Past talks
WINTER 2024
Joe Harrow (University of Kent)
April 22, 2024, 11AM-12 PM EDT
Zoom meeting ID: 965 2971 3727
Determinantal expressions for Ohyama polynomials
The Ohyama polynomials provide algebraic solutions of the D7 case of the Painleve III equation at a particular sequence of parameter values. It is known that many special function solutions of Painleve equations are expressed in terms of tau functions that can be written in the form of determinants, but until now such a representation for the Ohyama polynomials was not known. Here we present three different determinantal formulae for these polynomials: the first, in terms of Wronskian determinants related to a Darboux transformation for a Lax pair of KdV type; and the second, in terms of Hankel determinants, which is related to the Toda lattice. The third is for a slightly modified form of the polynomials and is given as a Wronskian of associated Laguerre polynomials.
A recording of the talk can be found at
April 1st, 2024, 4-5 PM EDT
East Hall EH1866
Dimers on a Riemann surface and compactified free field
In this talk I will be speaking about the dimer model sampled on a general Riemann surface. In this setup, the dimer height function becomes additively multivalued with a random monodromy. Given a sequence of graphs approximating the conformal structure of the surface in a suitable way, the underlying sequence of height functions is expected to converge to the compactified free field on the surface. Recently, this problem was addressed by Berestycki, Laslier and Ray in the case when a Riemann surface is approximated by Temperley graphs. Using various probabilistic methods, they obtained the following universal result: given that the random walk associated with these graphs converges to the Brownian motion on the surface (in an appropriate sense), the limit of height functions exists, is conformally invariant and does not depend on a particular sequence of graphs. However, the identification of the limit with the compactified free field was missing in this result. In my recent work I am trying to fill this gap by studying the same problem from the perspective of discrete complex analysis. For this purpose, I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain local geometric conditions. In this setup I obtain an analytic description of the limit which allows to identify it with a suitable version of the compactified free field; I also prove the convergence in some non-Temperlian cases when the surface is generic. A core part of this approach is the regularity theory on t-embeddings recently developed by Chelkak, Laslier and Russkikh, as well as an analytic technique linking the problem with Quillen determinant of a family of Cauchy-Riemann operators developed by Dubédat.
A recording of the talk can be found at
March 18, 2024, 4-5 PM EDT
Zoom meeting ID: 965 2971 3727
Noncommutative stochastic calculus
Noncommutative or free probability is a branch of mathematics that is useful for describing the large-N limits of many N×N random matrix models. In this theory, classical probability spaces are replaced by pairs (𝒜,𝜏), where 𝒜 is an (operator) algebra and 𝜏 : 𝒜 → ℂ is a certain kind of linear functional. In such a pair, 𝒜 and 𝜏 are conceptualized as the space of ``noncommutative random variables'' and the ``expectation'' functional on 𝒜, respectively. The analogy with classical probability goes much further; indeed, there are notions of distribution, independence, L^p-spaces, conditional expectation, and more. My talk will focus on my recent joint work with David Jekel and Todd Kemp on developing a noncommutative theory of stochastic calculus. I shall frame the discussion around some joint work in progress with Guillaume Cébron and Nicolas Gilliers: applications of the theory to the characterization of large-N limits of solutions to N×N matrix stochastic differential equations.
A recording of the talk can be found at
March 15, 2024, 9-10 AM EDT
Zoom meeting ID: 965 2971 3727
A new isomonodromy deformation equation with movable branch points
The isomonodromic deformation plays a universal role in connecting various research areas of mathematics and physics. In this talk, I am going to discuss the isomonodromy theory for a new class of Fuchsian-type elliptic second-order equations defined on the moduli space of elliptic curves with the parameter τ ∈ H (upper half-plane). We will observe that the isomonodromic deformation equation is governed by a new second-order nonlinear equation with a deep connection to the Painlevé VI equation but admits essentially different properties from PVI. Indeed, the new isomonodromic deformation equation, distinct from the Painlevé VI equation, admits so-called movable branch points that can be explicitly determined by their monodromy data.
A recording of the talk can be found at
March 11, 2024, 4-5 PM EST
East Hall EH1866
Long-time Asymptotics of the KdV Steplike Solutions
The long time asymptotics of the steplike solutions of the KdV equation on the constant backgrounds has long been well studied at the physical level of rigor. This talk will present some recent mathematically rigorous results that refine and justify the above-mentioned asymptotics. We will also compare the efficiency of the two most common methods --- the Nonlinear Steepest Descent and the Inverse Scattering Transform -- for obtaining the soliton asymptotics over a larger space-time domain and for the most general possible class of steplike initial data.
A recording of the talk can be found at
March 4, 2024, 11AM-12PM EST
Zoom meeting ID: 965 2971 3727
On the exact Gevrey order of formal Puiseux series solutions to the third Painlevé equation
I would like to speak about one rather old joint work with Andrey Vasilyev.
We consider the third Painlevé equation.
The Puiseux series formally satisfying it, asymptotically approximate of Gevrey order one solutions to this equation in sectors with the vertices at infinity.
A condition sufficient for the convergence of formal solutions of an ODE with analytic left-hand side is well-known.
On the other hand sufficient conditions for the divergence given in the same terms are unknown.
We present the family of values of the parameters such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.
A recording of the talk can be found at
February 19, 2024, 4-5 PM EST
East Hall EH1866
Ablowitz-Ladik lattice and the Circular β-ensemble: a (kind of) surprising connection
In this talk, we focus on the interplay between the theory of integrable systems, and random matrix theory.
