The UC Berkeley Combinatorics Seminar

DATE	SPEAKER	TITLE (click to show abstract)
September 3rd	Yuhan Jiang, UC Berkeley	The Ehrhart series of alcoved polytopes Alcoved polytopes are convex polytopes that are the closure of a union of alcoves in an affine Coxeter arrangement. They are rational polytopes and, therefore, have Ehrhart quasipolynomials. Here we describe a method for computing the generating function of the Ehrhart quasipolynomial, or Ehrhart series, of any alcoved polytope via a particular shelling order of its alcoves. We also show a connection between Early's decorated ordered set partitions and this shelling order for the hypersimplex \Delta_{2,n}.Additionally, I will discuss the alcoved triangulation of positroid polytopes. This talk will be based on my work on positroid polytopes and my joint work with Elisabeth Bullock on alcoved polytopes.
September 10th	Wencai Liu, Texas A&M	Flat bands for $\mathbb{Z}^d$-periodic graph operators In this talk, I will discuss applications of algebraic geometry and combinatorics to the spectral theory of $\mathbb{Z}^d$-periodic graph operators. A key focus will be on flat bands—eigenvalues corresponding to infinite-dimensional eigenspaces. I will present a complete characterization of the $\mathbb{Z}^d$-periodic graphs whose associated operators generically exhibit no flat bands. This work is partially joint with Matt Faust.
September 17th	Fu Liu, UC Davis	Symmetrizing polytopes and posets A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset (with a reasonable structure), determine whether there exists a polytope whose face poset is this poset. In 1990s, Kapranov defined a poset, as a hybrid between the face poset of the permutohedron and the associahedron, and he asked whether this poset is realizable. Shortly after his question was posed, Reiner and Ziegler provided a realization. Recently, Castillo and I gave a different construction as a deformations of nested permutohedra. Motivated by these work, we introduce and study the G-symmetrization G(P) of an arbitrary polytope P for any reflection group G. We show that the combinatorics, and moreover, the normal fan of such a symmetrization can be recovered from its refined fundamental fan, a decorated poset describing how the normal fan of P subdivides the fundamental chamber associated to the reflection group G. Our theoretical results provide a recipe for constructing a realization of a G-symmetric poset. In particular, I will present how to use this recipe to realize Kapranov's poset as a symmetrization of a carefully embeded associahedron. This is joint work with Federico Castillo.
September 24th	Lizzie Pratt, UC Berkeley	Exterior Cyclic Polytopes and Convexity of Amplituhedra In this talk, we introduce a new polytope called the exterior cyclic polytope. Our motivation comes from particle physics, and in particular from a geometric object called the amplituhedron, which lives in a Grassmannian Gr(k, r) and appears in calculations of particle scattering. The exterior cyclic polytope is the convex hull of the amplituhedron in the ambient Plücker space of Gr(k, r). We describe its face structure and faces, which in the case k=2 are controlled by a matroid called the hyperconnectivity matroid. Furthermore, we described the dual of the k=2 and r=4 polytope in terms of the twist map of Marsh and Scott, and use this to define a notion of dual amplituhedron.
October 1st	Mitsuki Hanada , UC Berkeley	Representation Theoretic Bases for $R_{n,\lambda,s}$ The $\Delta-$Springer modules $R_{n,\lambda,s}$,  defined by Griffin, are a generalization of the Type A coinvariant ring. These rings unify the stories of the generalized coinvariant rings $R_{n,k}$, introduced by Haglund—Rhoades—Shimozono, and the Garsia—Procesi rings $R_\lambda$. We give a monomial basis of $R_{n,\lambda,s}$ consisting of generalizations of Garsia—Stanton descent monomials. The basis simultaneously generalizes the descent bases of $R_{n,k}$ and $R_{\lambda}$. The construction of this basis is deeply connected with a combinatorial object called battery-powered tableaux, introduced by Gillespie--Griffin. We highlight the representation theoretic properties of this monomial basis by using it to give a direct combinatorial proof of the graded Frobenius character of $R_{n,\lambda,s}$ in terms of battery-powered tableaux, answering a question of Gillespie—Griffin. This is joint work with R.Chou.
October 8th (No Pretalk )	Vadim Gorin, UC Berkeley	Dunkl operators and random matrices Dunkl differential-difference operators are one-parameter deformations of the usual derivative. First studied for their remarkable commutativity and their role in the Calogero-Moser-Sutherland quantum many-body system, they have recently found surprising applications in random matrix theory. In the talk we will discuss the structural combinatorial properties of the Dunkl operators, witness the emergence of Catalan numbers, and see how the operators are used in the asymptotic problems for stochastic systems.
