Autumn 2025

Seminar time: Thursdays 1:50-2:45pm.
Location: Math Tower MW154

October 9: Shukun Wu (Indiana University Bloomington)

Title: Bilinear estimates in F_p with application in Roth-type result

Abstract: We improve an L^2 × L^2 -> L^2 estimate for a certain bilinear operator in the finite field of size p, where p is a prime sufficiently large. Our method carefully picks the variables to apply the Cauchy-Schwarz inequality. As a corollary, we show that there exists a quadratic progression x,x+y,x+y2 for nonzero y inside any subset of F_p of density >= p^{−1/8}. This is joint work with Necef Kavrut.

October 16: Autumn break (no seminar 🙁 ).

October 23: No Seminar

Note: Vadim Gorin (UC Berkeley) will give a colloquium related to Combinatorics on October 23.

Location and time: Scott Lab E001, Thurs Oct 23, 3:00 – 3:55pm.

Title of the talk: Dunkl operators and random matrices

October 30: Houcine Ben Dali (Harvard University)

Title: A combinatorial formula for interpolation Macdonald polynomials

Abstract: In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials,  defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials.  Like the Macdonald polynomials, these interpolation Macdonald polynomials are closely connected to the Hecke algebra, and admit nonsymmetric versions, which generalize the nonsymmetric Macdonald polynomials.  I will present a combinatorial formula for interpolation Macdonald polynomials in terms of signed multiline queues.  This formula generalizes the combinatorial formula for Macdonald polynomials in terms of multiline queues given by Corteel–Mandelshtam–Williams.  This is based on a joint work with Lauren Williams.

November 6: Cosmin Pohoata (Emory University) (Cancelled, the talk will be rescheduled to Spring 2026)

This is a joint Probability-Combinatorics seminar! Notice the unusual time!

Time and place: 10:20 — 11:15 at the usual Math Tower MW 154.

November 13: Marcus Michelen (Northwestern University)

This is a joint Probability-Combinatorics seminar!

Time and place: 10:20 — 11:15 at the usual Math Tower MW 154.

Title: Separation of roots of random polynomials

Abstract: Let f_n be a degree n polynomial with independent and identically distributed coefficients.  What do its roots tend to look like?  Classical results of Erdos-Turan and others tell us that most roots are near the unit circle and that they are approximately rotationally equidistributed.  Qualitatively, the roots tend to “repel” each other.  In order to quantify this repulsion, we will discuss the smallest distance m_n between pairs of roots and prove that m_n is of order m^{-5/4} and satisfies a non-degenerate limit theorem when rescaled.  This is based on joint work with Oren Yakir.

November 20: Tao Jiang (Miami University, OH) 

Title: Balanced supersaturation and its applications to enumeration problems

Abstract: In extremal combinatorics, supersaturation problems study how many copies of a given substructure F there are when a host set system is sufficiently dense. Recently, a lot of attention was focused on a balanced variant of the problem in which we wish to find a collection of copies of F that is not only dense but also covers every subset of edges “uniformly”. Such balanced supersaturation results have been used in conjunction with the powerful container method to yield much progress on the problem of enumerating F-free graphs or hypergraphs.

In this talk, we discuss two recent results on the enumeration problem obtained via the balanced supersaturation approach. Given an r-graph F, an r-graph G is F-free if it does not contain F as a subgraph. Let {\rm ex}_r(n,F) denote the maximum number of edges in an n-vertex F-free r-graph and let {\rm forb}(n,F) denote the number of n-vertex F-free r-graphs. We show for r\geq 4, {\rm forb} (n,F)=2^{(1+o(1)){\rm ex}_r(n,F)} for a large family of r-partite r-graphs known as 2-contractible hypertrees. This is the first known family of r-partite r-graphs achieving such a bound. For r\geq 5, it answers a conjecture of Balogh, Narayanan, and Skokan on the number of r-graphs not containing a given linear cycle. This is joint work with Sean Longbrake.

Let H be any tree poset of height k. Let La^*(n,H) denote the maximum size of a subfamily of the boolean lattice B_n that does not contain H as an induced subposet. Let {\rm forb^*}(n,H) denote the number of induced P-free subfamily of B_n. We show that {\rm forb^*}(n,H)=2^{(k-1+o(1))\binom{n}{n/2}}, and that with high probability the largest induced H-free subfamily of a p-random subfamily of B_n, for p\gg n^{-1} has size (k-1+o(1))p\binom{n}{n/2}. These strengthen or extend earlier work of Balogh, Garcia, and Wigal, and of Balogh, Mycroft, and Treglown, and of Collares and Morris. This is joint work with Sean Longbrake, Sam Spiro, and Liana Yepremyan.

November 27: Thanksgiving break (no seminar 🙁 ).