This connection was first realized by H. Spohn, who was able to compute the density of states for the Toda lattice by connecting it to the corresponding one of the Gaussian β ensemble, a well known random matrix model. The computation of this quantity enabled him to apply the theory of generalized hydrodynamics, so to compute the correlation functions for the Toda lattice.
In this talk, I consider another integrable model, namely the Ablowitz-Ladik lattice; I introduce the Generalized Gibbs ensemble for this lattice, and I relate it with the so-called Circular β ensemble, a classical random matrix model for unitary matrices. This allows us to compute explicitly the density of states for the Ablowitz-Ladik lattice in terms of the one of this random matrix ensemble.
This talk is mainly based on these two papers:
G. M. , and T. Grava: Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, circular β- ensemble and double confluent Heun equation. Communication in Mathematical Physics. DOI: 10.1007/s00220-023-04642-8
G. M., and R. Memin: Large Deviations for Ablowitz-Ladik lattice, and the Schur flow. Electronic Journal of Probability. DOI: 10.1214/23-EJP941
A recording of the talk can be found at
FALL 2023
November 27, 2023, 4-5 PM EST
Zoom meeting ID: 965 2971 3727
The intertwined derivative Schrödinger system of Calogero–Moser–Sutherland type
This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R. Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables. This new system is still an integrable extension and perturbation of the original CMSdNLS equations. Thanks to the conjugation acting method, I have also established the explicit expression of general solutions on the torus and on the real line in my work [hal-04227081].
You can find the slides here.
November 20, 2023, 4-5 PM EST
East Hall EH1866
Geometry of half-space log-gamma polymers
The half-space directed polymer is a variant of directed polymer that studies how polymers behave in the presence of an attractive wall. Depending on the strength of the boundary, the polymers are expected to have two distinct phases: the bound phase and the unbound phase. In this talk, I will focus on the half-space polymer model with log-gamma weights which makes the model integrable. I will describe our results in the unbound phase where we obtain KPZ exponents and in the bound phase where we obtain stochastic boundedness of the endpoint. Our proof proceeds by constructing the half-space log-gamma line ensemble which has a novel feature of attraction/repulsion at the boundaries. Based on two joint works: one with Guillaume Barraquand and Ivan Corwin, and one with Weitao Zhu.
A recording of the talk can be found at
November 13, 2023, 4-5 PM EST
East Hall EH1866
The upper tail behaviour of the KPZ equation and the CDRP via the tangent method
The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. Particularly important observables are the one- and multi-point distributions of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data. This is joint work with Shirshendu Ganguly. Time permitting, we will also highlight forthcoming work with Ganguly and Lingfu Zhang developing these techniques to obtain the scaling limit of the path measure in the associated model of the continuum directed random polymer (CDRP) (as well as in the zero temperature analogue of the directed landscape), under the conditioning of large point-to-point free energy. The zero temperature case establishes a conjecture of Liu and Wang.
A recording of the talk can be found at
November 6, 2023, 4-5 PM EST
Zoom meeting ID: 965 2971 3727
A Riemann-Hilbert approach to q-difference Painlevé VI
Almost since their very discovery over a century ago, it is known that the classical Painlevé equations govern monodromy preserving deformations of certain linear ODEs. This lies at the heart of the powerful Riemann-Hilbert approach to these equations. In this talk, I will discuss recent extensions of this approach to the q-difference setting, focusing on the q-analog of Painlevé VI derived by Jimbo and Sakai. I will show how, analogous to the classical theory, a corresponding monodromy manifold can be constructed and the global asymptotics of solutions can be derived by analysing associated Riemann-Hilbert problems. The special role of classical-function solutions in this framework will also be highlighted.
This is based on joint work with Nalini Joshi.
A recording of the talk can be found at
October 30, 2023, 4-5 PM EDT
East Hall EH1866
Field Theory of the Massive Ising Model
The broad conjecture that the critical Ising model in two dimensions is described by a Conformal Field Theory (CFT) has seen a number of rigorous justifications in recent years, notably including Chelkak, Hongler, and Izyurov's recent proof that the correlations between primary fields of the model converge under scaling limit to explicit CFT correlations. We prove an analogous result for the massive model, i.e. convergence of correlations between the spin, disorder, fermion, and energy fields under a near-critical scaling limit setting. The massive model, by definition, is not conformally invariant and therefore its correlations do not enjoy a same kind of explicit description as in the critical case; however, analogues of the critical operator product expansions (OPE's) persist and the correlations may be bosonized and identified with correlations in the massless Sine-Gordon model. Based on joint works with Chelkak, Izyurov, Webb, and others.
A recording of the talk can be found at
October 23, 2023, 4-5 PM EDT
East Hall EH1866
Bipartite spherical spin glass at critical temperature
October 9, 2023, 9-10 AM EDT
Zoom meeting ID: 96529713727
Moment matrix and C-Toda lattice hierarchy
In this talk, we focus on the reduction of the 2-dimensional Toda hierarchy. Based on the Gauss-Borel decomposition of the moment matrix and appropriate time evolution, we construct the corresponding Lax representation. Symmetric moment matrices are considered that lead to the differential relations between symmetric tau functions and 2d-Toda’s tau function. Motivated by the Cauchy two-matrix model, we study the rank one shift condition in the symmetric case, from which the C-Toda lattice hierarchy is found with the special Lax matrices and intergrable structure.
