Past

We construct triangle equivalences between singularity categories of
two-dimensional cyclic quotient singularities and singularity categories of
a new class of finite dimensional local algebras, which we call Knörrer
invariant algebras. In the hypersurface case, we recover a special case of Knörrer’s equivalence for (stable) categories of matrix factorisations.
We’ll then explain how this led us to study Ringel duality for
certain (ultra strongly) quasi-hereditary algebras.
This is based on joint work with Joe Karmazyn.

Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretised, they lead to ill-conditioned linear systems, often of huge dimensions: regularisation consists in replacing the original system by a nearby problem with better numerical properties, in order to find meaningful approximations of its solution. In this talk we will explore the regularisation properties of many iterative methods based on Krylov subspaces. After surveying some basic methods such as CGLS and GMRES, innovative approaches based on flexible variants of CGLS and GMRES will be presented, in order to efficiently enforce nonnegativity and sparsity into the solution.

Patlak-Keller-Segel equations 
\[
\begin{aligned}
u_t - L u &= - \mathop{\text{div}\,} (u \nabla v) \\
v_t - \Delta v &= u,
\end{aligned}
\]
where L is a dissipative operator, stem from mathematical chemistry and mathematical biology.
Their variants describe, among others, behaviour of chemotactic populations, including feeding strategies of zooplankton or of certain insects. Analytically, Patlak-Keller-Segel equations reveal quite rich dynamics and a delicate global smoothness vs. blowup dichotomy. 
We will discuss smoothness/blowup results for popular variants of the equations, focusing on the critical cases, where dissipative and aggregative forces seem to be in a balance. A part of this talk is based on joint results with Rafael Granero-Belinchon (Lyon).

The Thue-Morse sequence is perhaps the simplest example of an automatic sequence. Various pseudorandomness properties of this sequence have long been studied. During the talk, I will discuss a new result in this direction, asserting that the Gowers uniformity norms of the Thue-Morse sequence are small in a quantitative sense. Similar results hold for the Rudin-Shapiro sequence, as well as for a much wider class of automatic sequences which will be introduced during the talk.

The talk is partially based on joint work with Jakub Byszewski.

Using a variational approach applied to generalized Rayleigh functionals, we extend the concepts of singular values and singular functions to trivariate functions defined on a rectangular parallelepiped. We also consider eigenvalues and eigenfunctions for trivariate functions whose domain is a cube. For a general finite-rank trivariate function, we describe an algorithm for computing the canonical polyadic (CP) decomposition, provided that the CP factors are linearly independent in two variables. All these notions are computed using Chebfun. Application in finding the best rank-1 approximation of trivariate functions is investigated. We also prove that if the function is analytic and two-way orthogonally decomposable (odeco), then the CP values decay geometrically, and optimal finite-rank approximants converge at the same rate.
 

Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an
Euler-Poincare formula for the r-depth Bernstein projector. We establish an Euler-Poincare formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P.  This work is joint with Dan Barbasch and Dan Ciubotaru.
 

In this talk, I will talk about my current approach to model customer movements and in particular congestion inside supermarkets using queuing networks. As the research question for my project is ‘How should one design supermarkets to minimize congestion?’, I will then talk about my current progress in understanding how the network structure can affect this dynamics.

Linear elasticity can be rigorously derived from finite elasticity in the case of small loadings in terms of \Gamma-convergence. This was first done by Dal Maso-Negri-Percivale in the case of one-well energies with super-quadratic growth. This has been later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). I will discuss recent developments in the case when the distance between the wells is arbitrary. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions which turn out to be optimal. (This is joint work with Alicandro, Dal Maso and Lazzaroni.) 

Gaussian fields are prevalent throughout mathematics and the sciences, for instance in physics (wave-functions of high energy electrons), astronomy (cosmic microwave background radiation) and probability theory (connections to SLE, random tilings etc). Despite this, the geometry of such fields, for instance the connectivity properties of level sets, is poorly understood. In this talk I will discuss methods of extracting geometric information about levels sets of a planar Gaussian field through discrete observations of the field. In particular, I will present recent work that studies three such discretisation schemes, each tailored to extract geometric information about the levels set to a different level of precision, along with some applications.

Large-scale homology computations are often rendered tractable by discrete Morse theory. Every discrete Morse function on a given cell complex X produces a Morse chain complex whose chain groups are spanned by critical cells and whose homology is isomorphic to that of X. However, the space-level information is typically lost because very little is known about how critical cells are attached to each other. In this talk, we discretize a beautiful construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X from an overlaid discrete Morse function.

