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The charm of integrability – Honoring the scientific contributions of Alexander Its on the occasion of his 70th birthday

The goal of this conference is to bring together experts and young researchers, mathematicians and physicists, scientists with different backgrounds and different takes on integrable systems theory, to discuss the latest achievements in this dynamic field and to point at future research directions within the discipline. At the same time the meeting will serve to honour Alexander Its, a world renowned expert in the field, for his many ground-breaking contributions to the theory of integrable systems over the past 40 years, on the occasion of his 70th birthday.

The full list of invited speakers can be found below.

View the programme here.

The conference will be zoom live streamed. Please contact maths-conference-administrator@bristol.ac.uk for access details.

Registration

Registration has now closed.

Funding

Limited funds are available for financial support, covering accommodation, and priority will be given to early career researchers, especially to those that will present a poster at the poster session. To be considered for financial support, participants are asked to complete the financial support form after completing their registration. Funding will be allocated on a first-come-first-serve basis. The last date to be considered for funding is August 26th.

Venue and accommodation

The event will take place at the School of Mathematics, Fry Building, Woodland Road, Bristol. BS8 1UG.

Information on accommodation options can be found by visiting the Visit Bristol website.

Invited speakers:

Jinho Baik (University of Michigan)

Title: Multi-point distribution of (periodic) KPZ fixed points and differential equations

Abstract: The KPZ fixed point and its periodic version are two-dimensional random fields that are expected to be the universal limit of many random growth models and interacting particle systems. Their multi-point distributions were evaluated recently. We show that the involved Fredholm determinants can be expressed in terms of Its-Izergin-Korepin-Slavnov integrable operators. We then discuss the matrix integrable differential equations that these operators satisfy. These equations include matrix NLS with complex time, matrix mKdV, matrix KP, and multi-component KP hierarchy.

Estelle Basor (American Institute of Mathematics)

Title: Asymptotics of determinants of block Toeplitz matrices with symbols having jump discontinuities

Abstract: Well known classical theorems describe the asymptotics of finite Toeplitz matrices with scalar symbols of Fisher-Hartwig type. This talk will extend the classical results to the block case for symbols with jump discontinuities using an operator theory approach.

Marco Bertola (Concordia University)

Title: Dim retrograde solitons and degenerate Riemann surfaces.

I will report on recent work with Alexander Tovbis and Bob Jenkins on how to compute effective formulas for the partial degeneration of Theta functions on nodal surfaces. As an application we provide the study of solitons on stationary cnoidal background of the KdV equation. Some interesting physical phenomena include the fact that the “solitons” are now more like wave-packets with distinct group and phase velocities. The group velocity may be positive (as usual) or negative (retrograde solitons). Moreover the explicit formula allows for the study of the soliton-on-soliton scattering matrix.

Pavel Bleher (IUPUI School of Science)

Title: Ensembles of Random Matrices with Complex Potentials:Phase Diagrams and Topological Expansion

Abstract: We will discuss recent rigorous results on ensembles of random matrices withcomplex potentials, including topological expansion and phase diagrams in these ensembles inthe complex phase space of parameters. This is an ongoing project with Ahmad Barhoumi,Marco Bertola, Alfredo Dea ̃no, Roozbeh Gharakhloo, Ken McLaughlin, Alex Tovbis, andMaxim Yattselev.

Anne Boutet de Monvel (Université de Paris)

Title: TBA

Alexander Bobenko (Technische Universität Berlin)

Title: Integrability for geometry: Is a surface characterized by its metric and curvatures?

Abstract: We consider a classical problem in differential geometry, known as the Bonnet problem, whether a surface in three space is characterized by its metric and mean curvature function. Generically, the answer is yes. Special cases when it is not the case are classified. In the first part we consider Bonnet surfaces, which are surfaces (with non-constant mean curvature) possessing continuous families of isometries preserving mean curvature. Their global classification is given using the theory of Painleve equations. In the second part, which is a recent joint work with Tim Hoffmann and Andrew Sageman-Furnas, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. Integrable systems play a crucial role in this construction. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface.