A recording of the talk can be found at
October 2, 2023, 4-5 PM EDT
East Hall EH1866
Large deviations for the deformed polynuclear growth
The polynuclear growth model (PNG) is a prototypical example of random interface growth among the Kardar-Parisi-Zhang universality class. In this talk I will discuss a q-deformation of the PNG model recently introduced by Aggarwal-Borodin-Wheeler. We are mainly interested in the large time large deviations of the one-point distribution under narrow-wedge (droplet) initial data, i.e., the rare events that the height function at time t being much larger (upper tail) or much smaller (lower tail) than its expected value. Large deviation principles with speed t and t^2 are established for the upper and lower tails, respectively. The upper tail rate function is computed explicitly and is independent of q. The lower tail rate function is described through a variational problem and shows nontrivial q-dependence. Based on joint work with Matteo Mucciconi and Sayan Das.
A recording of the talk can be found at
September 25, 2023, 9-10 AM EDT
Zoom meeting ID: 96529713727
Gap probability near the cusp singularity in random matrix ensembles
We study the gap probability of finding no eigenvalues near the cusp singularity in random matrix ensembles. It was known that the cusp singularity leads to a new universal determinantal process characterized by the Pearcey kernel. By studying the Fredholm determinant of the Pearcey kernel, we establish an integral representation of the gap probability in terms of the Hamiltonian of a system of nonlinear differential equations. The large gap asymptotics are then derived in both the Pearcey process and the thinned Pearcey process where each particle in the process is removed independently with certain probability. The hard edge Pearcey process and higher order Airy processes will also be discussed in the talk.
The slides for the talk can be found here.
September 18, 2023, 4-5 PM EDT
East Hall EH1866
Random polynomials near the unit circle
It is well known that a random polynomial with iid coefficients has most of its roots close to the unit circle. Recently, we found that both the minimum modulus of the polynomial on the unit circle itself, as well as the closest root to the unit circle has a limiting exponential distribution. In the talk I will explain the joint mechanism behind these results.
The talk is based on joint works with Nicholas Cook, Hoi Nguyen and Ofer Zeitouni.
A recording of the talk can be found at
September 11, 2023, 4-5 PM EDT
East Hall EH1866
Global Rational Approximations of Functions with Factorially Divergent Asymptotic Series
Rational approximations of functions offer a rich mathematical theory. Touching subjects such as orthogonal polynomials, potential theory and of course differential equations. In this talk we will discuss a specific type of rational approximant, factorial expansions. In recent work with O. Costin and R. Costin we have developed a theory of dyadic expansions which improve the domain and rate of convergence when compared to the classical methods found in the literature. These results provide a general method for producing rational approximations of Borel summable series with locally integrable branch points. Surprisingly, these expansions capture the asymptoticly important Stokes phenomena. Additionally, we find applications in operator theory on Hilbert spaces providing new representations for (bounded and unbounded) positive and self-adjoint operators in terms of the semigroups and unitary groups they generate. Finally, as an example of an important application we discuss representing the tritronquée solutions of Painlevé’s first equation.
A recording of the talk can be found at
WINTER 2023
Alexander R. Its (Indiana University-Purdue University Indianapolis)
April 24, 2023
TT* - equations of Cecotti and Vafa, Riemann-Hilbert method and Iwasawa factorization.
In this talk, some interesting (we believe !) new features of the Riemann-Hilbert method of the asymptotic analysis of integrable systems will be discussed. These features have emerged during the study of the global solution of the tt* equations of Cecotti and Vafa which the speaker has been pursuing, for some time already, together with M. Guest and Chang-Shou Lin. The method which we use in this study is based on a combination of the isomonodrtomy technique and Iwasawa factorization from the theory of loop groups. This allows us to simplify significantly the asymptotic analysis of the tt* equations and, simultaneously, to bring a new light on some aspects of the well known relation between the Birkhoff-Grothendieck and Iwasawa factorizations. Another important link - to a 1980 paper by I. Krichever on the nonlinear analog of the d'Alembert's formula, will be also highlighted.
The talk represents an ongoing joint project of M. Guest and the speaker, and it is dedicated to the memory of Igor Krichever with whom the project was started.
A recording of the talk can be found at
Milivoje Lukić (Rice University)
April 17, 2023
Reflectionless canonical systems: almost periodicity and character-automorphic Fourier transforms
In the spectral theory of self-adjoint and unitary operators in one dimension (such as Schrodinger, Dirac, and Jacobi operators), a half-line operator is encoded by a Weyl function; for whole-line operators, the reflectionless property is a pseudocontinuation relation between the two half-line Weyl functions. We develop a comprehensive theory of reflectionless canonical systems with an arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theorem property. This generalizes, to an infinite gap setting, the constructions of finite gap quasiperiodic (algebro-geometric) solutions of stationary integrable hierarchies. Instead of theta functions on a compact Riemann surface, the construction is based on reproducing kernels of character-automorphic Hardy spaces in Widom domains with respect to Martin measure. We also construct unitary character-automorphic Fourier transforms which generalize the Paley-Wiener theorem. Finally, we find the correct notion of almost periodicity which holds in general for canonical system parameters in Arov gauge, and we prove generically optimal results for almost periodicity for Potapov-de Branges gauge, and Dirac operators. This is joint work with Roman Bessonov and Peter Yuditskii.
