Seminar

Title: Improving Implementations of CSIDH

Abstract: 

I will present on some papers related to implementations of CSIDH [1].

Title: Orthogonal Polynomials and the Addition Formula

Abstract:

Many questions about designs and codes in the unit sphere can be reduced to questions about members of a family of orthogonal polynomials, the so-called Gegenbauer polynomials.

Title: Undermining key exchange

Abstract:

There are many ways to undermine systems that rely on cryptography. An attacker could of course try to find flaws in the hardware and software implementing the system, or find mathematical weaknesses in the cryptographic algorithms.

Title: The sparsest matroids omitting an independent flat

Abstract:

Given integers $k,r \ge 1$, what is the smallest a rank-$r$ matroid can be that does not contain a $k$-element independent set that is also a flat? I will answer this question, characterize the extremal examples, and draw parallels with a problem in graph theory.

Title: Submodular function maximization

Abstract:

I will give the introductory talk for this term's topic for the Combinatorial-Optimization Reading group, namely submodular function maximization. Submodular functions arise in diverse contexts in combinatorial optimization.

Title: When are two Schur functions the same?

Abstract: 

A pair of skew shapes are said to be (skew) equivalent if they admit the same number of semistandard tableaux of any weight; i.e. if their associated skew Schur functions are equal.

Title: Monochromatic cycle partitions

Abstract:

A classic result of Erdős, Gyárfás, and Pyber states that the vertex set of every complete graph, whose edges have been coloured with r colours, can be covered by r² log r disjoint monochromatic cycles.

Title: Polynomial Spaces

Abstract:

We describe some simple machinery that enables us to derive upper bounds on the size of codes, and lower bounds on the size of designs in a quite general setting.

Title: Combinatorial questions motivated by Invariant Theory

Abstract:

We begin the talk by discussing a question in Invariant Theory: given a representation $V$ of a Lie group $G$, when if the invariant ring $k[V]^G$ a polynomial ring? We give a conjectural answer which we have verified for $SL_n$ and discuss some combinatorial questions motivated by the proof. This is joint work with Dan Edidin.

Title: Signings and induced subgraphs of the Hypercube

Abstract:

Just over a month ago, Hao Haung resolved the sensitivity conjecture, a 30 year-old question about the complexity of boolean functions.