Mattia Cafasso (Université Angers)

Title:The finite temperature discrete Bessel kernel and integrable equations

Abstract: Using the theory of discrete integrable operators and discrete Riemann-Hilbert problems, I will show that the largest particle distribution of the point process associated to the finite-temperature discrete Bessel kernel satisfies some interesting integrable equations such as, for example, the cylindrical Toda equation. This is a joint work with Giulio Ruzza.

Tom Claeys (UCLouvain)

Title:Weak and strong confinement in the Freud random matrix ensemble

Abstract: Eigenvalues of unitary invariant random matrices confined by a Freud weight $|x|^\beta$ exhibit a transition between classical random matrix statistics and Poisson statistics as $\beta$ decreases. We describe the gap probabilities in this ensemble as a function of $\beta$. The talk will be based on joint work with I. Krasovsky and A. Minakov.

Peter Clarkson (University of Kent)

Title: Orthogonal Polynomials and Symmetric Sextic Freud weights

Abstract: In this talk I will discuss orthogonal polynomials associated with symmetric sextic Freud weights. In particular I will describe properties of the recurrence coefficients in the three-term recurrence relation associated with these orthogonal polynomials. These recurrence coefficients satisfy a fourth-order discrete equation which is the second member of the first discrete Painleve hierarchy, also known as the string equation, and also satisfies a coupled system of second-order, nonlinear differential equations. The weight arises in the context of Hermitian matrix models and random matrices.This is joint work with Kerstin Jordaan (University of South Africa) and Ana Loureiro (University of Kent).

Tamara Grava (University of Bristol)

Title: Gibbs ensemble for Integrable Systems and random matrices

We consider discrete integrable systems with random initial data and connect them with the theory of random matrices. We will consider the Toda lattice whose Gibbs ensemble has been connected by H. Spohn to the Gaussian beta-ensemble at high temperature. We then consider the defocusing nonlinear Schrodinger equation in its integrable version, that is called Ablowitz Ladik lattice. In the random initial data setting the Lax matrix of the Ablowitz Ladik lattice turns into a random matrix that is related to the circular beta-ensemble at high temperature. We obtain the density of states of the random Lax matrix, when the size of the matrix goes to infinity, by establishing a mapping to the one-dimensional log-gas. The density of states is obtained via a particular solution of the double-confluent Heun equation.

Jon Keating (University of Oxford)

Title: Multifractal eigenfunctions in intermediate systems

Abstract: I will discuss two systems that are intermediate between integrability and chaos, and for which the quantum eigenfunctions exhibit multifractal scaling. This is joint work with Henrik Ueberschär.

Vladimir Korepin (Stony Brook University)

Title: Lattice Nonlinear Schrodinger Equation

Abstract: History, applications and open problems will be considered. We shall mainly pay attention to the quantum case.

Igor Krasovsky (Imperial College London)

Title: Asymptotics of the sine- and Airy-kernel determinants on two large intervals

Abstract: We consider the probability of 2 large gaps without eigenvalues in the local scaling limits in the bulk and at the edge of the spectrum of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constants in the relevant asymptotics. In the bulk, the asymptotics without determination of the constant were found in 1997 by Deift, Its, and Zhou (in the general case of n gaps).

To obtain our results, we used a differential identity with respect to the gap edges and the observation that when the gaps are far apart, they are asymptotically independent, and so we can make use of the known constant for the one-gap asymptotics. The integration of the differential identity was the most challenging task due to theta-functions involved in the formulae. This is a joint work with Benjamin Fahs (sine-kernel) and Theo-Harris Maroudas (Airy-kernel).

Arno Kuijlaars (KU Leuven)

Title: Critical measures on higher genus Riemann surface

Abstract:I will report on recent work with Marco Bertola and Alan Groot on the definition and properties of critical measures on compact Riemann surfaces.

Our work is inspired by random tilings with periodic weights that can be analyzed via matrix valued orthogonal polynomials. The matrix valued orthogonality can be viewed as scalar orthogonality on a Riemann surface, and the critical measures provide a step towards an understanding of their asymptotic behavior.

