Talks

We show that the partial sums of the long Plucker relations for pairs of weakly separated Plucker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher–Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat–Vishwakarma (2023). In fact we obtain a characterization of weak separability, by showing that no other pair of Plucker coordinates satisfies this property. Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley–Lieb immanants, and Plucker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. This is joint work with Soskin.

Condensation of topological defects is the foundation of the modern theory of bulk-boundary correspondence, also known as topological holography. In this talk, I discuss string condensation in 3+1D topological orders, which plays a role analogous to anyon condensation in 2+1D topological orders. I will demonstrate through examples how they correspond to 2+1D symmetry enrichd phases, including both gapped and gapless phases. Then I give a detailed analysis of string condensaiton in 3+1D discrete gauge theories. I compute the outcome of the condensation, namely the category of excitations surviving the condensation. The results suggest that a complete topological holography for 2+1D phases can only be established by taking into account all possible ways of condensing strings in the bulk 3+1D topological order.

The matter power spectrum on subgalactic scales is very weakly constrained so far. While inflation predicts a nearly scale-invariant primordial power spectrum down to very small scales, many new physics scenarios can lead to significantly different predictions, such as axion dark matter in the post-inflationary scenario, vector dark matter produced during inflation, early matter domination, kinetic misalignment axions, self-interacting dark matter, atomic dark matter, etc. Therefore, any successful measurement on the matter power spectrum tests inflation extensively and probes early universe dynamics and the nature of dark matter, making it a new frontier in cosmology and dark matter physics. We proposed observing fast radio bursts (FRB) with solar-system scale interferometry by sending radio telescopes to space, which allows us to greatly expand the sensitivity on the matter power spectrum from Mpc to AU scales. Two sightlines looking at the same FRB source can sample different regions of the Universe in the transverse direction and thus obtain an arrival time difference that depends on the matter power spectrum. Our calculations show that this setup will be sensitive to the scale-invariant power spectrum predicted by inflation on small scales and can also probe QCD axion miniclusters predicted in the post-inflationary scenario.

On April 4th, 2024, the Dark Energy Spectroscopic Instrument (DESI) released its first set of cosmological results based on measurements of the baryon acoustic oscillations (BAO) scale in the spatial distribution of galaxies and quasars, and in the Lyman-alpha forest. Those measurements constrain the expansion history of the Universe in the redshift range 0.1 < z < 4.16, with implications for studies of dark energy, neutrino cosmology and the Hubble constant. To make the most of the cosmological information content of the galaxy & quasar distributions, we now analyze the full shape of their power spectra, constraining the evolution of the large scale structure of the Universe over the range 0.1 < z < 2.1. In this talk, following a brief overview of the DESI instrument and observations, we will present the latest public results, discuss some of their main cosmological implications, highlight efforts towards the full shape results and expectations for the next data release.

 Our Cosmological Standard Model, LambdaCDM, is a remarkable success story. It describes our Universe’s evolution from the Big Bang until today in terms of only a small handful of parameters. Despite its many successes, LambdaCDM is not a fundamental theory. In particular, the microscopic origin of dark matter and dark energy remain among the greatest puzzles in modern physics. Of the two, dark energy poses a particularly vexing challenge, as we lack an understanding of the smallness of its value. At the same time, over the last decade, observations have revealed further cracks in the LambdaCDM model, manifesting as discrepancies between early and late universe determinations of its parameters.

In this lecture, I will first review the LambdaCDM model and establish why it is considered our best model of the Universe. In the second part, I will discuss the intriguing possibility that the cosmic tensions, referring to the observational and theoretical challenges mentioned above, are si...

Totally positive (TP) matrices are matrices in which each minor is positive. First introduced in 1930's by I. Schoenberg and F. Gantmakher and M. Krein, these matrices proved to be important  in many areas of pure and applied mathematics. The notion of total positivity was generalized by G. Lusztig in the context of reductive Lie groups and inspired the discovery  of cluster algebras by S. Fomin and A. Zelevinsky.

In this mini-course, I will first review some basic features of TP matrices, including their spectral properties and discuss some of their classical applications. Then I will focus on weighted networks parametrization of TP matrices due to A. Berenstein, S. Fomin and A. Zelevinsky. I will show how elementary transformations of planar networks lead to criteria of total positivity and important examples of mutations in the theory of cluster algebras. Finally, I will explain how particular sequences of mutations can be used to construct exactly solvable nonlinear dynamical sy...

A colored vertex model is a solution of the Yang--Baxter equation based on a higher-rank Lie algebra. These models generalize the famous six-vertex model, which may be viewed in terms of osculating lattice paths, to ensembles of colored paths. By studying certain partition functions within these models, one may define families of multivariate rational functions (or polynomials) with remarkable algebraic features. In these lectures, we will examine a number of these properties:
(a) Exchange relations under the Hecke algebra;
(b) Infinite summation identities of Cauchy-type;
(c) Orthogonality with respect to torus scalar products;
(d) Multiplication rules (combinatorial formulae for structure constants).

Our aim will be to show that all such properties arise very naturally within the algebraic framework provided by the vertex models. If time permits, applications to probability theory will be surveyed.

Combinatorics has constantly evolved from the mere counting of classes of objects to the study of their underlying algebraic or analytic properties, such as symmetries or deformations. This was fostered by interactions with in particular statistical physics, where the objects in the class form a statistical ensemble, where each realization comes with some probability. Integrable systems form a special subclass: that of systems with sufficiently many symmetries to be amenable to exact solutions. In this talk, we explore various basic combinatorial problems involving discrete surfaces, dimer models of cluster algebra, or two-dimensional vertex models, whose (discrete or continuum) integrability manifests itself in different manners: commuting operators, conservation laws, flat connections, quantum Yang-Baxter equation, etc. All lead to often simple and beautiful exact solutions.

It is well known that the box-ball system discovered by Takahashi and Satsuma can be obtained by the ultra-discrete analogue of the discrete integrable system, including both the ultra-discrete analogue of the KdV lattice and the ultra-discrete analogue of the Toda lattice. This mini-course will demonstrate that it is possible to derive extended models of the box-ball systems related to the relativistic Toda lattice and the fundamental Toda orbits, which are obtained from the theory of orthogonal polynomials and their extensions. We will first introduce an elementary procedure for deriving box-ball systems from discrete KP equations. Then, we will discuss the relationship between discrete Toda lattices and their extensions based on orthogonal polynomial theory, and outline the exact solutions and ultra-discretization procedures for these systems. Additionally, we will introduce the box-ball system on R, which is obtained by clarifying its relationship with the Pitman transformation in ...