Postgraduate Seminars Spring 2020

These seminars (unless otherwise stated) will take place on Thursdays at 1pm in Maths Room 706 on an almost weekly basis.

See the map for further details. Talks are being given by 2nd and 3rd year Mathematics PhD students for PhD students.

Tuesday 14 January 2020 at 12pm in Maths Room 707

Speaker: Andries Salm

Supervisor: Dr Lorenzo Foscolo

Title: Holonomy and the Berger Classification

In geometry people are often interested in spaces with some additional structure. Namely, they may consider Riemannian, (hyper) Kähler, Calabi-Yau or other exotic manifolds like G2 or Spin(7). In this seminar we explain how these manifolds are related by varying vectors along loops. We will see that this variation will not only abstractly classify our spaces, but gives the additional structures we are interested in.

Tuesday 21 January 2020 at 1pm in Maths Room 706

Speaker: Dimitris Lekkas

Supervisor: Prof Yiannis Petridis

Title: On a Hyperbolic lattice counting problem

Hyperbolic lattice counting problems concern the counting of the number of points in the orbit of a discrete group Γ that lie in a subset of the upper half plane Η. The discrete subgroup Γ is a subgroup of PSL_2(R) thought as the orientation preserving isometries on H, on which it acts by linear fractional transformations. One of the classical problems in the field, is the hyperbolic analogue of the Gauss circle problem of counting

N(X) = #{(a,b) in Z^2 | X= a^2+b^2}

which in the hyperbolic setting can be formulated as the counting of

N(X,z,w) = #{γ in Γ | 4dist(γz,w)+2< X}

In this talk, we will discuss the classical problem as well as some modifications of it, in particular the case when the group Γ is replaced by A\Γ/A, where A is a hyperbolic subgroup of Γ.

30 January 2020

Speaker: Petru Constantinescu

Supervisor: Prof Yiannis Petridis

Title: Spectral theory of automorphic forms

My aim is to give a very accessible talk where I motivate the very rich subject of automorphic forms. I will highlight the basic properties of the hyperbolic upper half-plane, the modular curve, the Laplace operator on this surface and finally the spectral decomposition of square-integrable functions. If time allows, I will highlight some applications to the theory of modular symbols.

06 February 2020

Speaker: Shujian Liao

Supervisor: Dr Hao Ni

Title: Learning SDEs using RNN with log-signature features

Learning a function of streamed data through evaluation is an important question in many scientific areas. The difficulty comes from the high frequency of the driven stream. Motivated by the numerical approximation theory of the stochastic differential equations and machine learning fundamentals, I will introduce a novel approach (Logsig-RNN) combining the log-signature as representations of streamed data and the recurrent neural network (RNN). The ability of the former to manage high-frequency streams and the latter to manage large scale nonlinear interactions allows hybrid algorithms that achieve higher accuracy, are quicker to train. By testing on various human recognition datasets, i.e. NTU RGB+D 120 action data and Chalearn 2013 gesture data, I will show our proposed approach achieved state-of-art accuracy with high efficiency and robustness.

13 February 2020

Speaker: Peter Hearnshaw

Supervisor: Prof Leonid Parnovski/Prof Alex Sobolev

Title: Introduction to unbounded operators

There are many ways to understand the Laplacian on domains in R^n. We can consider it being able to act in the classical sense on functions which are twice differentiable. At the other extreme we could consider its action on distributions which allows us to define it on functions with very low regularity such as discontinuous functions. It turns out we need to define it on an intermediate between these two extremes in order to understand it as self-adjoint and obtain a complete understanding of the operator via its unique spectral measure. I will mention bounded operators then show the vital role the domain plays in defining unbounded operators. Through a series of easy results it is possible to understand how we should handle these types of operators. I'll then show that if we're able to define them correctly, spectral theory offers a huge number of results about their behaviour.

20 February 2020



27 February 2020

Speaker: Antigoni Kleanthous

Supervisor: Prof Timo Betcke

Title: Accelerated Calderón Preconditioning for light scattering by ice crystals in cirrus clouds

The Boundary Element Method is a popular technique used to simulate electromagnetic scattering by multiple absorbing dielectric particles. Different formulations exist, where the Maxwell equations are written in terms of boundary integral equations on the boundary of the scatterers. One such formulation is the PMHCWT formulation which upon discretisation suffers from ill-conditioning leading to slow convergence of iterative solvers such as GMRES. Calderón preconditioning is a well-known method used to remedy this ill-conditioning by leveraging the so-called Calderon identities that regularise the operators used. This reduces the number of iterations required by the iterative solver but at the expense of an increased number of matrix-vector products per iteration. 

We found that for single-particle problems, the  method is often outperformed by a simple mass-matrix preconditioner. For multi-particle problems a block-diagonal Calderón preconditioner provides a significant reduction in computational cost. To accelerate assembly time and reduce memory requirements, we explore the capabilities of a bi-parametric implementation where two sets of parameters are used:

  • high accuracy and high quadrature orders are used for the operator along with RWG basis functions which are cheap to assemble
  • lower accuracy and minimum quadrature orders are used for the preconditioner along with BC functions which are 6-times more expensive to assemble.

We also explore the performance of the implementations when only near-field interactions are included in the preconditioner. 

Finally, I'll present some of the results from a 3-month internship at the Met Office, where we've used the above methods to simulate properties of light scattering by aggregates of ice crystals found in cirrus clouds.

05 March 2020

Speaker: Bruno Souza Roso

Supervisor: Prof Michael Singer

Title: Introduction to Seiberg-Witten Theory

Simon Donaldson's use of gauge theory to probe the topology of four-manifolds changed the landscape of low dimensional topology in the 80s; his main results came from studying moduli spaces of SU(2)-instantons which required difficult theorems from geometric analysis such as Uhlenbeck's theorems.  In the 90s, Edward Witten suggested that the same results should be recoverable through the Seiberg-Witten theory, which already was well known to physicists; the two key advantages over the use of instantons are the facts that the gauge group is abelian and the moduli space compact.  Soon enough, all results of Donaldson theory were re-derived through this simplified setup and numerous new results have been discovered since.  Today, Seiberg-Witten theory, both in its four-dimensional version and the Floer-theoretic three-dimensional version are indispensable tools in low dimensional topology and continue to gather substantial research interest.  The talk shall introduce the Seiberg-Witten equations on four-manifolds and outline the process needed to construct the Seiberg-Witten invariant from them.  To finish off, some applications shall be discussed.

12 March 2020