Oleg Lisovyy (Université de Tours)

Title: Perturbative connection formulas for Heun equations

Abstract: Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed in terms of the large order asymptotics of the corresponding power series. We demonstrate that for the usual, confluent and reduced confluent Heun equation, the series expansion of the relevant asymptotic amplitude in a suitable parameter can be systematically computed to arbitrary order. This allows to check a recent conjecture of Bonelli-Iossa-Panea Lichtig-Tanzini expressing the Heun connection matrix in terms of quasiclassical Virasoro conformal blocks.

Peter Miller (University of Michigan)

Title: On the algebraic solutions of the Painlev\’e-III (D$_7$) equation

Abstract: The D$_7$ degeneration of the Painlev\’e-III equation has solutions that are rational functions of $x^{1/3}$ for certain parameter values. We apply the isomonodromy method to obtain a Riemann-Hilbert representation of these solutions. We demonstrate the utility of this representation by analyzing rigorously the behavior of the solutions in the large parameter limit. This is joint work with Robert Buckingham (Cincinnati). If time permits, we will describe an interesting related calculation that we learned about 10 years ago from Alexander Its.

Beatrice Pelloni (Heriot Watt University)

Title: Novelty and surprises in the theory of third-order boundary value problems.

Abstract: I will review the results for differential operators an integrable PDEs of odd-order, when posed on bounded domains. The original motivation for this work was the desire to understand the solution of a famous integrable PDE, the Kortweg-deVries equation. The doors to these results have been unlocked by the understanding of the behaviour of linear third-order boundary values problems, that have been studied over the last 20 years by means of the Unified Transform. In some non-self-adjoint cases, this approach yields a spectral diagonalisation of the operator. More generally, I will highlight the dependence of these problems on the specific boundary conditions and how this differs fundamentally from the even-order case. Novel and surprising examples arise for “Dirichlet-type” boundary conditions, as well as for quasi-periodic and time-periodic ones. I will also comment on some results for the nonlinear KdV case.

Nicolai Reshetikhin (University of California, Berkeley)

Title: Superintegrable systems on moduli spaces of flat connections.

Abstract: After a short review of superintegrability I will focus on superintegrable systems on moduli spaces of flat connections and on quantization of these systems. Quantum Hamiltonians of such systems have simple meaning in terms of corresponding topological quantum field theory.

Nina Snaith (University of Bristol)

Title: A stochastic model for the spacing of replication origins on chromosomes

Abstract: In joint work with Huw Day we probe the statistical distribution of origins of DNA replication on eukaryotic chromosomes and develop a stochastic model for the replication process.

Leon Takhtajan (Stony Brook University)

Title: New supersymmetric localization principle

Abstract: I will present a new supersymmetric localization principle, with application to trace formulas for a full partition function. Unlike the standard localization principle, this new principle allowsto compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. This is a joint work with Changha Choi, arXiv:2112.07942.

$Self-similar fractal structures in tri-diagonal random matrices.$

Self-similar fractal structure

$Self-similar fractal structures in tri-diagonal random matrices.$

Images of fractal structures courtesy of Henry Taylor

Contacts

Please contact Louisa Bartoszewicz regarding any administrative aspects of the conference.

Organising committee:

Thomas Bothner (University of Bristol)

Tamara Grava (University of Bristol; International School for Advanced Studies (SISSA/IAS)

Ken McLaughlin (Colorado State University)

Andrei Prokhorov (University of Michigan)

BJB90 – Bryan Birch Celebratory Conference

UPDATE: BJB90 has been rescheduled to 22nd April 2022. All original registrants will be transferred to the new date.

A one day conference to celebrate the 90th birthday of Bryan Birch and his many lasting contributions to number theory.

Visit the event website for further information.

Biography

Bryan Birch was educated at Trinity College, Cambridge where as a doctoral student he proved Birch’s theorem, one of the results to come out of the Hardy–Littlewood circle method; it shows that odd-degree rational forms in a large enough set of variables must have zeroes.

