SCHEDULE 

Winter Term 1  (September - December 2026)

Sept 15, 2026:

Speaker: Jinting Liang (UBC)

Title: Log-Concavity and Log-Convexity via Distributive Lattices

Abstract: The FKG inequality is a powerful tool for proving inequalities in distributive lattices. We show how a special case, which we call the Order Ideal Lemma, can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving binomial coefficients, Catalan numbers, Fibonacci numbers, Stirling numbers of the second kind, order polynomials, and more. In the process we define some new and interesting distributive lattices. This is joint work with Bruce Sagan.

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Sept 22, 2026:

Speaker: John Lentfer (UC San Diego)

Title: Type B bosonic-fermionic coinvariant rings

Abstract: In the 1990's Garsia and Haiman introduced the diagonal coinvariant ring, which generalizes the classical coinvariant ring (the cohomology ring of the flag variety) to two sets of commuting variables. Since then, there has been much interest in studying generalizations with k sets of n commuting variables and j sets of n anticommuting variables, called a bosonic-fermionic coinvariant ring R_n^{(k,j)}, for various choices of small k and j. There is also an analogous story for the Weyl group of type B_n (the hyperoctahedral group); denote the type B bosonic-fermionic coinvariant ring by R_{B_n}^{(k,j)}.

In this talk, we will learn about the history of these families of coinvariant rings, paying attention to when the story is the same in both types A and B, and when they diverge. New results include an explicit formula for the GL(2) x B_n module structure of R_{B_n}^{(0,2)}, the sign character of R_{B_n}^{(0,3)}, and the standard characters in types A and B. This is based on joint work with Yuhan Jiang (UC Berkeley).

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Sept 29, 2026:

Speaker: Josh Turner (UBC)

Title:  A filtered basis for Haiman ideals

Abstract: We will talk about the ideal J generated by antisymmetric polynomials in two sets of variables and its powers, and discuss how these Haiman ideals are connected to link homology, geometry, and other combinatorial objects of interest (including diagonal coinvariants and q,t-Catalan numbers). We will then show how to reinterpret recursions for computing link homology into a method for constructing a free, filtered basis for J^d over C[y].

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Oct 6, 2026:

Speaker: Shubham Sinha (UBC)

 Title: Twisted quantum cohomology of Grassmannians

Abstract: Schubert calculus describes the cohomology ring of a Grassmannian Gr(r,n). Quantum cohomology is a q-deformation of this ring that encodes refined geometric information beyond the intersections of Schubert cycles. Its structure constants involve combinatorial tools such as Littlewood–Richardson coefficients and cores of partitions. In this talk, I will introduce subset sums of Schur polynomials that encode the combinatorics of the cohomology of Gr(r,n), as well as related (twisted) deformations of the ring. Time permitting, I will briefly describe the geometric significance of our construction. We further specialize these polynomials at the nth roots of unity and prove identities that are of independent interest. This talk is based on ongoing joint work with Arvind Ayyer.

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Oct 13, 2026:

No seminar

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Oct 20, 2026:

Mikhail Mazin (Kansas State)

Title: Hilbert schemes of monomial ideals and generalized q,t-Catalan polynomials.

Abstract: The classical q,t-Catalan polynomials can be computed as enumerations of subpartitions of staircase partitions with respect to the area and dinv statistics. The triangular q,t-Catalan polynomials are a generalization where the staircase partitions are replaced with more general triangular partitions. The celebrated Shuffle Theorems provide a connection between the (triangular) q,t-Catalan polynomials and the action of the so-called Negut's elements of the elliptic Hall algebra on symmetric functions. In turn, this allows one to generalize the q,t-Catalan polynomials beyond the case of triangular partitions. However, the formula obtained from the elliptic Hall algebra action is not a positive sum of monomials, but rather a complicated sum of rational functions. Nevertheless, it is known that for any partition lambda one obtains a polynomial with integer coefficients, and its specialization at t=1 counts subpartitions of lambda with respect to area. Furthermore, computer computations suggest that if lambda is concave, then the coefficients of the polynomial are non-negative. A natural question is then to generalize the second statistic, dinv, to the case of concave partitions.

We propose a geometric approach to generalizing the dinv statistic. Given a concave partition lambda we consider the Hilbert scheme of points of the monomial ideal corresponding to lambda. These Hilbert schemes are highly singular. However, they are embedded into the Hilbert scheme of points on the complex plane. Intersecting with Bialynicki-Birula cells, one obtains a decomposition enumerated by subpartitions of lambda. With some luck, one can hope that the pieces of the decomposition are affine cells and then use their dimensions to generalize the dinv statistic. If lambda is triangular, then this construction recovers the usual dinv statistic. In addition, the construction provides a generalization of the dinv statistic for the following families of partitions lambda: 1) hooks; 2) partitions with at most three rows; 3) partitions (k,2,1,1,...), i.e. hooks with an extra box (1,1). (Case 3) was developed Jacob Bucciarelli in his PhD thesis.)

In this talk I will introduce the Negut's operators, Shuffle Theorems, and the generalized q,t-Catalan polynomials. I will then turn to the proposed geometric model: the Hilbert schemes of monomial ideals, focusing mostly on the three series of cases mentioned above.

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Oct 27, 2026: 

Joint AG/DM Seminar: Katrina Honigs (SFU)

Nov 3, 2026:

Olga Silina (UBC)

Nov 10, 2026:  Midterm Break (no seminar)

Nov 17, 2026: 

Nov 24, 2026:

Jose Simental (UNAM)

Dec 1, 2026: 

 Eugene Gorsky (UC Davis)