The quantum Bruhat graph, initially introduced by Brenti, Fomin, and Postnikov, is a weighted directed graph defined on finite Weyl groups. It serves as a valuable tool for exploring the quantum cohomology ring of the flag variety. In this presentation, we present a combinatorial formula for the minimal weights between any pair of permutations within the quantum Bruhat graph. Furthermore, for an ordered pair of permutations $u$ and $v$, we introduce the tilted Richardson variety $T_{u,v}$ demonstrating its equivalence to the two-pointed curve neighborhood of opposite Schubert varieties $X_u$ and $X^v$ in the minimal degree $d$. We establish a Deodhar-like decomposition for tilted Richardson varieties, leveraging it to prove several results. This is joint work with Shiliang Gao and Yibo Gao.

The use of randomness in Ramsey Theory has been a key tool in non-constructive lower bounds for Ramsey Numbers.

In this talk, I will describe a new direction in Ramsey Theory, which employs pseudorandom graphs instead of random graphs, and leads to breakthroughs on long-standing open problems in the area.

The dynamics of a billiard ball on a triangular table can be studied by considering geodesic trajectories on an associated singular flat metric structure called a translation surface when the angles of the triangle are commensurable with pi. In the case of the isosceles right triangle, this surface is a torus, whose geodesic trajectories in any direction are either all periodic or all uniquely ergodic. Triangles satisfying such a dichotomy are called lattice triangles, and our work contributes to the ongoing classification of such triangles. We make use of a number-theoretic criterion of Mirzakhani and Wright to classify such triangles with a large obtuse angle. This work is joint with Anne Larsen and Chaya Norton.

Consider a birth and death process in which each individual gives birth at rate $\lambda$ and dies at rate $\mu$, so that the population size grows at rate $r = \lambda - \mu$. Lambert (2018) and Harris, Johnston, and Roberts (2020) came up with methods for describing the exact genealogy of a sample of size $n$ taken from this population after time $T$. We use the construction of Lambert, which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with the sample. This allows us to derive point and interval estimates for the growth rate $r$, which are valid when $T$ and $n$ are large. We apply this method to the problem of estimating the growth rate of clones in blood cancer. This is joint work with Kit Curtius, Brian Johnson, and Yubo Shuai.

The goal of this talk is to discuss new developments of random geometry.  We will focus on small fluctuations (small balls) for degenerate diffusions and their connection to sub-Riemannian geometry. In particular, such diffusions can be used to describe spectral properties of their (hypoelliptic) generators, where the lack of ellipticity makes the analytic approach more challenging. Moreover, we will discuss random loops on Riemann surfaces which can be described in terms of SLEk loop measures. These are measures on the space of simple loops, and we will provide their asymptotics on small balls. Lastly, we will focus on the geometry of random Laplace eigenfunctions on the sphere and their application to physics and statistics.

Each site x in Z is initially occupied by a particle of color -x. Across each bond (x,x+1) particles swap places at rate 1 or q<1 depending on whether they are in reverse order (e.g. color 2 then 1) or order (color 1 then 2). This process describes a bijection of Z-->Z which starts maximally in reverse order and randomly drifts towards being ordered. Another name for this model is the "colored asymmetric simple exclusion process". I will explain how to use the Yang-Baxter equation along with techniques involving Gibbs measures to extract the space-time scaling limit of this process, as well as a discrete time analog known as the "stochastic six vertex model". The limit is described by objects in the Kardar-Parisi-Zhang universality class, namely the Airy sheet, directed landscape and KPZ fixed point. This is joint work with Amol Aggarwal and Milind Hegde.

I’ll prove and discuss the isomorphism $CG \cong \prod M_{d_i}(C)$, its abstract and concrete forms, and the relationship between the centers of the two algebras.

Given a vector field on a complex plane C^2 with polynomial coefficients one would like to know if all the integral curves of this vector field are algebraic. The problem seems to be very difficult. I will discuss an approach to this problem using the reduction to characteristic p, which produces a conjectural answer. Also I will explain the relation with the famous Grothendieck-Katz p-curvature conjecture. Probably the main beauty of the subject is its completely elementary nature, which makes it accessible to a first year student.