He then worked with Peter Swinnerton-Dyer on computations relating to the Hasse–Weil L-functions of elliptic curves. They formulated their conjecture relating the rank of an elliptic curve to the order of a certain zero of an L-function; it has been an influence on the development of number theory since the mid 1960s. They later introduced modular symbols.

In later work he contributed to algebraic K-theory (Birch–Tate conjecture). He then formulated ideas on the role of Heegner points (he had been one of those reconsidering Kurt Heegner’s original work, on the class number one problem, which had not initially gained acceptance). Birch put together the context in which the Gross–Zagier theorem was proved. He was elected a Fellow of the Royal Society in 1972; was awarded the Senior Whitehead Prize in 1993 and the De Morgan Medal in 2007. In 2012 he became a fellow of the American Mathematical Society. In 2020 he was awarded the Royal Society’s Sylvester Medal for his work in driving the theory of elliptic curves through the Birch–Swinnerton-Dyer conjecture and the theory of Heegner points. The Birch–Swinnerton-Dyer conjecture is one of the Clay Mathematics Institute Millennium Problems.

Registration is now open.

If you have any questions, please contact heilbronn-coordinator@bristol.ac.uk.

Join the Heilbronn Event mailing list to keep up to date with our upcoming events.

Colloquium – Imre Leader

Monday 13th December 2021: 15.00 – 16.00 We are delighted to welcome Imre Leader, Professor of Pure Mathematics, University of Cambridge, to deliver a Heilbronn colloquium in person, at the University of Bristol. Title: Pursuit and Evasion Abstract: In Rado’s famous `lion and man’ problem, a man and a lion are in a circular arena. The […]

PhD opportunities in mathematics

Join us for our annual PhD opportunities in mathematics event. This event is open to all undergraduate and masters students who are considering applying for a PhD in mathematics.

This event will be held in G.10, Fry Building.

Programme

13:00 – 14:00 Graduate student panel

14:00 – 14:30 Industry speaker – Kathryn Leeming, British Geological Survey

14:30 – 15:00 Industry speaker – Laura

15:00 – 15:15 Break

15:15 – 15:45 Academic speaker – Asma Hassanezhad, University if Bristol

15:45 – 16:30 Information from the Post Grad team

16:30 – 17:00 Informal discussions

Register to attend

Please register for this event by completing the following form.

Contact information

For practical information please email maths-conference-administrator@bristol.ac.uk

Colloquium – Emmanuel Breuillard

14:00 – 15:00pm

Online, Zoom Webinar

We’re very excited to welcome Emmanuel Breuillard (University of Cambridge), as a Heilbronn Virtual Visiting Professor.

Title: Approximate groups

Abstract: Symmetry is one of the most fundamental concepts of science and mathematics. Group theory is the field of mathematics devoted to its study. In this lecture we will discuss various situations where symmetry is present only partially and where a notion approximate symmetry arises naturally. In group theory this leads to the notion of approximate group, which has been much studied in the last twenty years in connection with various fields including additive combinatorics, harmonic and fractal analysis, asymptotic finite group theory, geometric group theory, ergodic theory and even logic and model theory. The lecture will introduce this concept and present some recent developments.

For more information and to register please visit the event website.

For more information, please contact heilbronn-coordinator@bristol.ac.uk.

Join the Heilbronn Event mailing list to keep up to date with our upcoming events.

An additional talk will be given by Emmanuel Breuillard as part of the Ergodic Theory and Dynamical Systems Seminars. For further information please the event website.

Distinguished Lecture Series: Amie Wilkinson

Online, Zoom Webinars

We’re pleased to announce that Amie Wilkinson’s (University of Chicago) postponed Distinguished Lecture series from May 2020 has now been rescheduled to May 2021. The colloquia details will be confirmed shortly. Visit the event website for further information.

The talks will be over three days:

10 May 2021, 16:00 – 17:00

12 May 2021, 16:00 – 17:00

14 May 2021, 16:00 – 17:00

Registration is now open.

If you have any questions, please contact heilbronn-coordinator@bristol.ac.uk.

Join the Heilbronn Event mailing list to keep up to date with our upcoming events.

