Wed, 19 Oct 2016

16:00 - 17:00
C1

Kähler groups, residually free groups and subgroups of direct products of surface groups.

Claudio Llosa Isenrich
(Oxford University)
Abstract

A Kähler group is a group which can be realised as the fundamental group of a close Kähler manifold. We will prove that for a Kähler group $G$ we have that $G$ is residually free if and only if $G$ is a full subdirect product of a free abelian group and finitely many closed hyperbolic surface groups. We will then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler: We explain how to construct subgroups of direct products of surface groups which have even first Betti number but are not Kähler. All relevant notions will be explained in the talk.

Thu, 13 Oct 2016

14:00 - 15:00
L5

Optimization with occasionally accurate data

Prof. Coralia Cartis
(Oxford University)
Abstract


We present global rates of convergence for a general class of methods for nonconvex smooth optimization that include linesearch, trust-region and regularisation strategies, but that allow inaccurate problem information. Namely, we assume the local (first- or second-order) models of our function are only sufficiently accurate with a certain probability, and they can be arbitrarily poor otherwise. This framework subsumes certain stochastic gradient analyses and derivative-free techniques based on random sampling of function values. It can also be viewed as a robustness
assessment of deterministic methods and their resilience to inaccurate derivative computation such as due to processor failure in a distribute framework. We show that in terms of the order of the accuracy, the evaluation complexity of such methods is the same as their counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. Time permitting, we also discuss the case of inaccurate, probabilistic function value information, that arises in stochastic optimization. This work is joint with Katya Scheinberg (Lehigh University, USA).
 

Fri, 20 May 2016
10:00
N3.12

Hall Algebras and Green's theorem

Adam Gal
(Oxford University)
Abstract

Hall algebras are a deformation of the K-group (Grothendieck group) of an abelian category, which encode some information about non-trivial extensions in the category.
A main feature of Hall algebras is that in addition to the product (which deforms the product in the K-group) there is a natural coproduct, which in certain cases makes the Hall algebra a (braided) bi-algebra. This is the content of Green's theorem and supplies the main ingredient in a construction of quantum groups.

Tue, 10 May 2016
14:30
L6

Finite Reflection Groups and Graph Norms

Joonkyung Lee
(Oxford University)
Abstract

For any given graph H, we may define a natural corresponding functional ||.||_H. We then say that H is norming if ||.||_H is a semi-norm. A similar notion ||.||_{r(H)} is defined by || f ||_{r(H)}:=|| | f | ||_H and H is said to be weakly norming if ||.||_{r(H)} is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. Using results from the theory of finite reflection groups, we demonstrate that any graph which is edge-transitive under the action of a certain natural family of automorphisms is weakly norming. This result includes all previous examples of weakly norming graphs and adds many more. We also include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture. Joint work with David Conlon.

Thu, 16 Jun 2016
16:00
L6

Gaps Between Smooth Numbers

Roger Heath-Brown
(Oxford University)
Abstract

Let $a_1, \cdots, a_N$ be the sequence of y-smooth numbers up to x (i.e. composed only of primes up to y). When y is a small power of x, what can one say about the size of the gaps $a_{j+1}-a_j$? In particular, what about

$$\sum_1^N (a_{j+1}-a_j)^2?$$

Tue, 01 Mar 2016

15:00 - 16:00
L1

A "Simple" Answer to a "Not Quite Simple" Problem - The Prequel to A "Simple" Question

Kesavan Thanagopal
(Oxford University)
Abstract

In this seminar, I aim to go through the "main prequel" of the talk I gave during the first Advanced Class of this term, and provide a "simple" answer to Abraham Robinson's original question that he posed in 1973 regarding the (un)decidability of finitely generated extensions of undecidable fields. I will provide a quick introduction to, and some classical results from, the mathematical discipline of Field Arithmetic, and using these results show that one can construct undecidable (large) fields that have finitely generated extensions which are decidable. Of course, as I had mentioned in the advanced class, a counterexample to the "simple" question that I have been working on unfortunately does not seem to lie within this class of large fields. If time permits, I will provide a sneak peek into the possible "sequel" by briefly talking about what the main issue of solving the "simple" problem is, and how a "hide-and-seek" method might come in handy in tackling that problem.

Wed, 24 Feb 2016

16:00 - 17:00
C3

CAT(0) Boundaries

Robert Kropholler
(Oxford University)
Abstract

I will talk about the boundaries of CAT(0) groups giving definitions, some examples and will state some theorems. I may even prove something if there is time. 

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