 Monte Carlo methods are one of the main tools of modern statistics and applied mathematics. They are commonly used to approximate integrals, which allows statisticians to solve many tasks of interest such as making predictions or inferring parameter values of a given model. However, the recent surge in data available to scientists has led to an increase in the complexity of mathematical models, rendering them much more computationally expensive to evaluate. This has a particular bearing on Monte Carlo methods, which will tend to be much slower due to the high computational costs.

This talk will introduce a Monte Carlo integration scheme which makes use of properties of the integrand (e.g. smoothness or periodicity) in order to obtain fast convergence rates in the number of integrand evaluations. This will allow users to obtain much more precise estimates of integrals for a given number of model evaluations. Both theoretical properties of the methodology, including convergence rates, and practical issues, such as the tuning of parameters, will be discussed. Finally, the proposed algorithm will be illustrated on a Bayesian inverse problem for a PDE model of subsurface flow.

The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT).  Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. Methods from non-reductive GIT allow us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves of fixed singularity type.

A conformal field theory is characterised by the CFT data, namely the spectrum of scaling dimensions and OPE coefficients. The idea of the conformal bootstrap is to use associativity of the operator algebra together with the symmetries of the theory to constraint the CFT data. For the sector of operators with large spin one can actually use these ideas to obtain analytical results. It was recently understood how to set up a systematic expansion around this sector, leading to analytic results to all orders in inverse powers of the spin. We will show how to use this large spin perturbation theory to obtain analytic results for vast families of CFTs. Some of the applications include vector models, weakly coupled gauge theories and the computation of loops for scalar theories in AdS.

In Your Third Year & want to find out about opportunities for summer placements and future graduate study?

Why not visit Oxford and hear from graduate students about their research

Saturday 21 January 2017: 1-6pm

Mathematical Institute, University of Oxford

TALKS ON

• Dynamics of jumping elastic toys
• Vertex models in developmental biology
• Modelling of glass sheets
• Glimpse into the mathematics of information
• Network analysis of consumer data
• Complex singularities in jet and splash flows

Complementary Lunch & Drinks Reception - TRAVEL BURSARIES AVAILABLE (up to £50)

Please RSVP to @email

In Your Third Year & want to find out about opportunities for

summer placements and future graduate study?

Why not visit Oxford and hear from graduate students about their research

TALKS ON

Dynamics of jumping elastic toys

Vertex models in developmental biology

Modelling of glass sheets

Glimpse into the mathematics of information

Network analysis of consumer data

Complex singularities in jet and splash flows

Complementary Lunch & Drinks Reception - TRAVEL BURSARIES AVAILABLE (up to £50)

Please RSVP to @email

A continuum of expanders -- David Hume

Expanders are a holy grail of networking; robustly connected networks of arbitrary size which require minimal resources. Like the grail, they are also notoriously difficult to construct. In this talk I will introduce expanders, give a brief overview of just a few aspects of their diverse history, and outline a very recent result of mine, which states that there are a continuum of expanders with fundamentally different large-scale geometry.

What makes cities successful? A complex systems approach to modelling urban economies -- Neave O'Clery

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, congestion and density. Seeking to uncover the mechanisms behind the success of cities using novel tools from the mathematical and data sciences, this work uses network techniques to model the opportunity landscape of cities. Under the theory that cities move into new economic activities that share inputs with existing capabilities, path dependent industrial diversification can be described using a network of industries. Edges represent shared necessary capabilities, and are empirically estimated via flows of workers moving between industries. The position of a city in this network (i.e., the subnetwork of its current industries) will determine its future diversification potential. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities.

We develop this framework to explain the large variation in labour formality rates across cities in the developing world, using data from Colombia. We show that, as cities become larger, they move into increasingly complex industries as firms combine complementary capabilities derived from a more diverse pool of workers. We further show that a level of agglomeration equivalent to between 45 and 75 minutes of commuting time maximizes the ability of cities to generate formal employment using the variety of skills available. Our results suggest that rather than discouraging the expansion of metropolitan areas, cities should invest in transportation to enable firms to take advantage of urban diversity.

This talk will be based on joint work with Eduardo Lora and Andres Gomez at Harvard University.

Abstract: Triangulations of surfaces serve as important examples for cluster theory, with the natural operation of “diagonal flips” encoding mutation in cluster algebras and categories. In this talk we will focus on the combinatorics of mutation on marked surfaces with infinitely many marked points, which have gained importance recently with the rising interest in cluster algebras and categories of infinite rank. In this setting, it is no longer possible to reach any triangulation from any other triangulation in finitely many steps. We introduce the notion of mutation along infinite admissible sequences and show that this induces a preorder on the set of triangulations of a fixed infinitely marked surface. Finally, in the example of the completed infinity-gon we define transfinite mutations and show that any triangulation of the completed infinity-gon can be reached from any other of its triangulations via a transfinite mutation. The content of this talk is joint work with Karin Baur.