Schedule
Read this first:
The talk times have to be compatible with Australasian, European, and American time zones. As each of these continents has a few timezones within itself, all times below are UTC (Universal Time Coordinated, also known as GMT). In practice and during the month of June, to obtain your local time, you:
- add 1 hour for London/Lisbon
- add 2 hours for the time Paris/Rome/Brussels/etc.
- add 3 hours for Moscow
- add 9 hours for Tokyo
- add 10 hours for Melbourne
- subtract 7 hours for Los Angeles
- subtract 5 hours for Nashville/Chicago/Madison WI
- subtract 4 hours for New York/Boston
For example:
- Tuesday 7am UTC is Tuesday 9am in Paris, Tuesday 5pm in Melbourne
- Tuesday 00:00am UTC is Monday 7:00pm in Madison WI, Tuesday 2:00am in Paris, Tuesday 10:00am in Melbourne
Note on additional materials (slides, problem sets, etc.)
Mini-courses have additional materials (lecture notes, slides, problem sets, etc.) which you can find by clicking on the link of the course itself.
Slides for talks are also posted, whenever available, under the respective talk abstracts.
First week
The first week consists of a couple of mini-courses and talks held online via Zoom.
Monday June 7th (all times UTC)
23:45---00:00 Opening remarks (Note: this is just before Vadim Gorin's first mini-course below). Video
Tuesday June 8th (all times UTC)
(Note: the first two talks are on Monday evening in the US, Tuesday morning in Australia)
01:30am---02:15am Evgeni Dimitrov, Characterization of Gibbsian line ensembles
Abstract: Gibbsian line ensembles are natural objects that arise in statistical mechanics models of random tilings, directed polymers, random plane partitions and avoiding random walks/Brownian motions. In this talk I will discuss two recent results that characterize continuous Gibbsian line ensembles by their top curve and explain how they fit into a larger framework that can be used to establish KPZ universal scaling limits for line ensembles.
Slides: click here | Video: click here
07:00am---08:15am Grégory Schehr (mini-course), Some aspects of noninteracting trapped fermions at finite temperature
08:30am---09:15am Alexandre Krajenbrink, Fredholm determinants, finite-temperature Painlevé equations and a fresh look on the Zakharov--Shabat system
Abstract: As Fredholm determinants are more and more frequent in the context of stochastic integrability, I discuss in this talk the existence of a common framework in many integrable systems where they appear. This consists in a hierarchy of equations, akin to the Zakharov-Shabat system, connecting an integro-differential extension of the Painlevé II hierarchy, the finite-time solutions of the Kardar-Parisi-Zhang equation and multi-critical fermions at finite temperature.
Wednesday June 9th (all times UTC)
(Note: the first two talks are on Tuesday evening in the US, Wednesday morning in Australia)
01:30am---02:15am Alisa Knizel, Stationary measure for the open KPZ equation
Abstract: In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process and asymptotics of q-Gamma functions. This construction is a joint work with Ivan Corwin.
Slides: click here
02:30am---03:00am Social time
07:00am---7:45am Sofia Tarricone, Higher order finite temperature Airy kernels and an integro-differential Painlevé II hierarchy
Abstract: In this talk we will analyse Fredholm determinants of a finite temperature version of the higher order Airy kernels that recently appeared in the description of a model for free fermions in anharmonic trap studied by Le Doussal, Majumdar and Schehr. The main result gives an expression of these Fredholm determinants in terms of distinguished solutions of an integro-differential Painlevé II hierarchy, whose first members were recently computed by Krajenbrink. Moreover, our result generalizes the case n = 1, already studied some years ago by Amir, Corwin and Quastel. 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 a joint work with Thomas Bothner and Mattia Cafasso, avaiable at https://arxiv.org/pdf/2101.03557.pdf.