Colloquium – Larry Guth

16:00 – 17:00pm

Online, Zoom Webinar

We’re very excited to welcome Larry Guth (Massachusetts Institute of Technology), as a Heilbronn Virtual Visiting Professor.

Title: Local smoothing for the wave equation

Abstract: The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions.

For more information and to register please visit the event website.

For more information, please contact heilbronn-coordinator@bristol.ac.uk.

Join the Heilbronn Event mailing list to keep up to date with our upcoming events.

Colloquium – Kaisa Matomäki

16:00 – 17:00pm

Online, Zoom Webinar

We’re very excited to welcome Kaisa Matomäki (University of Turku, Finland), as a Heilbronn Virtual Visiting Professor. For further information and to register for the colloquium, please visit the event website.

Title: On primes and other interesting sequences in short intervals

Abstract: By the prime number theorem, the number of primes up to $x$ is known to be asymptotically $x/\log x$. This suggests that whenever $H \leq x$ is reasonably large, the interval $[x, x+H]$ contains about $H/\log x$ primes. I will discuss what is known and what is not known about primes and almost primes (i.e. numbers with only few prime factors) in short intervals.
I will also talk about the Riemann zeta function and the Liouville function (defined, for an integer $n$, to be $+1$ or $-1$ depending on whether $n$ has an even or odd number of prime factors), both of which are closely connected to the prime numbers.

Registration is now open, please click here to register.

For more information, please contact heilbronn-coordinator@bristol.ac.uk.

Join the Heilbronn Event mailing list to keep up to date with our upcoming events.

Young Researchers in Mathematics 2021

This online conference will take place from Monday 6- Wednesday 9 June 2021.

Young Researchers in Mathematics is the conference for all PhD students! We want to welcome each and every early-career mathematician to this conference, where you can (virtually) meet researchers from all areas in a friendly environment.

YRM is the perfect opportunity to give talks about your maths, whether it be introductory or your own results. We also invite you to the plenary talks, which showcase a wide range of mathematics happening in the UK now.

Whether this is the first or final year of your PhD, this is the conference for you!

Registration is now open! The deadline for contributed talks is Monday 24 May 2021. Please click here to register.

For questions please email: yrm-2020 at bristol dot ac dot uk.

Random Matrices and Integrable Systems

Fry Building We’re very excited to welcome Random Matrices and Integrable Systems online event between 16-18th June 2021.

The past decade has seen enormous progress in understanding the behaviour of large random matrices and interacting particle systems.

A large variety of complementary methods have emerged to study these systems, and in particular methods originating from integrable systems.

This workshop is the first of a series of workshops organized within the PIICQ network “Integrable probability, classical and quantum integrability”.

This event has now passed. Presentations can be viewed via the following link.

Confirmed speakers

Giulio Ruzza (Université’ Catholique de Louvain)

Title: Airy kernel determinants, Korteweg-de Vries equation, and nonlocal Painlevé equations

Abstract: I will report on a joint work with Mattia Cafasso and Tom Claeys. We consider Fredholm determinants arising from conditional expectations of the Airy point process and show by a 2×2 Riemann-Hilbert method that they solve the Korteweg-de Vries (KdV) equation. This approach also allows

*to give a streamlined proof that the associated wave function satisfies a nonlocal version of the Painlevé II equation, originally discovered in the context of the Kardar-Parisi-Zhang narrow wedge solution by Amir, Corwin, and Quastel, and

*to study the initial data of the corresponding KdV solutions, the latter being unbounded and thus not belonging to the classes of KdV solutions usually studied in the literature.

