Seminar

Title: Rotation-invariant web bases from hourglass plabic graphs and symmetrized six-vertex configurations

Abstract: Many combinatorial objects with strikingly good enumerative formulae also have remarkable dynamical behavior and underlying algebraic structure. In this talk, we consider the promotion action on certain rectangular tableaux and explain its small, predictable order by reinterpreting promotion as a rotation in disguise. We show how the search for this visual explanation of combinatorial dynamics led to the solution of an algebraic problem involving web bases and a generalization of the six-vertex model of statistical physics. We find that this framework includes several unexpected combinatorial objects of interest: alternating sign matrices, plane partitions, and their symmetry classes (joint with Ashleigh Adams). This talk is based primarily on joint work with Christian Gaetz, Stephan Pfannerer, Oliver Pechenik, and Joshua P. Swanson.

Abstract: In this talk, we will cover the paper of Svensson and Tarnawski that shows that perfect matching in general (non-bipartite) graphs in is quasi-NC. Similar to the work of Fenner, Gurjar and Thierauf (covered earlier in the reading group), the approach is to derandomize the isolation lemma for the perfect matching polytope by applying weight functions to iteratively restrict to subfaces of the polytope. However, the perfect matching polytope in general graphs is not as well-structured as it is in bipartite graphs. The faces of the polytope no longer correspond to subgraphs and now involve additional tight odd set constraints that need to be dealt with. This makes it so that a cycle with non-zero circulation may still exist in the support of the new face. Additionally, the existence of odd cycles in the graph breaks the cycle counting argument used in the paper of Fenner, Gurjar, Thierauf. We will see how Svensson and Tarnawski deal with these issues in the talk.

Title: LP based approximation algorithm for an stochastic matching problem

Abstract: Consider the random graph model where each edge e has a fixed weight w_e and it is independently present in the graph with probability p_e​. Given these probabilities, we want to construct a maximum weight matching in the graph. One can only determine if an edge is present by querying it, and if an edge is present, it must be irrevocably included in the matching. Additionally, each vertex i can be queried no more than t_i times. The goal is to device an adaptive policy (algorithm) to query the edges of the graph one by one in order to maximize the expected weight of the matching.

In this talk we present an elegant LP-based constant-factor approximation algorithm with respect to the optimal adaptive policy for the problem.

This is one of the results due to Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra, in their paper "When LP is the Cure for your Matching Woes" from 2011.

Title: Hypercube decompositions and combinatorial invariance for Kazhdan-Lusztig polynomials

Abstract: Kazhdan-Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture, due to Lusztig and to Dyer, suggests that they only depend on the combinatorics of Bruhat order. I'll describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals in the symmetric group. This significantly generalizes the main previously known case of the conjecture, that of lower intervals.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

Title:Coefficient Positivity and Analytic Combinatorics

 Abstract: Given a rational function analytic at the origin, are its power series coefficients positive? We first motivate this question with historical background, examples and connections to decidability questions in combinatorics. We then present a method for proving positivity for a certain class of multivariate rational functions, by constructing explicit error bounds for an asymptotic expansion using analytic combinatorics in several variables. After applying our method to a couple of examples in the literature, including re-proving a (recently proved) conjecture of Gillis, Reznick and Zeilberger from 1983, we discuss the feasibility of completing certain steps in our analysis using methods from computer algebra.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm

Title: Purifying arbitrarily noisy quantum states

Abstract: Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. For constant initial error parameter and dimension, our procedure has sample complexity asymptotically optimal in the final error parameter, and almost matches the known optimal protocol for qubits. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.  Joint work with Andrew Childs, Honghao Fu, Zhi Li, Maris Ozols, Vedang Vyas. 

Title: Combinatorial Nullstellensatz and the Erdős box problem

Abstract: In the talk, I will show how Lasoń’s generalization of Alon’s Combinatorial Nullstellensatz can be used to obtain lower bounds on Turán numbers of complete r-partite r-uniform hypergraphs. As an example, I will give a short and simple explicit construction of a hypergraph free of copies of the complete r-partite r-uniform  hypergraph with parts of size 2, thereby providing a lower bound for the so-called Erdős box problem. This asymptotically matches best known bounds when r ≤ 4. 

Title: Non-measurable colourings avoiding large distances

Abstract: "In 1983, F{\"u}rstenberg, Katznelson, and Weiss proved that for every finite measurable colouring of the plane, there exists a $d_0$ such that for every $d \ge d_0$ there is a monochromatic pair of points at distance $d$. In contrast to this, we show that there is a finite colouring avoiding arbitrarily large distances.

Joint work with Rutger Campbell."

Title: Diophantine tuples, graphs, and additive combinatorics

Abstract: In this talk, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine tuples and Diophantine graphs. First, we show that if a nontrivial shift of a multiplicative subgroup $G$ contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, or the clique number of Diophantine graphs, which is the largest size of a set such that each pairwise product of its elements is $n$ less than a $k$-th power, refining the recent result of Dixit, Kim, and Murty.  One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest.  

We also present a significant progress towards a conjecture of S\'{a}rk\"{o}zy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes $p$, the set $\{x^2-1: x \in \F_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in $\F_p$ non-trivially.

Graph Property Testing using the Container Method

Abstract: The Graph and Hypergraph Container Methods have recently been used to obtain multiple striking results across different areas of mathematics. In this talk, we will see how the graph container method is particularly well-suited for the study of some fundamental problems in graph property testing.

The main problem we will discuss in the talk is the Independent Set Testing problem introduced by Goldreich, Goldwasser, and Ron (1998). In this problem, we are given oracle access to a graph on $n$ vertices that either (i) contains an independent set on $\rho n$ vertices, or (ii) is $\epsilon$-far from the property in the sense that at least $\epsilon n^2$ edges must be removed from the graph to make it have an independent set of this size. We will introduce a new container lemma for the latter class of graphs and we will show how this lemma can be used to obtain a near-optimal solution to the Independent Set Testing problem. We will also discuss how variants and extensions of the new container lemma can be used to prove a variety of other results in property testing.

This is joint work with Cameron Seth.