Pure Mathematics
All seminars will take place in the McCrea Building, Room 219, on Tuesdays at 2 pm, unless stated otherwise. Tea will be served after the seminar at 3pm in Room 237 of the McCrea Building. All are welcome!
Summer 2013
26th June at 11am in C219 Steven Galbraith (University of Auckland) A tutorial on recent work of Joux on the discrete logarithm problem
Spring 2013
8th January Iain Moffat (Royal Holloway) Duality and equivalence of graphs in surfaces
15th January Igor Wigman (King's College London) The distribution of defect and other nonlinear functionals of random spherical harmonics
22nd January Carolyn Chun (Brunel University) Fundamental questions in matroids
29th January Chris Gill (University of Oxford) Multiplicities and vertices in tensor products of Young modules
5th February Mark Walters (QMUL) Optimal Resistor Networks
12th February Stephen Huggett (University of Plymouth) Newton, the geometer
19th February Brita Nucinkis (Royal Holloway) On Richard Thompson's groups and their generalisations
26th February Bill Jackson (QMUL) Chromatic polynomials
5th March Ashot Minasyan (University of Southampton) Hyperbolically embedded subgroups in groups acting on trees
12th March Martin Loebl (Charles University, Prague) On Rota's bases conjecture
19th March Francesco Matucci (Paris) The conjugacy problem in extensions of Thompson's group
21st March Conchita Martinez-Perez (Zaragoza) Isomorphisms between Brin-Higman-Thompson groups (extra seminar at 2pm Thursday 21st March in McCrea 336)
Abstract:
Iain Moffatt: This talk revolves around two fundamental constructions in graph theory: duals and medial graphs. There are a host of well-known relations between duals and medial graphs of graphs drawn in the plane. By considering these relations we will be led to the working principle that duality and equality of plane graphs are equivalent concepts. It is then natural to ask what happens when we change our notion of equality. In this talk we will see how isomorphism of abstract graphs corresponds to an extension of duality called twisted duality, and how twisted duality extends the fundamental relations between duals and medial graphs from graphs in the plane to graphs in other surfaces. We will then go on to see how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials, including the chromatic polynomial, the Penrose polynomial, and topological Tutte polynomials.
Igor Wigman: This work is joint with Domenico Marinucci. Partially motivated by questions in mathematical physics and cosmology, we study the distribution of "defect" (or "signed measure") of high degree random spherical harmonics. We were able to determine the asymptotic shape of the defect variance precisely, and moreover prove a version of Central Limit Theorem for its distribution; our techniques yield similar results for other functionals, provided they satisfy some generic condition. Our proofs combine asymptotic analysis of the Legendre polynomials, together with a recently developed inequality of Nourdin-Peccati, based on the Malliavin Calculus. In this talk I plan to introduce the subject and discuss, at least briefly, the proofs of the main results.
Carolyn Chun : A matroid is a mathematical structure that generalizes the notion of linear independence in a matrix. In this talk, I will discuss recent progress on the most compelling open question in matroid theory, Rota’s conjecture.
Christopher Gill: One of the main outstanding questions in the representation theory of the symmetric groups, and the Schur algebras, is to determine the decomposition numbers. Recently Cohen, Hemmer, and Nakano have shown that determining the multiplicities of the direct summands occurring in a tensor product of Young modules is equivalent to determining these decomposition numbers. In this talk I will describe recent work giving certain restrictions on the vertices of direct summands in such a tensor product, and some reductions theorems for these multiplicities .
Mark Walters: Suppose we have a finite graph. We can view this as a resistor network where each edge has unit resistance. We can then calculate the resistance between any two vertices and ask questions like `which graph with n vertices and m edges minimises the average resistance between pairs of vertices?' We are not able to fully answer this question but we can show that the obvious answer is not correct. In this talk we will discuss the obvious answer, why it is wrong, and why the correct solution seems hard to find.