Karol Kozlowski (ENS Lyon)

Title: Convergence of the form factor series in the Sinh-Gordon quantum field theory in 1+1 dimensions

Abstract: Fredholm determinants of id + K with K an integrable integral operator arise, in particular, as representations for the correlation functions in the simplest instance of quantum integrable models: those explicitly equivalent to a model of free, viz. non-interacting, fermions. Various series of multiple integrals, namely those whose n th summand is given by a n-fold integral, have been obtained for the more complex, truly interacting, quantum integrable models. For numerous reasons, these series provide one with a new class of special functions which naturally generalise the one subordinate to Fredholm determinants of integrable integral operators. However, the status of these representations is still conjectural in that the question of convergence of such series is basically open. The resolution of this convergence problem constitutes thus a necessary step so as to put the theory of correlation functions in truly interacting quantum intgerable models on rigorous grounds but also so as to push further the boarders of special function theory. In this talk, I will report on the recent progress I achieved on the mentioned convergence problem in the case of the simplest non-trivial truly interacting quantum integrable model: the 1+1 dimensional Sinh-Gordon quantum field theory. After reviewing the physical origin of the problem, I will describe the technique that allows one to prove the convergence of the class of series of multiple integral which represent the correlation functions in this model. The proof amounts to obtaining a sufficiently sharp estimate on the leading large-n behaviour of the n-fold integral arising in this context. This appeared possible by refining some of the techniques that were fruitful in the analysis of the large-n behaviour of integrals over the spectrum of n × n random Hermitian matrices.

Sofia Taricone (Angers)

Title: Higher order finite temperature Airy kernels and an integro-differential Painlev´e II hierarchy.

Abstract: In this talk we will study Fredholm determinants of a finite temperature version of the higher order Airy kernels that recently appeared in statistical mechanics literature. The main result is an expression of these Fredholm determinants in terms of distinguished solutions of an integrodifferential Painlev´e II hierarchy. Our result generalizes the case n = 1, already studied by Amir Corwin and Quastel some years ago for a special choice of the weight function. This latter can be seen as a generalization of the well known formula connecting the Tracy-Widom distribution for GUE and the Hastings-McLeod solution of the Pailev´e II equation. The proof of our result, for generic n, relies on the study of some operator-valued Riemann-Hilbert problem that builds up the bridge between the description of the Fredholm determinants and the derivation of a Lax pair for this new integro-differential hierarchy. The talk is based on a joint work with Thomas Bothner and Mattia Cafasso, avaiable at https://arxiv.org/pdf/2101. 03557.pdf.

Jérémie Bouttier (CEA Saclay/ENS de Lyon)

Title: Multicritical Schur measures

Abstract: Schur measures are random integer partitions, that map to determinantal point processes. We explain how to construct such measures whose edge behavior (asymptotic distribution of the largest parts) is governed by a higher-order analogue of the usual Airy ensemble/Tracy-Widom GUE distribution. This “multicritical” analogue was previously encountered in models of fermions in non-harmonic traps, considered by Le Doussal, Majumdar and Schehr. These authors noted a coincidental connection with unitary random matrix models, which our construction explains via an exact mapping.

Mattia Cafasso (Angers)

Title: The generating function for the higher order Airy point processes and a vector-valued Painlevé II hierarchy.

Abstract: During this talk I will present an ongoing joint work with Sofia Tarricone, in which we show that the Fredholm determinants associated to the higher order Airy kernels with several discontinuities admit a Tracy-Widom formula in terms of a (new?) vector-valued Painlevé II hierarchy. The algebraic structure of this hierarchy is quite similar to the one associated to the integro-differential Painlevé II hierarchy we studied with Thomas Bothner and Sofia Tarricone, and which will be illustrated by Sofia Tarricone during her talk.

Guido Mazzuca (SISSA, Trieste)

Title: Correlation functions for short-range oscillator

Abstract: Correlation functions are related to transport properties of the materials. We study the correlation functions of a chain of harmonic oscillators with short range interactions when the data is sampled according to the Gibbs measure. By introducing a local generalization of spacing that reduces to relative distance between neighboring oscillators when the potential has only nearest neighbor interactions, we are able to calculate the correlation functions.
Such functions display rapid oscillations that travel along the chain.
We determine the scaling in time of the fastest traveling peaks that are asymptotically described by the Airy function.

This talk is main based on a joint work with T. Grava, T. Kriecherbauer, and K.T-R. McLaughling. https://link.springer.com/article/10.1007%2Fs10955-021-02735-z

Jinho Baik (University of Michigan Ann Arbor)

Title: Multi-time distributions of the KPZ fixed point

Abstract: The KPZ fixed point is a two-dimensional random field which is expected to be the universal limit of many random growth models and interacting particle systems that belong to the KPZ universality class. The one-point distribution of the KPZ fixed point for the step initial condition is the Tracy-Widom distribution and the one-dimensional slice in the spatial direction is distributed as the Airy process.