A recording of the talk can be found at https://youtu.be/ZEcHCixsCbU
Tomas Lasic Latimer (University of Sydney)
February 6, 2023
Asymptotics of q-orthogonal polynomials
This talk focuses on q-orthogonal polynomials, orthogonal polynomials whose orthogonality condition is supported on the discrete lattice: q^k, for integer k . We investigate the behaviour of such polynomials in the case k>0 (q<1) and, if time permits, we study what happens when we relax this condition and allow k to be negative. Using the RHP framework we deduce the asymptotic behaviour of these polynomials as their degree tends to infinity, as well as other properties such as uniqueness.
A recording of the talk can be found at https://youtu.be/5HTbvRjAtbk
Raj Rao Nadakuditi (University of Michigan)
March 20, 2023
Improved very sparse matrix completing using an intentionally randomized "asymmetric SVD"
Joint work with: Charles Bordenave (University Aix-Marseille)
Simon Coste (Université de Paris P7, LPSM)
We consider the matrix completion problem in the very sparse regime where, on average, a constant number of entries of the matrix are observed per row (or column). In this very sparse regime, we cannot expect to have perfect recovery and the celebrated nuclear norm based matrix completion fails because the singular value decomposition (SVD) of the underlying very sparse matrix completely breaks down.
We demonstrate that it is indeed possible to reliably recover the matrix. The key idea is the use of a ``randomized asymmetric SVD'' (which we will define) to find informative singular vectors in this regime in a way that the SVD cannot.
We provide sharp theoretical analysis of the phenomenon, including a prediction of the lower limits of statistical recovery and demonstrate the efficacy of the new method(s) using simulations.
A recording of the talk can be found at https://youtu.be/nlSmfwla6L4
Zhipeng Liu (University of Kansas)
March 13, 2023
Some exact formulas of the KPZ fixed point and directed landscape
In the past twenty years, there have been huge developments in the study of the Kardar-Parisi-Zhang (KPZ) universality class, which is a broad class of physical and probabilistic models including one-dimensional interface growth processes, interacting particle systems and polymers in random environments, etc. It is broadly believed and partially proved, that all the models share the universal scaling exponents and have the same asymptotic behaviors. The height functions of models in the KPZ universality class are expected to converge to a limiting space-time fluctuation field, which is called the KPZ fixed point. Moreover, there is a random “directed metric” on the space-time plane that is expected to govern all the models in the KPZ universality class. This “directed metric” is called the directed landscape. Both the KPZ fixed point and the directed landscape are central objects in the study of the KPZ universality class, while they were only characterized/constructed very recently [MQR21, DOV18].
In this talk, we will discuss some exact formulas of distributions in these two random fields. These exact formulas are in terms of an infinite sum of multiple contour integrals, which are analogous to the Fredholm determinant expansions. We will show some surprising probabilistic properties of the KPZ fixed point and the directed landscape using the exact formulas. Some of the results are based on joint work with Yizao Wang and Ray Zhang.
A recording of the talk can be found at https://youtu.be/I4zWZXU2ZTY
Brian Hall (University of Notre Dame)
March 6, 2023
Heat flow, random matrices, and random polynomials
It is a classical result that if you apply the backward heat flow to a polynomial with all real roots, the new polynomial will still have all real roots. Recent results have then shown a connection to random matrix theory. Suppose X is a Hermitian random matrix with characteristic polynomial p and Y is a random matrix chosen from the Gaussian Unitary Ensemble. Then applying the backward heat flow to p gives a polynomial whose roots resemble the eigenvalues of X+Y.
Things get much more interesting if we apply the forward heat operator to a polynomial with real roots, or if we apply a heat operator (forward or backward) to a polynomial with complex roots. I will discuss a conjecture of mine with Ching Wei Ho in which applying the heat operator to the characteristic polynomial of one random matrix model gives a new polynomial whose zeros resemble the eigenvalues of a second random matrix model. Then I will describe a general conjecture about the evolution of zeros of polynomials under the heat flow. Finally, I will describe some recent rigorous results in this direction, obtained with Ching Wei Ho, Jonas Jalowy, and Zakhar Kabluchko.
A recording of the talk can be found at https://youtu.be/iuyWrdZYfqA
Dmitry Chelkak (University of Michigan)
March 27, 2023
Magnetization in the planar Ising model and orthogonal polynomials
We will discuss a 'synthetic' proof of explicit formulae for the magnetization in the 'zig-zag layered' planar Ising model on Z^2 that is based upon an observation that the values of certain fermionic observables equal the coefficients of an orthogonal polynomial. (In the homogeneous case, this version of classical proofs dating back to Onsager, Yang and McCoy-Wu, among other simplifications, bypasses Toeplitz determinants as an intermediate step in the computations.) The main new result is a formula for the magnetizations in the m-th column of a layered model in the zig-zag half-plane via Hankel determinants constructed from the spectral measure of a certain Jacobi matrix that encodes interaction parameters between the columns. This result holds in full generality and leads to open questions on asymptotics of such determinants in several setups of interest. Based upon arXiv:1904.09168, joint work with Clément Hongler and Rémy Mahfouf.
