Seminar time: Thursdays 1:50-2:45pm.
Location: Enarson Classroom Building EC0322.

Title: Nilsequences on General Additive Patterns

Abstract: We will begin by briefly introducing the use of higher-order Fourier analysis in additive combinatorics for a general audience. In particular, we will discuss the arithmetic regularity lemma and how it identifies a certain class of arithmetically-structured functions—nilsequences—as extremal objects for problems in additive combinatorics. We will then explore how it has recently come to light that the analysis of nilsequences on certain additive patterns—those that satisfy a certain algebraic criterion known as the flag condition—is easier than in the general case, and discuss some recent ideas and developments in overcoming the difficulties that arise when the additive pattern of interest is not flag.

September 12: No seminar

September 19: No seminar

September 26: Zihan Zhang (OSU)

Title: Combinatorics in Error-Correcting Codes: List Decodability of Reed–Solomon Codes

Abstract: This talk will introduce some key concepts and foundational goals in the theory of error-correcting codes, a crucial area in information theory, communications, and cryptography. We will begin by quickly outlining fundamental principles in coding theory. Then, our focus will then shift to the combinatorial problem of list decoding, specifically in the context of Reed–Solomon (RS) codes. We will survey some most recent breakthroughs in understanding the optimal combinatorial list decodability of RS codes. This talk is designed for both newcomers and experts in the field, offering an up-to-date perspective on one of the most actively researched areas in coding theory.

October 3: Christopher Donnay (OSU)

Title: The asymptotics of redistricting the $n\times n$ grid

Abstract: Redistricting is the act of dividing a region into districts for electoral representation. Motivated by this application, we study two questions: How many ways are there to partition the $n\times n$ grid into $n$ contiguous districts of equal size? How many of these partitions are “compact”? We give asymptotic bounds on the number of plans: a lower bound of roughly $1.41^{n^2}$ and an upper bound of roughly $3.21^{n^2}$ . We then use the lower bound to show that most plans are not compact. This is joint work with Matthew Kahle.

October 10: Fall Break, No Seminar.

October 17: Sam Hopkins (Howard)

Title: Upho posets

Abstract: A partially ordered set is called upper homogeneous, or “upho,” if every principal order filter is isomorphic to the whole poset. This class of fractal-like posets was recently introduced by Stanley. Our first observation is that the rank generating function of a (finite type N-graded) upho poset is the reciprocal of its characteristic generating function. This means that each upho lattice has associated to it a finite graded lattice, called its core, that determines its rank generating function. With an eye towards classifying upho lattices, we investigate which finite graded lattices arise as cores, providing both positive and negative results. Our overall goal for this talk is to advertise upho posets, and especially upho lattices, which we believe are a natural and rich class of posets deserving of further attention. Essentially no background knowledge will be assumed, and we also hope to highlight several open problems.

October 24: Peter Van Hintum (IAS)

Title: Stability of the Brunn–Minkowski inequality and sets of small doubling

Abstract: The Brunn–Minkowski inequality is a foundational result in convex geometry asserting that if sets A and B in $\mathbb{R}^n$ have the same volume and $t$ is some number in $(0,1)$ , then $|tA+(1-t)B|>|A|$ . We’ll prove the following folklore conjecture on the stability of the Brunn–Minkowski inequality that if in fact $|tA+(1-t)B|<(1+d) |A|$ , then there is a common convex hull $K$ containing A and B so that $|K|<(1+ O(\sqrt{d}))|A|$ and $|co(A)|<(1+O(d))|A|$ . We’ll additionally describe how this result neatly fits in the additive combinatorics study of sets with small sumset. (based on joint work with Alessio Figalli and Marius Tiba)

October 31: Yifan Jing (OSU)

Title: A projective Szemerédi–Trotter theorem

Abstract: The Szemerédi–Trotter theorem asserts that the solutions of an algebraic curve from a given set in a field of characteristic 0 exhibit a power-saving improvement over the Schwartz–Zippel bound, provided that the curve satisfies a $K_{s,t}$ -free condition and the set is close to being a cube with equal side lengths. We establish a group-action version of the Szemerédi–Trotter theorem over any field, extending an earlier result by Bourgain. As an application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$ , where $k = \mathbb{F}_p^n$ or $\mathbb{C}$ , which can be interpreted as a quantitative Elekes–Szabó-type theorem without a general position assumption. (based on joint work with Tingxiang Zou)

November 7: Konstantin Matveev (Rutgers)

Title: Several meetings with Pieri formulas

Abstract: Pieri formulas are certain identities in the algebra of symmetric functions. I will talk about several results where they play a central role, in particular, about Macdonald-Littlewood-Richardson coefficients, Macdonald-positive homomorphisms, and the interplay of stochastic six-vertex models and the Robinson-Schensted-Knuth algorithm.

November 14: Vlad Kobzar (OSU)

Title: Lower Bound on the Block-Diagonal SDP Relaxation for the Clique Number of the Paley Graph

Abstract: We establish a lower bound on the value of the block-diagonal semidefinite program (SDP) relaxation for the clique number of the Paley graph of prime order. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; the Paley graph is a deterministic graph where two vertices are connected if their difference is a quadratic residue in a finite field with the number of elements given by certain primes and prime powers. Improving on the current $O(\sqrt{p})$ upper bound for the clique number of Paley graphs of prime order is a classic open problem in number theory and additive combinatorics. Kunisky and Yu (CCC 2023) provided numerical evidence that the upper bound given by the sum-of-squares relaxation of degree 4 (SOS-4) is growing at an order smaller than square root of $p$ and proved that the value of this relaxation is lower bounded as $\Omega (p^{1/3})$ . Gvozdenovic, Laurent and Vallentin introduced the block-diagonal hierarchy ( $L^t$ ) of SDPs that are weaker than the SOS SDPs; the values of these block-diagonal SDPs of degree 2 ( $L^2$ ) bound from above the values of the corresponding SOS-4 relaxations, and the $\Omega (p^{1/3})$ lower bound also applies to the $L^2$ relaxations. Building on the above-mentioned work, using Feige-Krauthgamer pseudomoments, we show that the $L^t$ relaxations are lower bounded by $2^{1-t}\sqrt{p}$ , at the leading order as $p$ gets large, and therefore these relaxations do not break the $\sqrt{p}$ barrier. Since the $L^t$ hierarchy is stronger than the Lovasz-Schrijver hierarchy, our lower bound also applies to the latter hierarchy. We also study the subgraphs (localizations) of the Paley graphs induced on a set of vertices extending a clique of a given size $a$ to a maximal clique. We prove that interchanging localization degree $a$ and relaxation degree $t$ are equivalent for the purpose of our lower bound, which is consistent with the localization-relaxation trade-off conjectured by Kunisky (Exp. Math. 2024). This is joint work with Yifan Wang and Yanling Shen (Columbia University).

November 28: Thanksgiving Break, No Seminar.

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Autumn 2024

Seminar time: Thursdays 1:50-2:45pm.
Location: Enarson Classroom Building EC0322.