Recently the multi-time, multi-position distribution functions are computed explicitly by Johansson and Rahman and also independently by Liu. We discuss these distribution functions and the associated integrable differential equations.

Theo Assiotis (University of Edinburgh)

Title: Ergodic decomposition of p-adic Hua measures on infinite p-adic matrices

Abstract: Neretin constructed an analogue of the Hua (or Cauchy type) measures on infinite p-adic matrices. Later Bufetov and Qiu classified the ergodic measures on infinite p-adic matrices for the natural action of $GL(\infty,\mathbb{Z}_p) \times GL(\infty,\mathbb{Z}_p)$. In this talk I will describe explicitly how the Hua measures of Neretin decompose into the ergodic measures of Bufetov and Qiu. This involves the scaling limit of an analogue of the singular values in this setting and some random partitions.

Laure Dumaz (ENS Paris)

Title: Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization.

Abstract: We consider the 1-dimensional continuous Schrodinger operator – d^2/d^x^2 + B’(x) on an interval of size L where the potential B’ is a white noise. We study the entire spectrum of this operator in the large L limit. We prove the joint convergence of the eigenvalues and of the eigenvectors and describe the limiting shape of the eigenvectors for all energies. When the energy is much smaller than L, we find that we are in the localized phase and the eigenvalues are distributed as a Poisson point process. The transition towards delocalization holds for large eigenvalues of order L. In this regime, we show the convergence at the level of operators. The limiting operator is acting on R^2-valued functions and is of the form “J \partial_t + 2*2 noise matrix” (where J is the matrix ((0, -1)(1, 0))), a form which already appeared as a conjecture by Edelman Sutton (2006) for limiting random matrices. Joint works with Cyril Labbé.

Nick Simm (University of Sussex)

Talk 1 Title: Characteristic polynomials of non-Hermitian matrices, dualities and Painleve transcendents

Abstract 1: I will discuss recent results on moments of characteristic polynomials of non-Hermitian random matrices. Various duality formulae are obtained and used to make a connection to Painlev’e transcendents. This allows us to conjecture the asymptotics of the partition function in more complicated models, such as the critical lemniscate. This is joint work with Alfredo Deano (Madrid).

Talk 2 Title: Secular coefficients of characteristic polynomials and holomorphic multiplicative chaos

Abstract 2: I will discuss the secular coefficients of random unitary matrices, a problem first investigated by Diaconis and Gamburd in 2004. Studying the C\beta E version, we obtain asymptotic limiting distributions for the coefficients and study their tightness. This is done by a link with Gaussian multiplicative chaos and our work motivates the introduction of a holomorphic counterpart of the chaos. This is joint work with Joseph Najnudel (Bristol) and Elliot Paquette (McGill).

Talk 3 Title: Real eigenvalues of real random matrices: new results for product ensembles

Abstract 3: Real eigenvalues of random matrices (such as real Ginibre matrices) were first investigated by Edelman et al. (1994). I will discuss two particular models that have been of recent interest, namely products of independent real Ginibre matrices and products of independent truncated orthogonal random matrices. The latter model concerns a recent collaboration with Alex Little (Bristol) and Francesco Mezzadri (Bristol).

Tom Claeys (Université' Catholique de Louvain)

Title: Airy kernel determinants, the KdV equation, and conditional point processes

Abstract: A family of unbounded solutions to the Korteweg-de Vries equation arose recently in the context of the Kardar-Parisi-Zhang equation. These solutions can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and are also related to an integro-differential Painlevé II equation. In my talk, I will will give an overview of these developments, and I will discuss in more detail the initial data of this family of solutions and the connection with conditional Airy point processes.