Slides: click here | Video: click here
08:00am---8:45am Matteo Mucciconi, KPZ solvable models and free fermions at positive temperature
Abstract: We report on a combinatorial construction that allows to relate marginal distributions of the q-Whittaker and the periodic Schur measures. The periodic Schur measure is a generalization of the Schur measure introduced by Borodin in 2006 and that models lozenge tilings in a cylindrical domain. Its free fermionic origin yields a nice mathematical structure and its correlations are determinantal. The q-Whittaker measure, introduced by Borodin and Corwin is another generalization of the Schur measure, which has found application in the rigorous description of KPZ models. Since mathematical properties of q-Whittaker polynomials are much more complicated than the Schur polynomials, the q-Whittaker measure is a more difficult object to handle. Using a bijective combinatorial approach we are able to relate the theories of Schur and q-Whittaker polynomials producing a remarkable correspondence between the two measures. Byproducts of our construction are determinantal and pfaffian formulas for stochastic systems in full and half space whose derivation bypasses complicated Bethe Ansatz calculations. Our arguments pivot around a combination of various theories, which had not yet been used in integrable probability, that include Kirillov-Reshetikhin crystals, Demazure modules, the Box-Ball system or the skew RSK correspondence. This is a joint work with Takashi Imamura and Tomohiro Sasamoto. https://arxiv.org/abs/2106.11913; https://arxiv.org/abs/2106.11922
Thursday June 10th (all times UTC)
07:00am---08:15am Jérémie Bouttier (mini-course), Some aspects of noninteracting trapped fermions at finite temperature
08:30am---9:15am Harriet Walsh, Multicriticality for random partitions and random unitary matrices
Abstract: We introduce multicritical measures on random partitions by tuning polynomial Hermitian Schur measures to have a 1/(2n+1) critical exponents, generalising the generic 1/3 exponent associated with KPZ universality. Their asymptotic edge distributions are higher-order analogues of the Tracy—Widom GUE distribution, which were first observed by Le Doussal, Majumdar and Schehr for the edge momenta of trapped fermions. We find explicit examples of these measures, and compute limit shapes. By relating Schur measures to unitary matrices, we study the phase transitions corresponding to these new edge statistics, and explain a connection with certain multicritical models motivated by gauge theory. Based on joint work with Dan Betea and Jérémie Bouttier.
Slides: click here | Video: click here
09:30am---10:00am Social time
Friday June 11th (all times UTC)
(Note: the first two talks are on Thursday evening in the US, Friday morning in Australia)
01:30am---02:15am Andrew Gitlin, A vertex model for LLT polynomials
Abstract: We describe a novel Yang--Baxter integrable vertex model, from which we construct a certain class of partition functions that are equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we are able to prove many properties of these polynomials, including symmetry and a Cauchy identity. We are also able to extend our techniques to construct a vertex model for super-symmetric LLT polynomials. This is based on joint work with Sylvie Corteel, David Keating, and Jeremy Meza.
Slides: click here
08:00am---09:15am Jérémie Bouttier & Grégory Schehr (office hour/discussion session), Some aspects of noninteracting trapped fermions at finite temperature
09:30am---10:15am Nikos Zygouras, Geometric Burge correspondence, polymer replicas and symmetries
Abstract: The Burge correspondence is a variant of the better known Robinson--Schensted--Knuth (RSK) correspondence. We construct a geometric lifting of the Burge correspondence as a composition of local birational maps, relate it to the already established geometric lifting of RSK and use it to study the law of log-gamma polymers in a persymmetric environment. The latter leads also to the computation of the law of polymer replicas as well as to a distributional symmetry between polymers in a symmetric and in a persymmetric log-gamma environment. Based on joint work with Elia Bisi and Neil O’Connell.
Second week
The second week consists of one mini-course and a few talks, all via Zoom.
Tuesday June 15th (all times UTC)
(Note: the first talk is on Monday evening in the US, Tuesday morning in Australia)
00:00am---00:45am GaYee Park, Naruse hook formula for linear extensions of mobile posets
Abstract: Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. We give a Naruse type hook-length formula to count linear extensions of such posets for a major index $q$-analogue. We also give an inversion index $q$-analogue of the Naruse formula for mobile tree posets.
