Since 2014 I have been organising a math-physics colloquium twice per year. Here is an archive of the current and past events. Math/Physics Colloquium 7 June, 2019 Knots, q-series, and Modularity Don Zagier (MPI, Bonn): Schwarzschild meets Ramanujan: From quantum black holes to mock modular forms Boris Pioline (UPMC, Paris):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, 2018Jesper Jacobsen Four-point functions in the Fortuin-Kasteleyn cluster model(ENS Paris):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 Counting self-avoiding paths using discrete holomorphic functions(IHES):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) Small quantum computers and big classical dataAram Harrow (MIT): Compressed sensing, phase retrieval, and the Clifford group David Gross (U. of Cologne):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:: Quantum number theory? Some speculative remarksProf. Minhyong Kim (U. Oxford) 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. : Modularity and 3-manifolds Prof. Sergei Gukov (Caltech) 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:: Modular double of the SL_q(2,R) and its applicationsProf. L. Faddeev (St Petersburg) 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. |

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