This problem was motivated by some questions about the design of statistical experiments (and has some surprising applications in chemistry) but this talk will not assume any statistical knowledge.
Stephen Huggett: We describe some of Newton's most profound geometrical discoveries, arguing that by thinking of him as a geometer we gain a deep insight into his peculiar genius. We pay particular attention to Newton's work on the organic construction, which deserves to be better known, being a classical geometrical construction of the Cremona transformation (1862). Newton was aware of its importance in geometry, using it to generate algebraic curves, including those with singularities. This is joint work with Nicole Bloye.
Brita Nucinkis : Richard Thompson's groups F, T and V are groups of homeomorphisms of the unit interval, the circle and the Cantor-set respectively. In this talk I will describe how one can view these groups and their generalisations as groups of automorphisms of so called Cantor-algebras, and how this viewpoint can be used to derive cohomological finiteness properties.
Bill Jackson: I will describe some of my favourite results and open problems for the chromatic polynomial and its relatives: flow polynomials, characteristic polynomials, Tutte polynomials and Potts model partition functions. I will also outline some techniques for working with these polynomials.
Ashot Minasyan: The concept of a hyperbolically embedded subgroup was introduced in a recent paper of Dahmani, Guirardel and Osin, where it was used to solve a number of open problems about the mapping class groups of closed surfaces and the outer automorphism groups of free groups.
I will discuss groups with such hyperbolically embedded subgroups and their similarities with relatively hyperbolic groups. I will then talk about a criterion for the existence of hyperbolically embedded subgroups in groups that act on simplicial trees.
Martin Loebl: The long-standing Rota's bases conjecture asserts that the columns of arbitrary n non-singular nxn matrices can be partitioned into n sets such that each set forms a non-singular matrix, and each set is a transversal, i.e., it has exactly one column of each of the original matrices.The long-standing Alon Tarsi conjecture asserts that for n even, the sum of sign(L), L nxn latin square, is non-zero, where sign(L) is the product of the 2n permutations given by the rows and the columns of L. Note that the analogous sum is zero for n odd.
In the 90's it was proven that the Alon Tarsi Conjecture implies the Rota's bases conjecture for n even. I will speak about our recent result asserting that for each n (odd or even), Rota's bases conjecture is implied by the assertion that the sum of sign(L'), L' nxn latin square with 1st row and 1st column equal to the identity permutation, is non-zero. Note that for n even, this condition is equivalent to the Alon Tarsi conjecture. The starting point in our reasoning is a non-commutative generalization of a formula of Shmuel Onn. We also establish a link between the Rota's bases conjecture and the Ryser Brualdi Stein conjecture. This is joint work with Ron Aharoni.
Francesco Matucci:
In a recent paper, Bogopolski, Martino and Ventura develop a criterion to study the conjugacy problem in extension of groups. This is achieved by reducing to the study of two other decision problems: the twisted conjugacy problem and the orbit decidability problem. We describe a simplified point of view of the conjugacy in Thompson's group F which allows us to attack these decision problems. We produce examples of extensions of F where the conjugacy problem is solvable and others where it is unsolvable. We also prove related results on twisted conjugacy class and discuss possible generalizations. This is joint work with Jose' Burillo and Enric Ventura.
Conchita Martinez-Perez: This is a joint work with Warren Dicks. We review arguments in the literature that together with a new result yield the complete classification up to isomorphism of the Brin-Higman-Thompson groups.
Steven Galbraith: Antoine Joux has recently developed new algorithmic ideas for index calculus algorithms for the discrete logarithm problem (DLP) in finite fields. This computational problem is central to public key cryptography. On May 21 he announced a solution to the DLP in GF(2^6168)^*. This was a major computational milestone. Last Tuesday he (in collaboration with others) announced a paper claiming that the algorithm runs in "quasi-polynomial" time (*arXiv:1306.4244). *I attended two lectures in Paris last week on this work. In this talk I will sketch these new ideas, to the best of my current understanding.*
Previous seminars
Summer 2012
Autumn 2012