Events

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: 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: Wires, bits, and the cost of sorting

Abstract: How hard is it to sort a list of n integers? A basic course on algorithms says it's O(n log n) time, but what if the list is enormous - so big you would need to cover the surface of the moon just to store it?

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.

Title: Nash-Williams Orientation for Infinite Graphs

Abstract: Nash-Williams proved in 1960 that an edge-connectivity of 2k is sufficient for a finite graph to admit a k-arc-connected orientation and conjectured that the same holds for infinite graphs. We show that the conjecture is true for locally finite graphs with countably many ends.

This is joint work with Max Pitz and Marcel Koloschin.

Title: Online edge colouring

Abstract: Vizing's Theorem provides an algorithm that edge colors any graph of maximum degree Δ can be edge-colored using Δ+1 colors, which is necessary for some graphs, and at most one higher than necessary for any graph. In online settings, the trivial greedy algorithm requires 2Δ-1 colors, and Bar-Noy, Motwani and Naor in the early 90s showed that this is best possible, at least in the low-degree regime.