Neil O’Connell (University College, Dublin)

Title: Interacting diffusions on positive definite matrices

Abstract: We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

Jon Keating (University of Oxford)

Title: Moments of moments and Gaussian Multiplicative Chaos

Abstract: I will discuss connections between the moments of moments of the characteristic polynomials of random unitary (CUE) matrices and Gaussian Multiplicative Chaos, focusing in particular on the critical-subcritical case.

Thomas Bothner (University of Bristol)

Title: A Riemann-Hilbert approach to Fredholm determinants of integral Hankel composition operators

Abstract: These three talks will highlight a novel way of characterizing Fredholm determinants of Hankel composition operators via Riemann-Hilbert problems

Emilia Alvarez (University of Bristol)

Title: Moments of the logarithmic derivative of characteristic polynomials from SO(N) and USp(2N)

Abstract: I will discuss work with Nina Snaith on asymptotics of moments of the logarithmic derivative of characteristic polynomials of orthogonal SO(N) and symplectic USp(2N) random matrices, evaluated near the point 1. The leading order behaviour in this regime as N tends to infinity is governed by the likelihood that the matrices in each ensemble have an eigenvalue at or near the point 1. These results follow work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, where they compute these asymptotics in the case of unitary random matrices.

Bhargavi Jonnadula (University of Bristol)

Title: Symmetric function theory and moments of characteristic polynomials

Abstract: Symmetric function theory has proven to play a crucial role in random matrix theory to study fundamental quantities such as the joint moments of traces and correlations of the characteristic polynomial. For Hermitian ensembles, we compute exact formulae for moments of characteristic polynomials using symmetric function theory, which further allows us to compute large matrix asymptotics. We comment on the universal properties of the moments of characteristic polynomials for the Gaussian unitary ensemble and how to recover the semi-circle law in the large matrix limit.

This is joint work with Jon Keating and Francesco Mezzadri.

Massimo Gisonni (SISSA, Trieste)

Title: Correlators of the Jacobi Unitary Ensemble

Abstract: The Jacobi Unitary Ensemble (JUE) arises as the Beta distribution over Hermitian matrices. In this talk, we focus on the computation and interpretation of its correlators. Respectively, on one side the Matrix Resolvent method, introduced by Bertola, Di, Dubrovin, provides closed formulae for the cumulant functions of the JUE. On the other, work of Harnad et al. on KP-hypergeometric tau functions allows us to express their topological expansion in terms of triple monotone Hurwitz numbers.

This talk is based on joint work with T.Grava and G.Ruzza.

Programme

Wednesday 16 June, 2021

08:55 – 09:00 – Welcome

09:00 – 09:50 – Nick Simm

10:00 – 10:50 – Thomas Bothner

11:00 – 11:30 – Break

11:30 – 12:00 – Bhargavi Jonnadula

12:10 – 12:40 – Emilia Alvarez

12:50 – 15:30 – Lunch

15:30 – 16:00 – Laure Dumaz

16:10 – 16:40 – Theo Assiotis

16:50 – 17:10 – Break

17:10 – 17:40 – Karol Kozlowski

17:50 – 18:20 – Guido Mazzuca

Thursday 17 June, 2021

08:55 – 09:00 – Welcome

09:00 – 09:50 – Nick Simm

10:00 – 10:50 – Thomas Bothner

11:00 – 11:30 – Break

11:30 – 12:00 – Jon Keating

12:10 – 13:30 – Lunch

13:30 – 14:00 – Neil O’Connell

14:10 – 14:40 – Jérémie Bouttier

14:50 – 15:30 – Break

15:30 – 16:00 – Massimo Gisonni

16:10 – 16:40 – Jinho Baik

Friday 18 June, 2021

08:55 – 09:00 – Welcome

09:00 – 09:50 – Nick Simm

10:00 – 10:50 – Thomas Bothner

11:00 – 11:30 – Break

11:30 – 12:00 – Mattia Cafasso

12:10 – 13:30 – Lunch

13:30 – 14:00 – Sofia Tarricone

14:10 – 14:40 – Giulio Ruzza

14:50 – 15:30 – Break

15:30 – 16:00 – Tom Claeys

Organisers

Francesco Mezzadri
Joseph Najnudel
Tamara Grava
Nina Snaith

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