Slides: click here | Video: click here
01:00am---01:30am Social time
Wednesday June 16th (all times UTC)
(Note: the two talks are on Tuesday evening in the US, Wednesday morning in Australia)
00:00am---00:45am Marianna Russkikh, Lozenge tilings and the Gaussian free field on a cylinder
Abstract: We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured by Gorin for tiling models on planar domains with holes. This talk is based on joint work with Andrew Ahn and Roger Van Peski.
Slides: click here | Video: click here
01:00am---01:45am Travis Scrimshaw, Refined dual Grothendieck polynomials and last-passage percolation
Abstract: The last passage percolation (LPP) process is taking the maximum over all paths in a random matrix. Here we will consider the random matrix entry $w_{ij}$ to have a geometric distribution with parameter t_i x_j. When t_i = x_j = q^{1/2}, then this corresponds to measuring the movement of particles in the totally asymmetric simple exclusion process. Recently, Yeliussizov constructed a bijection between the random matrices and plane partitions, which showed that the (up to a renormalization factor) distribution is given by a specialized refined dual Grothendieck polynomial, which arose from geometry through K-theoretic Schubert calculus. In this talk, we will show that Yeliussizov's bijection holds for the general parameter case and that skew refined dual Grothendieck polynomials describe the transition probabilities. We provide a number of identities involving skew refined dual Grothendieck polynomials. This is joint work with Kohei Motegi https://arxiv.org/abs/2012.15011.
Thursday June 17th (all times UTC)
(Note: the first two talks are on Wednesday evening in the US, Thursday morning in Australia)
00:00am---00:45am Alejandro Morales, On the Okounkov--Olshanski formula for standard tableaux of skew shapes
Abstract: The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two q-analogues for reverse plane partitions, which complements work by Stanley and Chen for semistandard tableaux. We also give applications and several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. This is joint work with Daniel Zhu.
Slides: click here
01:00am---01:45am Weiying Guo, Comparison of non-symmetric Macdonald polynomials
Abstract: Non-symmetric (symmetric) Macdonald polynomials have been known to arise from many different aspects in mathematical physics, combinatorics and also probability theory i.e., the vertex model and the asymmetric simple exclusion process. In this talk, we are going to give an aspect of the non-symmetric Macdonald from the double affine Hecke algebra (DAHA) and how various formulas of the non-symmetric Macdonald polynomials are related i.e., the alcove walk formula (RY08) and the non-attacking filling formula (HHL07). This work is joint with Arun Ram.
Slides: click here
08:30am---09:15am William Mead, The two-species totally asymmetric simple exclusion process
Abstract: The asymmetric simple exclusion process is a stochastic model of indistinguishable particles with KPZ-like limiting behaviour. Very few mathematically rigorous results are known for the limiting behaviour of multi-species models. In this talk we consider a model in which particles of a second species may overtake the other species, which is related to a higher rank analogue of the stochastic six-vertex model. We give an exact integral formula for the transition probability on the infinite one-dimensional lattice, and from this derive expressions for bulk crossing probabilities and discuss their implications. Based on joint work with Jan de Gier and Michael Wheeler.
Friday June 18th (all times UTC)
00:00---00:45am Valentin Buciumaș, Integrable models and p-adic representation theory
Abstract: I will start with a very brief overview on the representation theory of p-adic groups. The objective is to explain how special functions like Schur polynomials, Hall-Littlewood polynomials and their generalizations appear in this context. We will briefly go over the Satake isomorphism and the Casselman Shalika formula. I will then present certain integrable systems and explain how they may be used to study such special functions. The focus will be on metaplectic Whittaker functions for GL_r over a p-adic field. This is joint work with Brubaker, Bump and Gustafsson.
Slides: click here | Video: click here
01:00---01:45am Alexandr Garbali, Symmetric functions and quantum toroidal gl_1
Abstract: The basic representation of the quantum toroidal algebra gl_1 is given on the space of symmetric functions. This representation can be expressed either by the action of the algebra in the basis of Macdonald functions or in the basis of power sum symmetric functions. After explaining these details I will pass to the associated integrable vertex model which is defined by the R-matrix. I will then discuss connections between the R-matrix and Macdonald operators.