Events

Title: Nakajima quiver varietites and irreducible components of Springer fibers

Abstract:

Springer fibers and Nakajima quiver varieties are amongst the most important objects in geometric representation theory. While Springer fibers can be used to geometrically construct and classify irreducible representations of Weyl groups, Nakajima quiver varieties play a key role in the geometric representation theory of Kac--Moody Lie algebras.

Title: The e-positivity of chromatic symmetric functions

Abstract:

The chromatic polynomial was generalized to the chromatic symmetric function by Stanley in his seminal 1995 paper. This function is currently experiencing a flourishing renaissance, in particular the study of the positivity of chromatic symmetric functions when expanded into the basis of elementary symmetric functions, that is, e-positivity.

Title: Symmetry of random graphs

Abstract:

A graph is said asymmetric if its automorphism group contains only the identity permutation. Otherwise, it is said symmetric. In this talk we will survey symmetry of Erdos-Renyi random graphs, of random regular graphs, and of random graphs with moderate degree sequences.

Title: Weighted generating functions for weighted chord diagrams

Abstract:

Motivated by the universal property of the Connes-Kreimer Hopf algebra of rooted trees and Hopf subalgebras arising from so-called combinatorial Dyson-Schwinger equations, we introduce a class of two-variable recursive functional equations involving Hochschild 1-cocycle operators.

Title: A Strongly Polynomial Label-Correcting Algorithm for Linear Systems with Two Variables per Inequality

Abstract:

In this talk, I will present a strongly polynomial label-correcting algorithm for solving the feasibility of linear systems with two variables per inequality. The algorithm is based on the Newton–Dinkelbach method for fractional combinatorial optimization, and extends previous work of Madani (2002).

Title: Matroids, tropical geometry, and positivity

Abstract: 

The theory of matroids -- a class of combinatorial objects which simultaneously generalize graphs as well as vectors in a vector space -- was pioneered by William Tutte in his 1948 PhD thesis. Matroids are also closely connected to the Grassmannian and the tropical Grassmannian.  In recent years, mathematicians and physicists have been exploring positive notions of all of these objects, finding applications to scattering amplitudes and shallow water waves.  In my talk I will give an introduction to matroids, tropical geometry, and positivity, and survey some of the beautiful results and applications.

Title: An Upper Bound on Graphical Partitions

Abstract:

An integer partition is called graphical if it can be realized as the size-ordered degree sequence of a simple graph (with no loops or multiple edges).  In his 1736 paper on the Königsberg bridge problem, arguably the origin of graph theory, Euler gave a necessary condition for a partition to be graphical: its sum must be even.

Title: One Dollar Each Eliminates Envy

Abstract:

We study the fair division of a collection of $m$ indivisible goods amongst a set of $n$ agents. Whilst envy-free allocations typically do not exist in the indivisible goods setting, envy-freeness can be achieved if some amount of a divisible good (money) is introduced.

Title: Halting Time is Predictable for Large Models: A Universality Property and Average-case Analysis

Abstract:

Average-case analysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis, it is more representative of the typical behavior of an algorithm, but remains largely unexplored in optimization. One difficulty is that the analysis can depend on the probability distribution of the inputs to the model.

Title: Graph coloring of graphs with large girth is hard for the Nullstellensatz

Abstract:

In this talk we will discuss a method to solve combinatorial problems using hierarchies of systems of linear equations using Hilbert's Nullstellensatz. In particular, we will study the behaviour of these hierarchies for deciding the non-$k$-colorabilty of graphs.