The seminars take place on Wednesdays from 2-3 pm during term time in Arts LT1. They are aimed to appeal to most of our staff and PhD students. They are followed by coffee, tea and biscuits in the McCrea building (next to room 0-08). Most times we will go to dinner afterwards.

Everyone is welcome!

Schedule:

17. January:  Thomas Webster (RHUL)

Title: On the sum of the number of divisors of integral binary quadratic forms

Abstract: The divisor function d(x) of a positive integer x is the number of positive divisors it has. What happens when you sum d(x) for x up to some big number X? How fast does it grow as X gets larger and what are the main terms? What happens when instead of every positive integer, we only consider positive integers that can be expressed as ax²+bxy+cy² for some values of x and y, when x and y are less than or equal to X, and a, b and c are constants? This talk is about the asymptotic expressions I have obtained for the sum of the divisor function of these more general quadratic forms, the literature surrounding similar results and the methods I use to produce these theorems.

24. January: Maud de Visscher (City University London)

Title: p-Kazhdan Lusztig polynomials in type (AxA, A)

Abstract: Kazhdan-Lusztig polynomial were defined by Kazhdan and Lusztig in the 1970’s when studying deformations of Coxeter groups. They have wide ranging applications to many areas of Representation theory and Geometry. There is an algorithm for computing them but, in general, no closed combinatorial formula is known. Building on the work of Soergel, major advances in our understanding of these polynomials was made in the last decade by Elias-Williamson and Libendinsky-Williamson when they ‘categorified’ these polynomials using the diagrammatic Hecke category.This category can be defined over any field, leading to the definition of modular versions of the Kazhdan-Lusztig polynomials. 

In this talk, I will focus on the KL polynomials associated to the symmetric group and its maximal parabolic subgroups. We will see that in this case the modular KL polynomials coincide with the ordinary ones and explicit combinatorial formulas can be given for these.I will also briefly mention how these results can be extended further to all Hermitian symmetric pairs. This talk is based on joint work with C. Bowman, N. Farrell, A. Hazi, E. Norton and C. Stroppel.

31. January: Scott Baldridge (Louisiana State University)

Title: Using quantum states to understand the four-color theorem

Abstract: The four-color theorem states that a bridgeless plane graph is four-colorable, that is, its faces can be colored with four colors so that no two adjacent faces share the same color. This was a notoriously difficult theorem that took over a century to prove. In this talk, we generate vector spaces from certain diagrams of a graph with a map between them and show that the dimension of the kernel of this map is equal to the number of ways to four-color the graph. We then generalize this calculation to a homology theory and in doing so make an interesting discovery: the four-color theorem is really about all of the smooth closed surfaces a graph embeds into and the relationships between those surfaces.

The homology theory is based upon a topological quantum field theory. The diagrams generated from the graph represent the possible quantum states of the graph and the homology is, in some sense, the vacuum expectation value of this system. It gets wonderfully complicated from this point on, but we will suppress this aspect from the talk and instead show a fun application of how to link the Euler characteristic of the homology to the famous Penrose polynomial of a plane graph.

This talk will be hands-on and the ideas will be explained through the calculation of easy examples! My goal is to attract students and mathematicians to this area by making the ideas as intuitive as possible. Topologists and representation theorists are encouraged to attend also—these homologies sit at the intersection of topology, representation theory, and graph theory.

This is joint work with Ben McCarty.

7. Feburary: Bianca Marchionna (Universität Bielefeld)

Title: Suborbit zeta functions and groups acting on trees

Abstract: Given a group G acting transitively on a set X, it is common to look at the suborbits, i.e. the orbits in X of a prescribed stabilizer subgroup G_x. In several cases, for every n≥1, only finitely many - say a_n - of them have size n. A way to study the sequence {a_n} is via the Dirichlet series zeta_{G, G_x}(s)=∑_n≥1a_n n^-s.
The talk aims at introducing zeta_{G, G_x}(s) and how it might relate to invariants or structural properties of G. We mainly focus on the case in which G acts on a tree T and X is, for instance, a vertex G-orbit. This favourable setting allows us to reformulate the problem into a more accessible one, involving only the geometry and the combinatorics carried by X.

6. March: Jan Petr (University of Cambridge)

Title: Towards odd-sunflowers: size-aware families and lightnings

Abstract: Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and Pälvolgyi, we initiate the study of size-aware families: a family F in P[n] is said to be size-aware if each A in F contains at most |A| other elements of F as a subset. We show that the maximum size of a size-aware family is attained by the middle two layers of the hypercube. The situation becomes more complicated when we require the size-aware family to be also intersecting. The talk will include a recapitulation of a few classic results about extremal problems concerning set systems.

 

13. March: Lawk Mineh (University of Southampton)

Title: Tilings in groups

Abstract: A subset of a group is called a tile if the group can be covered by a collection of disjoint translates of the subset. A longstanding question asks whether each finite subset can be extended to a finite file in every group. Until recently, little progress was made on this question and relatively few examples were known to satisfy this property. A new method of tiling in the hyperbolic groups was developed by Akhmedov last year, which we extend to the much broader setting of acylindrically hyperbolic groups. This is a joint work with Joe MacManus.

 

20. March: Yacong Zhou (RHUL)

Title: Lower Bounds for Maximum Weight Bisections of Graphs with Bounded Degrees

Abstract: A cut is a partition of the vertices of a graph into two disjoint subsets. A bisection in a graph is a cut in which the number of vertices in the two parts differs by at most 1. The MAX-BISECTION (MAX-CUT) problem is to find a bisection (cut) that maximizes the number of edges in the bisection. Both of these two problems are known to be NP-hard. Thus, many researchers worked on finding some good lower bounds for the maximum bisection (or cut) in graphs. In this talk, I will introduce some new lower bounds for the maximum weight of bisections of edge-weighted graphs with a bounded maximum degree, which improve a bound proved by Lee, Loh, and Sudakov for (unweighted) maximum bisections in graphs whose maximum degree is either even or equals 3, and for almost all graphs. I will show that a tight lower bound for the maximum size of bisections in 3-regular graphs obtained by Bollobás and Scott can be extended to weighted subcubic graphs. I will also talk about edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs with only one exceptional graph. This talk is based on a joint work with Stefanie Gerke, Gregory Gutin, and Anders Yeo.

 

27. March: Dilton Mayhew (University of Leeds) 

This seminar will start at 3pm!

Title: Monadic second-order definability of classes of matroids

Abstract: Matroids can be seen as abstractions of geometrical configurations. Classic examples arise from finite collections of vectors in a vector space. Any such matroid is said to be representable. In this case,  we can think of the matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic.

Monadic second-order logic is a formal language useful for applications in the theory of graphs and matroids. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle's Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects).

Given Courcelle's Theorem and other similar results, it is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder.

This talk will contain a brief introduction to matroids.