October 15th	Gabriel Raposo, UC Berkeley	Fluctuations of random standard Young Tableaux We will introduce the Young generating function and use it to characterize the law of large numbers and the central limit theorem behaviors for random partitions. As an application of these results, we present a framework to obtain conditional Gaussian Free Field fluctuations for height functions associated with random standard Young tableau. To prove these results we develop algebraic formulas for operators on the Gelfand–Tsetlin algebra of the symmetric group.
October 22nd	Peter Bürgisser, TU Berlin	Probabilistic intersection rings of homogeneous spaces Suppose X_1, ….,X_s are submanifolds of a compact homogeneous space M, in general position, with finite intersection. We may think of M as a real or complex projective space, or a Grassmann manifold. The signed count of intersection points can be described in terms of the real cohomology algebra of M: intersection corresponds to multiplication. Integral geometry provides methods to compute the expectation of the total number of intersection points when the X_i are moved at random with respect to the Haar measure. We motivate and outline the functorial construction of a probabilistic algebra, whose multiplication mirrors the intersection of randomly moved submanifolds. The goal is to understand the expected total number of intersection points. The elements of this algebra are classes of zonoids (certain convex bodies) in the exterior algebra, and their multiplication is induced by the wedge product. This algebra contains the cohomology algebra as a direct summand. There is an intimate relationship with Alesker’s multiplication of valuations of convex bodies.
October 29th	Pranav Enugandla, UC Berkeley	Clasped Web bases for SL(4) invariants of tensor spaces In the late 90’s Kuperberg developed a web basis for the invariant space of tensor products of irreducible modules for SL(2) and SL(3), providing a diagrammatic calculus for homomorphism spaces in the representation category. Since then, webs have found applications to cluster algebras, dimer models, and quantum topology. In 2025, a web basis for tensor products of fundamental representations for SL(4) was constructed by Gaetz, Pechenik, Pfannerer, Striker, and Swanson using hourglass plabic graphs. I will talk about joint work with Christian Gaetz, in which we extend Kuperberg’s ideas to describe clasped web bases for invariants of tensor products of arbitrary irreducible SL(4) representations.
November 5th	Yelena Mandelshtam, University of Michigan	KP solitons from algebraic curves and the positive Grassmannian The Kadomtsev–Petviashvili (KP) equation is an important nonlinear PDE in the theory of integrable systems, with rich families of solutions arising both from algebraic geometry and from combinatorics. On one hand, Krichever showed how to build solutions from algebraic curves using Riemann theta functions. On the other, Kodama and Williams connected soliton solutions to the geometry of the positive Grassmannian. In this talk I will describe recent and ongoing work with various collaborators, where we study what happens when algebraic curves degenerate tropically. In this limit, theta-function solutions collapse to soliton solutions, and we can track how the geometry of the tropical curve manifests in the combinatorial structure of the soliton. This provides a new bridge between the algebro-geometric and combinatorial approaches to KP solutions.
November 12th	Grant Barkley, University of Michigan	Extended weak order Weak order is a partial order on a Coxeter group which orders elements by containment of their inversion sets. For finite Coxeter groups, weak order is a lattice, while for infinite Coxeter groups this is no longer the case. Extended weak order is a bigger poset, introduced by Matthew Dyer, which conjecturally recovers the lattice property. It is the containment order on "biclosed sets" of roots in a root system. Conjecturally, biclosed sets can be used to give generalized Bruhat decompositions and describe the lattice of torsion classes for preprojective algebras. We will discuss progress on these and other conjectures relating biclosed sets to geometry, algebra, and combinatorics.
November 19th	Laura Pierson	Orbit lengths for promotion on 2-row and near-hook tableaux Promotion has been well-studied for rectangular standard Young tableaux, in which case the orbit lengths divide the total number of boxes and are described by a cyclic sieving phenomenon (CSP), but little is known about the orbit lengths for tableaux of general shape. We approach this problem by building a stable sequence of tableaux where we fix the bottom portion and add extra boxes to the first row to get n total boxes, with n varying. We show that for 2-row tableaux with a fixed bottom row, the orbit lengths are divisors of certain monic polynomials in n, with degree generally equal to the number of distinct lengths of runs of consecutive numbers in the bottom row. For the subsets of 2-row tableaux where all runs have the same length, we show that the orbit lengths are characterized by a CSP polynomial that is a slightly modified version of the major index generating function, like in the rectangle case. We also show that for any stable sequence of tableaux, the orbit lengths are linear in n as long as all non-first-row entries differ from each other by at least 2, which asymptotically happens for almost all tableaux in the limit as n → ∞. We also calculate the orbit lengths for near-hook tableaux, which are divisors of certain linear or quadratic polynomials in n.
November 26th	No seminar - Thanksgiving
December 3rd	Dustin Ross, SFSU
December 10th

Older seminar webpages: Spring 2025 Fall 2024 Spring 2024 Fall 2023 Spring 2023 Fall 2022 Spring 2022 Fall 2021 Spring 2021 Fall 2020 2018-2020