A recording of the talk can be found at https://youtu.be/GFqcaH0JGe8
Youyi Huang (Texas Tech University)
January 23, 2023
Entropy fluctuation formulas of fermionic Gaussian states
We study the statistical behavior of quantum entanglement in bipartite systems over fermionic Gaussian states as measured by von Neumann entropy. The formulas of average von Neumann entropy with and without particle number constrains have been recently obtained, whereas the main results of this work are the exact yet explicit formulas of variances for both cases. For the latter case of no particle number constrain, the results resolve a recent conjecture on the corresponding variance. Different than the existing methods in computing variances over other generic state models, proving the results of this work relies on a new simplification framework. The framework consists of a set of new tools in simplifying finite summations of what we refer to as dummy summation and re-summation techniques. As a byproduct, the proposed framework leads to various new transformation formulas of hypergeometric functions. This talk is based on the joint work with Lu Wei available at https://arxiv.org/abs/2211.16709
A recording of the talk can be found here: https://youtu.be/sGdpLRdiorI
Ilia Gaiur (University of Toronto)
January 9, 2023
Irregular isomonodromic deformations: Hamiltonian aspects
We study isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (aka truncated current algebra). In my talk I will explain how to choose isomonodromic times in irregular situations, and how this choice may be explained from the Poisson point of view. Such choice covers a wide class of isomonodromic systems such as classical Painlevé transcendents as well as higher order ones and matrix Painlevé systems. I will also introduce a general formula for Hamiltonians of isomonodromic flows. Talk is based on the thesis of the speaker and the results obtained in a collaboration with M. Mazzocco and V. Roubtsov (arXiv:2106.13760).
A recording of the talk can be found here: https://youtu.be/eHjkKmV0EqI
FALL 2022
Maria Ntekoume (Rice University)
December 5, 2022
Critical well-posedness for the derivative nonlinear Schrödinger equation on the line
Benjamin Eichinger (TU Wien)
November 28, 2022
Eigenvalue asymptotics for continuum Schrödinger operators
Gaultier Lambert (KTH)
November 21, 2022
Applications of the theory of Gaussian multiplicative chaos to random matrices.
Log-correlated fields are a class of stochastic processes which describe the fluctuations of some key observables in different probabilistic models in dimension 1 and 2 such as characteristic polynomials of random matrices. Gaussian multiplicative chaos is a renormalization procedure which aims at defining the exponential of a Log-correlated field in the form of a family of random measures. These random measures can be thought of as describing the extreme values of the underlying field. In this talk, I will review the theory of multiplicative chaos and report on some applications to the characteristic polynomial of the Ginibre ensemble. If time permits, I will also a connection to Laughlin theory for the fractional Hall effect and a few open problems.
A recording of the talk can be found here:
Klara Courteaut (KTH)
November 14th, 2022
Rate of convergence of linear statistics for the compact classical groups - from Berry-Esseen to super-exponential
Bjorn Berntson (KTH)
November 7, 2022
Nonlocal nonlinear Schrödinger equations and Calogero-Moser systems
The integrability and applicability of the nonlinear Schrödinger (NLS) equation have inspired the development of various integrability-preserving NLS relative equations. I will discuss a new such equation that interpolates between the mixed Manakov system and a known nonlocal NLS equation containing a Hilbert transform. This new system admits families of exact analytic solutions parameterized by the solutions of hyperbolic and elliptic Calogero-Moser many-body systems. Special attention is given to the second family of solutions, which features the interaction of periodic waves against a dynamic background. This is joint work with Alexander Fagerlund (KTH Physics).
A recording of the talk can be found here:
Marcelo Campos (IMPA)
October 31, 2022
The least singular value of a random symmetric matrix
Ovidiu Costin (Ohio State University)
October 24, 2022
Approximating analytic functions
In problems of high complexity in mathematics and physics it is often the case that a solution can only be generated as a finite number of terms of a perturbation series at certain special points and a practical problem of considerable importance is extracting as much information as possible from this partial data.
One of the oldest techniques developed for this purpose, a technique of remarkable efficacy is that of Padé approximants, which are capable of extrapolating power series much beyond their radius of convergence.
A question we ask is: what is the optimal strategy to approximate the underlying function from Taylor series data, and if optimality cannot be achieved in practice, what are the most efficient near-optimal methods? The optimality question turns out to be well-posed in settings often encountered in applications (such as Borel transforms of divergent series). We describe this optimal procedure which relies on the uniformization of the Riemann surface of the associated function. The accuracy of the optimal method far exceeds that of classical methods such as Padé approximants (which however retain the advantage of simplicity and flexibility).
Another important question is, what is the sensitivity of approximation methods to numerical inaccuracies? We answer this question as well, and in the process we discover remarkable universality phenomena. The mathematically predicted sensitivity to noise lines up sharply with numerical experiments.
A recording of the talk can be found here:
Norm Levenberg (Indiana University)
October 10, 2022
An extremal problem for the Bergman kernel of orthogonal polynomials
The notion of optimal measures associated to a compact set K in Euclidean space is related to that of optimal designs in statistics. This can be reformulated in terms of an optimization problem related to Bergman functions associated to measures on K. After a brief motivational discussion, we discuss another optimization problem involving these Bergman functions which itself is related to finding polynomials of extremal growth for K at a point outside of K. Using potential theory and estimates for Faber polynomials, we prove an asymptotic result regarding this problem when K is a planar compact set bounded by a sufficiently regular closed curve.
A recording of the talk can be found here: https://youtu.be/mXk77_n3P9Y
Christophe Charlier (Lund University)
October 3, 2022
Gap probabilities and planar Fisher-Hartwig singularities in the random normal matrix model
In the first part of this talk, I will present recent results about large gap asymptotics on annuli in the random normal matrix model. In this two-dimensional setting, the theta function emerges in the asymptotics in a novel way which I will discuss (it is not related to Riemann-Hilbert problems).
