The PM-seminar takes place on Wednesday at 2pm in McCrea 219.

Here is a campus plan.

September 26:

Abstract: The graph of isogenies of supersingular elliptic curves has many

applications in computational number theory and public key

cryptography. I will present some of these applications and I will also

discuss some open problems.

October 3:

Abstract: We will introduce the concepts mentioned in the title, explain why they

are interesting and show a connection between them. Thereby we will see examples

of simple locally compact groups without lattices; finitely generated, simple amenable

groups and the famous Higman-Thompson groups.

October 10:

Abstract: There are (at least) 3 competing notions of "dimension" for discrete groups:

cohomological dimension, geometric dimension (the smallest dimension of a classifying

space), and Lusternik-Schnirelmann category (of a classifying space). Theorems of

Eilenberg-Ganea and Stallings and Swan from the 1950’s and 60's imply that these all

coincide, except for the possible existence of a group with cat=cd=2 and gd=3.

I will discuss equivariant generalisations of these theorems to the setting of groups with

operators. The statements involve Bredon cohomological dimension with respect to families

of subgroups, which I'll define during the talk.

October 17:

Abstract: Ashot Minasyan and I construct groups that establish the result in the title,

resolving a question that has been around for almost 30 years. I will start by explaining

the phrases `CAT(0)' and `biautomatic'. After that I will talk about our groups and why

they have the properties that we claim.

October 18:

(!!note change of day, time and venue!!)

October 24: No seminar!

October 31:

The celebrated Hasse-Minkowski local-global principle relates the existence of zeros of

quadratic forms over the field of rational numbers, and more generally number fields ('global'),

to the existence of zeros over various completions ('local'), namely the real and complex numbers,

and the p-adics, where analytic methods can be applied. A similar local-global principle holds in

positive characteristic, where the 'local' fields are the fields of formal Laurent series over finite

fields. However, while it is classical that for each local field K of characteristic zero there is an

algorithm that determines whether a given system of polynomial equations has a common zero in

K, and even more generally whether a given first-order sentence in the language of rings holds in

K, for Laurent series fields over finite fields the existence of such algorithms is only partially

understood. In this talk I will report on what is known about this, which will lead us from number

theory and algebraic geometry to the model theory of valued fields.

November 7:

Abstract: We will consider some questions in classical knot theory, and see how surfaces

play a part. In particular, we will look at how knot theory compares with the physical

world, and how hard it is to untangle a knot.

November 14:

Abstract: This talk is about semigroup C*-algebras, i.e., C*-algebras generated by left regular

representations of semigroups. After a general introduction, we will focus on classes of one-relator

monoids and Artin monoids of finite type. We present structural results and K-theory computations

for the corresponding semigroup C*-algebras. This is joint work with Tron Omland and Jack Spielberg.

November 21:

Abstract

November 28:

Abstract: We study a polynomial with connections to correspondence colouring, a recent generalization

of list-colouring, and the Unique Games Conjecture. Given a graph G and an assignment of elements of

the symmetric group S_r to the edge of G, we define a cover graph: there are sets of r vertices corresponding

each vertex of G, called fibres, and for each edge uv, we add a perfect matching between the fibres

corresponding to u and v. A transversal subgraph of the cover is an induced subgraph which has exactly

one vertex in each fibre. In this setting, we can associate correspondence colourings with transversal cocliques

and unique label covers with transversal copies of G.

We define a polynomial which enumerates the transversal subgraphs of G with k edges. We show that this

polynomial satisfies a contraction deletion formula and use this to study the evaluation of this polynomial

at -r-1. This is joint work with Chris Godsil and Gordon Royle.

December 5:

Abstract: I will start with a few words about the joint history of math and biology. I will touch on the

work of Fibonacci and Volterra. What I then will talk about in detail is a neat biological problem: the

paradoxical existence of recombination hotspots in the genome. I will start with a model for that

system based on two alleles, and I will explain how we analysed the behaviour. This will lead me

to the existence of a homoclinic cycle, and the analysis of the stability of the homoclinic cycle. As

a finale I will then take that same model but now for more alleles, explain how the homoclinic cycles

join up to form a homoclinic network in the vicinity of which we find chaotic dynamics (which is

where the horseshoes come in). After that I will interpret these results in terms of biology

December 12:

Abstract: In this talk I will give an overview of what is known about conjugacy growth in infinite discrete

groups and the formal series associated with it. In particular, I will highlight the connections between the

rationality or lack of rationality of these series to analytic combinatorics and geometry.