Future students

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: Multiplicative Weight Update (MWU) Method for Solving Packing/Covering LPs

Abstract: I will present an overview of the MWU technique for solving Packing/Covering LPs easily and efficiently. The talk will be based on a combination of results from the following resources: (1) https://epubs.siam.org/doi/pdf/10.1137/1.9781611976014.11 (2) https://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf

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: Diagonal coefficients, graph invariants with the symmetries of Feynman integrals, and the proof of the c_2 completion conjecture

Abstract: In a scalar field theory the contribution of a Feynman diagram to the beta function of the theory, the Feynman period, can be written as an integral in terms of the (dual) Kirchhoff polynomial of the graph.

Title: Hypergraph Matchings Avoiding Forbidden Submatchings

Abstract: We overview a general theory for finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of forbidden submatchings (which we view as a hypergraph H where V(H)=E(G)).

Title: Linear relations and Lorentzian property of chromatic symmetric functions

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

Abstract: The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian Wachs q-analogue (q-CSF) have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013).

Title: Distance-regular graphs that support a uniform structure

Abstract: Given a connected bipartite graph $G$, the adjacency matrix $A$ of $G$ can be decomposed as  $A=L+R$, where $L=L(x)$ and $R=R(x)$ are respectively the  lowering and the raising matrices with respect to a certain vertex $x$. The graph $G$ has a \textit{uniform structure} with respect to $x$ if the matrices $RL^2$, $LRL$, $L^2R$, and $L$ satisfy a certain linear dependency.

Title: Extended Schur Functions and Bases Related by Involutions

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

Abstract: The extended Schur basis and the shin basis generalize the Schur functions to the dual algebras of the quasisymmetric functions and the noncommutative symmetric functions. We define a creation operator and a Jacobi-Trudi rule for certain shin functions and show that a similar matrix determinant expression does not exist for every shin function.

Title: Kemeny’s constant and random walks on graphs

Abstract: Kemeny's constant is an interesting and useful quantifier of how well-connected the states of a Markov chain are. This comes to the forefront when the Markov chain in question is a random walk on a graph, in which case Kemeny's constant is interpreted as a measure of how `well-connected' the graph is. Though it was first introduced in the 1960s, interest in this concept has recently exploded. This talk will provide an introduction to Markov chains, an overview of the history of Kemeny’s constant, discussion of some applications, and a survey of recent results, with an emphasis on those that are extensions or generalizations of simple random walks on graphs, to complement Sooyeong’s talk from two weeks ago.