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Title:ÌýPolynomial Spaces

Abstract:

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

Title:ÌýCombinatorial 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:ÌýMonochromatic cycle partitions

Abstract:

A classic result of ErdÅ‘s, 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:ÌýOrthogonal Polynomials and the Addition Formula

Abstract:

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

Title:ÌýWhen are two Schur functions the same?

Abstract:Ìý

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:ÌýThe 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:ÌýUndermining key exchange

Abstract:

There are many ways to undermine systems that rely on cryptography.ÌýAn 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:ÌýImproving Implementations of CSIDH

Abstract:Ìý

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

Title:ÌýLP bounds and the addition formula

Abstract:

I will show how we can derive useful bounds on the size of codes and designs, using linear programming and the addition formula.

Title:ÌýChord diagrams, generating functions, and qft

Abstract:

I'll talk about some joint work with Julien Courtiel where some nice enumerative combinatorics tells us something about how gauge theories are such special quantum field theories.