Events

Title: Quantum Automorphism Groups of Trees

Abstract: We give a characterisation of quantum automorphism groups of trees. In particular, for every tree, we show how to iteratively construct its quantum automorphism group using free products and free wreath products. This can be considered a quantum version of Jordan’s theorem for the automorphism groups of trees. This is one of the first characterisations of quantum automorphism groups of a natural class of graphs with quantum symmetry. This talk is based on the following preprint: https://arxiv.org/abs/2311.04891

Title: Filtered deformations of commutative algebras

Note: There will be no pre-seminar.

Abstract: We’ll look at different ways of deforming the multiplicative structure of “classical” algebras to obtain new algebras and explain how this algebraic structure can often be understood combinatorially.  We’ll then look at a special class of algebras one can produce this way called filtered deformations and we’ll discuss a conjecture of Etingof which asserts that in positive characteristic that filtered deformations of commutative rings should be in some natural sense very close to being commutative themselves. Not much background will be assumed.

Title: Several Gyrafas-Sumner-type results for treewidth

Abstract: For a graph parameter which goes unbounded at both ends of a robust sparsity spectrum, one naturally asks if it is bounded anywhere in the middle. This is modelled after a well-known conjecture of Gyarfas and Sumner, asking the above question for the chromatic number as the parameter and the girth as the measure of sparsity. We look through this lens at the treewidth as the parameter and present a number of recent results answering the corresponding question in various settings. If time permits, we’ll try to sketch a proof as well as some directions for future work.

This is joint work with Bogdan Alecu, Maria Chudnovsky, Sophie Spirkl and partly with Tara Abrishami.

Title: Online Convex Optimization

Abstract: Online learning (OL) is a theoretical framework for learning with data online. Moreover, we usually make no assumptions on the distribution of the data, allowing it even to be adversarial to the learner. Maybe surprisingly, we can still design algorithms that, in some sense, “successfully learn” in this setting. This level of generality makes many of the ideas, algorithms, and techniques from OL useful in applications in theoretical computer science, optimization in machine learning, and control. In this talk I will give a brief introduction to the key concepts in online learning and  mention a few topics within or adjacent to online learning that I believe cover fundamental ideas in OL and/or with interesting open research questions.

Title: On the intersection density of transitive groups with degree 3p

Abstract: Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, a subset $\mathcal{F}\subset G$ is intersecting if any two elements of $\mathcal{F}$ agree on some elements of $\Omega$. The \emph{intersection density} of $G$ is the rational number $\rho(G)$ given by the maximum ratio $\frac{|\mathcal{F}|}{|G|/|\Omega|}$, where $\mathcal{F}$ runs through all intersecting sets of $G$.

Most results on the intersection density focus on particular families of transitive groups. One can look at problems on the intersection density from another perspective. Given an integer $n\geq 3$, we would like to determine the possible intersection densities of transitive groups of degree $n$. This problem turns out to be extremely difficult even in the case where $n$ is a product of two primes.

Title: Polarization operators in superspace

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:00 pm.

Abstract: The classic coinvariant space is a graded representation of the symmetric group with deep connections to permutation statistics and Hall-Littlewood polynomials. Its generalization, the diagonal harmonics, has a rich connection to Macdonald polynomials and the q,t-Catalan numbers. In this talk, we consider the variant of the classical coinvariant story in the superspace. We briefly survey its connections and recent developments. We introduce polarization operators and discuss a new basis for its alternating component. We also discuss a folklore on cocharge and propose a basis for the super harmonics.

Title: Sublinear Regret 101

Abstract: The purpose of this talk is to look at specific algorithms for online convex optimization attaining sublinear regret. In this way, we can get more familiar with the online learning setup and with the ingredients necessary to attain sublinear regret.

Title:  A threshold for fractional Sudoku completion

Abstract: The popular puzzle game Sudoku presents a player with a 9-by-9 grid having some numbers filled in a few of the cells.  The player must finish filling in numbers from 1 to 9 so that every row, column, and 3-by-3 box contains each of these numbers exactly once.  We can extend Sudoku so that the boxes are $h$-by-$w$, and the overall array is $n$-by-$n$, where $n=hw$.  The puzzle is now similar to completing a latin square of order n, except of course that Sudoku has an additional box condition.

Title: The Container method for Enumerating Triangle-free graphs

Abstract: The container method is a tool for bounding the number of independent sets in graphs and can be generalized to hypergraphs. It utilizes the observation that independent sets are found in clusters or “containers” within the graph. One application of this method is to bound the number of finite objects with some forbidden substructure. For example, we can bound the number of graphs on N vertices that do not contain K₃ as a subgraph, that is, the number of Triangle-free graphs on N vertices. We will take a look at finding this bound using the container method.

Title: Prophets, philosophers, and online algorithms

Abstract: In online Bayesian selection problems, a seller is faced with a stream of buyers arriving sequentially. Each arriving buyer makes, for each item on sale, a take-it-or-leave-it offer drawn from some known distribution, which the seller must immediately either accept or decline. It is common to compare the seller's benefit to the optimal offline solution, obtained by a "prophet'' who knows the future. However, the seller might not even be able to compete with the optimal online algorithm, which may be computationally intractable to run. In this case the optimal online algorithm is obtained by a "philosopher'' with sufficient time to think (compute).  In this talk I will discuss some developments on online algorithms efficiently approximating this philosopher.