In the second part of the talk, I will discuss about determinants with circular root- and jump-type singularities. These determinants are of interest in the study of the eigenvalue moduli of random normal matrices, but so far determinants with circular root-type singularities have been unexplored. I will show that such singular determinants have a novel type of asymptotic behavior described in terms of the so-called associated Hermite polynomials.
A recording of the talk can be found here:
Liying Li (University of Michigan)
September 19, 2023
Stationary solutions for 1D Burgers equations and KPZ scaling
In the first part, we will talk about the stationary solutions for 1D stochastic Burgers equations and their ergodic properties. We will classify all the ergodic components, establish the "one force---one solution" principle, and obtain the inviscid limit. The key objects to study are the infinite geodesics and infinite-volume polymer measures in random environments, and the ergodic results have their counterparts in the geodesic/polymer language. In the second part, we will present a random point field model that is motivated by the coalescing and monotone structure of the optimal paths in random environments that arise in many KPZ models. The 2/3 transversal exponent from the KPZ scaling becomes a natural parameter for the renormalization action in this model, and can be potentially extended to values other than 2/3. Some preliminary results are given.
A recording of the talk can be found here: https://youtu.be/l3NocFvw7GQ
WINTER 2022
Jonathan Husson (University of Michigan)
April 11, 2022
Asymptotics of spherical integrals and large deviations of the largest eigenvalues for random matrices.
The Harish-Chandra-Itzykson-Zuber integral, also called spherical integral is defined as the expectation of exp(Tr(AUBU*)) for A and B two self adjoint matrices and U Haar-distributed on the unitary/orthogonal/symplectic group. It was initially introduced by Harish-Chandra to study Lie groups. Since then, it has known many kinds of applications, from physics to statistical learning. In this talk we will study the asymptotics of these integrals when one of the matrices remains of finite rank. We will also see how to derive from these asymptotics large deviation principles for the largest eigenvalues for random matrix models that satisfy a sub-Gaussian bound.
A recording of the talk can be found here:
Giorgio Young (Rice University)
April 4, 2022
Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows
We consider the Cauchy problem for the KdV hierarchy -- a family of integrable PDEs with a Lax pair representation involving one-dimensional Schrodinger operators -- under a local in time boundedness assumption on the solution.
For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency.
This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craig-type conditions. This is joint work with Milivoje Lukić.
A recording of the talk can be found here: https://youtu.be/5qcxmuShBjg
Nicholas Simm (University of Sussex)
March 28, 2022
Secular coefficients of random unitary matrices
I will discuss the coefficients of the characteristic polynomial of a random unitary matrix. These are commonly known as secular coefficients, and their study was initiated by Diaconis and Gamburd in 2004 who showed that their moments are linked to the combinatorics of magic squares. I will discuss generalisations of this result and some new connections to random permutations and to Gaussian multiplicative chaos theory. This enables us to write down limit theorems for the coefficients in certain circumstances and to partially resolve an open problem of Diaconis and Gamburd who asked about this limit behaviour. This is joint work with Joseph Najnudel (Université Côte d'Azur) and Elliot Paquette (McGill University).
A recording of the talk can be found here:
Sofia Tarricone (Université Catholique de Louvain-la-Neuve)
March 21, 2022
Higher order finite temperature Airy kernels and an integro-differential Painlevé II hierarchy
In this talk we will study Fredholm determinants of integral operators acting through 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 integro-differential Painlevé II hierarchy. Our result generalizes the case n = 1, already studied by Amir, Corwin and Quastel in 2011. 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 Painlevé 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 the joint work with Thomas Bothner and Mattia Cafasso, available at: https://imstat.org/journals-and-publications/annales-de-linstitut-henri-poincare/annales-de-linstitut-henri-poincare-accepted-papers/
A recording of the talk can be found here: https://youtu.be/Pfk5tzHBjLc
Guilherme Silva (Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo)
March 14, 2022
Multiplicative statistics of random matrices and the integro-differential Painlevé II equation
We consider a large family of multiplicative statistics of eigenvalues of hermitian matrix models. We prove that they converge to an universal multiplicative statistics of the Airy2 point process which, in turn, is described in terms of a particular solution to the integro-differential Painlevé II equation (shortly int-diff PII). The same solution to this int-diff PII appeared for the first time in the description of the narrow wedge solution to the KPZ equation, so our results connect the KPZ equation in finite time with random matrix theory in an universal way.
We work under a one-cut regular assumption on the potential, and also under mild and natural assumptions on the multiplicative statistics. But as we also plan to explain, our approach indicates that families of integrable systems other than the int-diff PII may appear when considering multiplicative statistics associated with critical potentials.
The talk is based on joint work with Promit Ghosal (MIT).
A recording of the talk can be found here:
Roozbeh Gharakhloo (Colorado State University)
March 7, 2022
Phase Diagram and Topological Expansion in the Complex Quartic Random Matrix Model
Chiara Franceschini (University of Modena)
February 21, 2022
Algebraic approach to stochastic duality for Markov processes
In this talk I will give a review of the notion of duality theory for Markov processes and, in particular, interacting particle systems. I will show the connection with Lie algebras via some examples and conclude with some applications in the context of hydrodynamic limit and the characterization of the non equilibrium stationary measure.
