible to study, for example, the spectrum of the integers $Spec (\mathbb{Z})$ as a geometric object with considerable complexity. Many analogies with physical theories can be built out of this perspective, out of which we will concentrate mostly on one: the correspondence between the representation theory of Galois groups and gauge theory. We speculate on the possibility of motivating difficult conjectures in number theory by way of quantum field theory.
Prof. Sergei Gukov (Caltech) : Modularity and 3-manifolds
Abstract: In this talk I will describe a new bridge between number theory and low-dimensional topology, motivated by physics (more precisely, by the recent work https://arxiv.org/pdf/1605.07615.pdf). In topology, one way to characterize spaces is by their fundamental group, which unfortunately is too complicated in general and, therefore, one might prefer to consider its representations into various Lie groups. In the case of 2-manifolds, it turns out that such representation spaces for a group G and its Langlands dual provide examples of mirror manifolds. In this talk we explore the question: What is the analogue for 3-manifolds?
Math/Physics Colloquium 06 Dec, 2016:
Prof. Balázs Szendöi (U. Oxford)
Title: "Sheaf-theoretic partition functions: combinatorics, quantisation and representation theory"
Abstract: I will give an overview of some aspects of Donaldson-Thomas theory, the theory associating enumerative invariants to moduli spaces of sheaves on Calabi-Yau threefolds - one possible mathematical approach to the definition of "BPS state counts". I will concentrate on the examples of affine three-space and the resolved conifold geometry - the first one, while degenerate from a physics point of view, is easiest to handle mathematically, while the latter, although more complicated, is still possible to analyse fully. The mathematical tool of localisation will allow us to write our partition functions as interesting combinatorial generating series. One can then "quantise" the theory, and in fact in two different ways: one involves refining (or "quantizing") the invariants, whereas the other involves deforming the underlying geometry in a non-commutative "quantum" direction. If time permits, I will also make a connection to representation theory.
Prof. Marcos Mariño (U. Genève)
Title: Stringy Geometry and Quantum Mechanics
Abstract: In string theory, point particles become extended objects, and our usual notions of geometry have to be generalized to some form of "stringy” geometry. In this talk I will review how stringy geometry has led to new developments in enumerative geometry, like mirror symmetry. This is however only half of the story: stringy geometry should be itself generalized to a full quantum geometry, incorporating the effects of interacting strings. I will summarize some recent developments in which this geometry is encoded in simple quantum mechanical systems, providing in this way a correspondence between enumerative geometry and spectral theory.
Math/Physics Colloquium 02/10/2015:
Terry Gannon (U. of Alberta): Finite conformal field theories for pretty dumb people,
Matthias Gaberdiel (ETH Zürich) :Higher Spins & Strings
Math/Physics Colloquium 09/04/2015:
Dr. T. Dimofte (IAS): A 3d Bridge Between Physics and Mathematics
Prof. L. Faddeev (St Petersburg) : Modular double of the SL_q(2,R) and its applications
Math/Physics Colloquium 20/10/0214:
David Hernandez (Institut de Mathematiques de Jussieu - Paris Rive Gauche, Paris 7)
Title: Baxter's relations and spectra of quantum integrable systems.
Abstract: In his seminar work, Baxter established that the spectrum of
six (and eight)-vertex model can be described in terms of polynomials and
of the famous Baxter's QT-relation. Frenkel-Reshetikhin conjectured that
there is an analog form for the spectrum of more general quantum
integrable systems (more precisely, generalizing the XXZ model, whose
spectrum is the same as that of the six-vertex model).
We will present our recent proof (with E. Frenkel) of this conjecture for
arbitrary untwisted quantum affine algebras. Our approach is based on the
study of prefundamental representations we constructed previously with M.
Jimbo. We establish generalized Baxter's as relations in the Grothendieck
ring of a category O containing the prefundamental representations.
Vasily Pestun (IHES)
Title: Gauge theories and quantum groups
Abstract: Recently new relations have been found that link together the
representation theory of Yangians or quantum affine algebras, in
particular the ring of q-characters, the quantization of complex
completely integrable systems, such as moduli space of periodic
monopoles or Hitchin systems, and the dynamics of equivariantly
deformed d=4 N=2 supersymmetric gauge theories. I will give an
introductory overview of the topic and state the new results.