Algebra & Combinatorics Seminar:   2025–26

Abstract.

Abstract. We generalize the seminal polynomial partitioning theorems of Guth and Katz (2015) to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n - k - r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemeredi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves.

These results, together with some of my other recent work (e.g., bounding the number of distinct distances on plane Pfaffian curves), are steps in a larger program - pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk. This talk is based on multiple joint works with Saugata Basu, Antonio Lerario, Martin Lotz, Adam Sheffer, and Nicolai Vorobjov.

Abstract. In 2017, Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of $K$-theoretic quantized Coulomb branches associated with certain 3D $N = 4$ quiver gauge theories. We describe a new geometrical construction of a deformation of one of these shifted quantum affine algebras as the Hall algebra of the category of restricted representations of the Lie algebra $\mathfrak{sl}_2$ over a finite field. The main tool we use is an equivalence of categories by Rudakov that relates the above category to that of representations over a certain quiver modulo relations. This is joint work with Peter Samuelson.

Abstract. I will give a gentle introduction to inequalities connecting symmetric polynomials and majorization. These have been studied by Maclaurin and Newton (1700s), Schlomilch (1800s), Gantmacher, Muirhead, Schur (1900s), and several others. Recently, Cuttler–Greene–Skandera and Sra (2010s) characterized majorization via inequalities involving Schur polynomials, elementary symmetric polynomials, and power sums. With Tao (2021), we analogously characterized weak majorization via Schur polynomials.

After discussing these and additional developments, I will speak about recent joint work with Hong Chen and Siddhartha Sahi, in which we conjecturally extend the above results, to characterize (weak) majorization via Jack and Macdonald polynomials. We prove these conjectures for all partitions with two parts.

Abstract. We prove an inductive formula to construct a path from the highest weight element to any given vertex in the crystal graph of the polytope realization of the Kirillov–Reshetikhin crystal $KR^{i,m}$ of type $A$ introduced by Kus in 2013. For $i \leq 2$ or $i \geq n-1$, we provide explicit formulas of the same by only using the lowering crystal operators and in those cases, using these paths, we determine the explicit image of any polytope under the affine crystal isomorphisms between the polytope and the tableau realizations of the Kirillov–Reshetikhin crystals. This is a joint work with Dipnit Biswas.

Abstract. Consider random integer partitions $\lambda$ obeying the Poissonized Plancherel Measure of parameter $t^2$. We establish, through Riemann–Hilbert techniques, asymptotics of the multiplicative averages \begin{equation} Q(t,s)=\mathbb{E}\left[ \prod_{i\ge 1} \left(1+q^{s+i-\lambda_i}\right)^{-1} \right], \end{equation} for any $q \in (0,1)$ fixed and $s,t$ large. We compute explicitly the rate function $\mathcal{F}_q(x) = - \lim_{t \to \infty} t^{-2} \log Q(t,xt)$ which is expressed in a closed form through Weierstrass elliptic functions. The equilibrium measure of such a Riemann–Hilbert problem presents, in general, saturated regions and it undergoes two third-order phase transitions of different nature, which we describe. Applications of our results include an explicit characterization of tail probabilities of the height function of the $q$-deformed polynuclear growth and of the edge of the positive temperature discrete Bessel point process.

Abstract. I will present a (cohomological) Fourier–Mukai transform for tropical abelian varieties and provide some applications, including a generalized Poincaré formula for any nondegenerate line bundle on tropical abelian varieties, generalizing Beauville's identities for complex abelian varieties. Based on joint work with Farbod Shokrieh.