A recording of the talk can be found here:
Ryosuke Odoi (Waseda University)
February 7, 2022
The constant problem of the tt*-Toda equations
The tt* (topological-anti topological fusion) equations arose in the work of Cecotti and Vafa on supersymmetric quantum field theory, and the tt*-Toda equations are a special case of these equations. Solutions of the tt*-Toda equations can be considered as certain isomonodromic deformations of meromorphic connections. They can be parametrized by two kinds of data. One comes from the asymptotic behavior of the solutions, and the other one comes from the monodromy including Stokes matrices. Two kinds of data correspond to each other via the Riemann-Hilbert correspondence. We will see that this correspondence is symplectic and give a generating function explicitly, and we will see an application to the constant problem.
A recording of the talk can be found here: https://youtu.be/HwtI1gSli_g
Giulio Ruzza (Université Catholique de Louvain)
January 31, 2022
Airy process and Korteweg-de Vries equation
I will report on joint works with M. Cafasso, C. Charlier, T. Claeys, in which we study a new class of Korteweg-de Vries solutions. They are built out of certain multiplicative expectations of the Airy point process and they generalize the self-similar solution associated with the Hastings-McLeod Painlevé II transcendent; in general they are associated with a specific solution, again characterized by an Airy limiting behavior, of an "integro-differential" generalization of the Painlevé II equation. The solutions are unbounded and the classical scattering-inverse scattering theory cannot be applied; however, they can be characterized through a Riemann-Hilbert problem, which allows to study rigorously and precisely their small time asymptotics, which we do uniformly in the space variable. A special case of the construction provides refined tail asymptotics for a specific solution ("narrow-wedge solution") of the Kardar-Parisi-Zhang stochastic equation. Depending on time I will comment on some more recent generalizations.
A recording of the talk can be found here:
Arieh Iserles (University of Cambridge)
January 24, 2022
Fast approximation on the real line
Louise Gassot (ICERM)
January 10, 2022
Zero-dispersion limit for the Benjamin-Ono equation on the torus.
We discuss the zero-dispersion limit for the Benjamin-Ono equation on the torus given a single well initial data. We prove that there exist approximate initial data converging to the initial data, such that the corresponding solutions admit a weak limit as the dispersion parameter tends to zero. The weak limit is expressed in terms of the multivalued solution of the inviscid Burgers equation obtained by the method of characteristics. We construct our approximation by using the Birkhoff coordinates of the initial data, introduced by Gérard, Kappeler and Topalov. In the case of the cosine initial data, we completely justify this approximation by proving an asymptotic expansion of the Birkhoff coordinates.
A recording of the talk can be found here:
FALL 2021
Manuela Girotti (Saint Mary's University)
November 29, 2021
Asymptotic Analysis of the Interaction Between a Soliton and a Regular Gas of Solitons (a.k.a. Gulliver and the Lilliputians)
We analyze the case of a (dense) mKdV soliton gas and its large time behaviour in the presence of a single tracer soliton. The solution, which can be expressed in terms of Fredholm determinants, can be decomposed as the sum of the background gas solution (an elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the kinetic velocity equations and the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves.
This is a joint work with Tamara Grava (U. Bristol/SISSA), Bob Jenkins (UCF), Ken McLaughlin (CSU) and Alexander Minakov (U. Karlova).
A recording of the talk can be found here: https://youtu.be/h3jxyYUaGMQ
Oleg Lisovyi (Université de Tours)
November 22, 2021
From Painlevé functions to Heun accessory parameters and back
The reconstruction of linear ODEs from their monodromy leads to two interesting classes of special functions: Heun accessory parameters (for the simplest 2nd order scalar ODEs) and Painlevé functions (for the simplest 1st order 2 by 2 linear systems). I will discuss several approaches to computation of these functions, such as Hill and Widom determinant, continued fractions and combinatorial series. After recalling a classical relation between Heun and Painlevé equations, I will explain how it leads to identities expressing Heun accessory parameters in terms of Painlevé functions and vice versa.
A recording of the talk can be found here:
Christopher Lustri (Macquarie University)
November 15, 2021
Exponential Asymptotics in Discrete Equations
Stokes' Phenomenon refers to apparently discontinuous changes in exponentially small asymptotic terms in different regions of the complex plane separated by Stokes curves. Studying this change directly is challenging, as conventional asymptotic series methods cannot capture exponentially small terms. I will discuss exponential asymptotic methods, which are able to examine these effects. These methods reveal that Stokes' Phenomenon is not discontinuous; it is a smooth effect that occurs in the neighbourhood of Stokes curves.
I will show how exponential asymptotic methods can be extended to study discrete problems. Using these ideas, I will show how Stokes' Phenomenon can be studied certain solutions to the first discrete Painlevé equation, and a few other related discrete problems. I will finish by speculating on how I hope to exploit integrability to improve this type of analysis.
A recording of the talk can be found here: https://youtu.be/KNTXCkkTVUs
Maxim Derevyagin (University of Connecticut)
November 1, 2021
Discrete Darboux Transformations And Orthogonal Polynomials
Two basic discrete Darboux transformations in the theory of orthogonal polynomials are called Geronimus and Christoffel transformations. The consistency relation for those two gives the discrete Toda equation, a discrete integrable system, and it can also be considered as a relation between the elements of the Padé table.
In this talk, we are going to review the basics of discrete Darboux transformations for orthogonal polynomials. Then we'll show how such transformations can lead to Sobolev orthogonal polynomials, exceptional orthogonal polynomials, and indefinite orthogonal polynomials. Some associated asymptotic results for orthogonal polynomials and convergence results for underlying Padé approximants will be presented as well.
