Future students

Title: A Short Introduction to Projective Geometry

Abstract:

Basically, see the title. I will be considering the real and complex cases mainly, because that is
what is needed in quantum physics.

Title: RSKy Business

Abstract:

We sketch the Robinson-Schensted-Knuth algorithm, then use it to glimpse into the representation theory of some classical combinatorial objects.

Title: I'll prove the Fundamental Theorem of Algebra

Abstract:

Gauss proved that complex polynomials always admit a root. I'll explain how he came to that conclusion and present a proof that rigorizes his argument in a nice way. You will only need to know a little vector calculus and the intermediate value theorem.

Title: The 8-connected excluded minors for the class of quasi-graphic matroids

Abstract:

The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both lifted-graphic matroids and frame matroids, each of which generalises the class of graphic matroids.

Title: Enumeration in quantum algebras

Abstract:

Many of the classical algebras that occur in algebraic geometry and other mathematical fields have natural quantizations; that is, one can deform the multiplication rule using a parameter q, which has the property that when we specialize q at 1 we recover the classical object.  As part of the general goal of understanding the representation theory of these rings, one often wants to understand the prime spectra of these algebras, that is, the collection of prime ideals in these rigs. 

Title: The 1,2,3-Conjecture and a related problem for bipartite graphs

Abstract:

Title: Quantum Walks and State Transfer

Abstract:

Quantum walks are the quantum analogues of classical random walks and can be used to model quantum computations. If a quantum walker starts at a vertex of a graph and after some length of time, has probability 1 of being found at a different vertex, we say there is perfect state transfer between the two vertices.

Title: The quantum and private capacities of quantum channels, and the solution in the low-noise regime

Abstract:

We first summarize background on the quantum capacity of a quantum channel, and explain why we know very little about this fundamental quantity, even for the qubit depolarizing channel (the quantum analogue of the binary symmetric channel) despite 20 years of effort by the community.

Title: Symmetric group characters as symmetric functions

Abstract:

The irreducible polynomial representations have Schur polynomials as characters. Similarly there is an in-homogeneous basis of symmetric functions that we call the "irreducible character basis" which are characters of irreducible symmetric group modules when the symmetric group is represented as the subgroup of permutation matrices.

Title: Using eigenvalues to bound the independence number of a graph

Abstract:

Finding a maximum independent set (or clique) in an arbitrary graph has been shown to be NP-hard.  As any independent set gives a lower bound on the independence number, determining an upper bound is usually more useful.