A recording of the talk can be found here:
Brian Simanek (Baylor University)
October 25, 2021
Universality Limits for Orthogonal Polynomials
We will consider the scaling limits of polynomial reproducing kernels for measures on the real line. For many years there has been considerable research to find the weakest assumptions that one can place on a measure that allows one to prove that these rescaled kernels converge to the sinc kernel. Our main result will provide the weakest conditions that have yet been found. In particular, it will demonstrate that one only needs local conditions on the measure. We will also settle a conjecture of Avila, Last, and Simon by showing that convergence holds at almost every point in the essential support of the absolutely continuous part of the measure.
A recording of the talk can be found here: https://youtu.be/9_NhfLEBQGo
Jeffrey Oregero (MSRI)
October 11, 2021
The focusing nonlinear Schrödinger equation on the circle: spectral theory, elliptic finite-gap potentials, and soliton gases.
One of the prototypical integrable nonlinear evolution equations is the nonlinear Schrödinger (NLS) equation, which is a universal model for weakly nonlinear dispersive wave packets, and as such it arises in a variety of physical settings, including deep water, optics, acoustics, plasmas, condensed matter, etc. A key role in many studies of the NLS equation is played by the Zakharov-Shabat (ZS) spectral problem. This is because the associated ZS operator, which is a first-order matrix differential operator, makes up the first half of the Lax pair of the NLS equation. There are two variants of the NLS equation, referred to as focusing and defocusing, respectively. The corresponding ZS operators are also referred to as focusing and defocusing. In optics, the focusing NLS equation arises when the refraction increases with increasing wavelength, i.e., in the case of anomalous dispersion. Solutions of the focusing and defocusing NLS equation have very different physical behavior. In turn, these differences reflect a markedly different mathematical structure. In particular, the ZS operator for the defocusing NLS equation is self-adjoint, while that for the focusing NLS equation is not. In this seminar I will discuss (i) the existence of an explicit two-parameter family of elliptic finite-gap potentials of the focusing ZS operator, and (ii) soliton gases in the semiclassical limit of the focusing NLS equation on the circle.
A recording of the talk can be found here:
Fudong Wang (University of Central Florida)
October 4, 2021
A dbar-steepest descent analysis for the long-time asymptotic behavior of oscillatory Riemann-Hilbert problems
We will discuss the long-time asymptotic behavior of oscillatory Riemann-Hilbert problems (RHPs) arising in the mKdV hierarchy (reducing from the AKNS hierarchy). Our analysis is based on the idea of dbar-steepest descent. We will consider RHPs generated from the inverse scattering transform of the AKNS hierarchy with weighted Sobolev initial data. The asymptotic formula for three regions (Oscillating, fast decaying, Painleve region) of the spatial and temporal dependent variables will be presented.
A recording of the talk can be found here: https://youtu.be/FlqG23dk9sg
Patrik Nabelek (Oregon State University)
September 27, 2021
Weakly Nonlinear Longwave Models and Solitons
One of the prototypical integrable nonlinear evolution equations is the nonlinear In this talk I will discuss (1+1) D completely integrable partial differential equations that; in particular, I will consider the Korteweg--de Vries equation and the Kaup--Broer system with both weak and strong capillarity. I will discuss solitary waves or 1-soliton solutions to these equations, discuss a nonlinear superposition principal that allows the calculation of what are called N-soliton solutions, and then consider various classes of solutions that can be produced as a limit of these solutions and N goes to infinity. I will also discuss the periodic and quasiperiodic solutions. I will discuss recent approaches based on the theory of Riemann--Hilbert problems and singular integral equations. I will give some motivation for this work based on the goal of modeling a soliton gas, and provide a brief literature review of some recent experimental results on soliton gasses in hydrodynamics.
A recording of the talk can be found here:
Mateusz Piorkowski (MSRI)
September 13, 2021
Long-time asymptotics of KdV dispersive shock wave via Riemann-Hilbert problems
In this talk we will summarize a recent paper on the KdV equation with steplike initial data. The focus lies on the Deift-Zhou nonlinear steepest descent analysis in the transition region, where solutions converge to a modulated elliptic (Its-Matveev) solution. We state the corresponding Riemann-Hilbert problem, as well as the global parametrix (model) problem. Surprisingly, the global parametrix problem has in general no matrix valued solution. We thus have to rely on a vector-valued model solution and compare it directly to the exact solution. For this we rely on the work of Zhou on Fredholm index theory for singular integral operators.
A recording of the talk can be found here: https://youtu.be/VtUjTOjBjwE
WINTER 2021
Nikos Hatzizisis (University of Crete)
April 19, 2021
A semiclassical WKB problem for the Dirac operator with a decaying potential
We shall be interested in the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a fairly smooth -but not necessarily analytic- potential decaying at infinity. In particular, using ideas and methods going back to Langer and Olver, we provide a rigorous semiclassical analysis of the scattering coefficients, the Bohr-Sommerfeld condition for the location of the EVs and their corresponding norming constants.
A recording of the talk can be found here: https://youtu.be/cb5MZfwdckw
Guilherme Silva (Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo)
April 12, 2021
Limiting distribution of periodic TASEP
We discuss some recently found results on the one-point distribution of the periodic TASEP. In particular, we discuss similarities and distinctions between the integrable structures of this distribution and its non-periodic counterpart, the Tracy-Widom distribution.
A recording of the talk can be found here: