Thu, 24 Oct 2019
13:00

Industrial agglomeration and diversification

Dr Samuel Heroy
(University of Oxford)
Abstract

As early as the 1920's Marshall suggested that firms co-locate in cities to reduce the costs of moving goods, people, and ideas. These 'forces of agglomeration' have given rise, for example, to the high tech clusters of San Francisco and Boston, and the automobile cluster in Detroit. Yet, despite its importance for city planners and industrial policy-makers, until recently there has been little success in estimating the relative importance of each Marshallian channel to the location decisions of firms.
Here we explore a burgeoning literature that aims to exploit the co-location patterns of industries in cities in order to disentangle the relationship between industry co-agglomeration and customer/supplier, labour and idea sharing. Building on previous approaches that focus on across- and between-industry estimates, we propose a network-based method to estimate the relative importance of each Marshallian channel at a meso scale. Specifically, we use a community detection technique to construct a hierarchical decomposition of the full set of industries into clusters based on co-agglomeration patterns, and show that these industry clusters exhibit distinct patterns in terms of their relative reliance on individual Marshallian channels.

The second part is to use industry relatedness, which we measure via a similar metric to co-location, to better understand the association of industrial emissions to city-industry agglomeration. Specifically, we see that industrial emissions (which are the largest source of greenhouse emissions in the US) are highly tied to certain industries, and furthermore that communities in the industry relatedness network tend to explain the tendency of particular industry clusters to produce emissions. This is important, because it limits cities' abilities to move to a greener industry basket as some cities may be more or less constrained to highly polluting industry clusters, while others have more potential for diversification away from polluting industries.

Thu, 24 Oct 2019

12:00 - 13:00
L4

Structure theory of RCD spaces up to codimension 1

Daniele Semola
(Scuola Normale Superiore di Pisa)
Abstract

The aim of this talk is to give an overview about the structure theory of finite dimensional RCD metric measure spaces. I will first focus on rectifiability, existence, uniqueness and constancy of the dimension of tangents up to negligible sets.
Then I will motivate why boundaries of sets of finite perimeter are natural codimension one objects to look at in this framework and present some recent structure results obtained in their study.
This is based on joint works with Luigi Ambrosio, Elia Bruè and Enrico Pasqualetto.
 

Wed, 23 Oct 2019
16:00
C1

Surfaces via subsurfaces: an introduction to Masur-Minsky

Harry Petyt
(University of Bristol)
Abstract

The mapping class group of a surface is a group of homeomorphisms of that surface, and these groups have been very well studied in the last 50 years. The talk will be focused on a way to understand such a group by looking at the subsurfaces of the corresponding surface; this is the so-called "Masur-Minsky hierarchy machinery". We'll finish with a non-technical discussion of hierarchically hyperbolic groups, which are a popular area of current research, and of which mapping class groups are important motivating examples. No prior knowledge of the objects involved will be assumed.

Tue, 22 Oct 2019

15:30 - 16:30
L6

Asymptotics of Toeplitz determinants with Fisher-Hartwig singularities and applications to random matrix theory

Benjamin Fahs
(Imperial College London)
Abstract

We discuss asymptotics of Toeplitz determinants with Fisher--Hartwig singularities, and give an overview of past and more recent results.
Applications include the study of asymptotics of certain statistics of the characteristic polynomial of the Circular Unitary Ensemble (CUE) of random matrices. In particular recent results in the study of Toeplitz determinants allow for a proof of a conjecture by Fyodorov and Keating on moments of averages of the characteristic polynomial of the CUE.
 

Tue, 22 Oct 2019

15:30 - 16:30
L4

Stability conditions and spectral networks

Fabian Haiden
(Oxford)
Abstract

Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.

Tue, 22 Oct 2019

14:30 - 15:00
L5

An optimal polynomial approximation of Brownian motion

James Foster
(Oxford)
Abstract

In this talk, I will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. Most notably, the coefficients obtained from this expansion are independent Gaussian random variables. This will enable us to generate approximate Brownian paths by matching certain polynomial moments. To conclude the talk, I will discuss related works and applications to numerical methods for SDEs.
 

Tue, 22 Oct 2019
14:15
L4

Representations associated to gradations of colour Lie algebras

Philippe Meyer
(Oxford University)
Abstract

The notion of colour Lie algebra, introduced by Ree (1960), generalises notions of Lie algebra and Lie superalgebra. From an orthogonal representation V of a quadratic colour Lie algebra g, we give various ways of constructing a colour Lie algebra g’ whose bracket extends the bracket of g and the action of g on V. A first possibility is to consider g’=g⊕V and requires the cancellation of an invariant studied by Kostant (1999). Another construction is possible when the representation is ``special’’ and in this case the extension is of the form g’=g⊕sl(2,k)⊕V⊗k^2. Covariants are associated to special representations and satisfy to particular identities generalising properties studied by Mathews (1911) on binary cubics. The 7-dimensional fundamental representation of a Lie algebra of type G_2 and the 8-dimensional spinor representation of a Lie algebra of type so(7) are examples of special representations.

Tue, 22 Oct 2019

14:00 - 15:00
L6

Homomorphisms from the torus

Matthew Jenssen
(Oxford)
Further Information

We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus Z_m^n, where m is even, to any fixed graph. Our main result establishes the "phase coexistence" phenomenon in a strong form: it shows that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some "dominant phase". This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper q-colourings of Z_m^n (so in particular, the discrete hypercube). For the proof we develop a `Cluster Expansion Method', which we expect to have further applications, by combining machinery from statistical physics, entropy and graph containers. This is joint work with Peter Keevash.
 

 
Tue, 22 Oct 2019

14:00 - 14:30
L5

A neural network based policy iteration algorithm with global H^2 -superlinear convergence for stochastic games on domains

Yufei Zhang
(Oxford)
Abstract

In this work, we propose a class of numerical schemes for solving semilinear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H^2 -norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem and the perpetual American option pricing problems are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.
 

Tue, 22 Oct 2019

12:45 - 14:00
C5

Numerical Simulations using Approximate Random Numbers

Oliver Sheridan-Methven
((Oxford University))
Abstract

Introducing cheap function proxies for quickly producing approximate random numbers, we show convergence of modified numerical schemes, and coupling between approximation and discretisation errors. We bound the cumulative roundoff error introduced by floating-point calculations, valid for 16-bit half-precision (FP16). We combine approximate distributions and reduced-precisions into a nested simulation framework (via multilevel Monte Carlo), demonstrating performance improvements achieved without losing accuracy. These simulations predominantly perform most of their calculations in very low precisions. We will highlight the motivations and design choices appropriate for SVE and FP16 capable hardware, and present numerical results on Arm, Intel, and NVIDIA based hardware.

 

Tue, 22 Oct 2019
12:00
L4

Differential equations for one-loop string integrals

Oliver Schlotterer
(Uppsala)
Abstract

In this talk, I will describe new mathematical structures in the low-energy  expansion of one-loop string amplitudes. The insertion of external states on the open- and closed-string worldsheets requires integration over punctures on a cylinder boundary and a torus, respectively. Suitable bases of such integrals will be shown to obey simple first-order differential equations in the modular parameter of the surface. These differential equations will be exploited  to perform the integrals order by order in the inverse string tension, similar to modern strategies for dimensionally regulated Feynman integrals. Our method manifests the appearance of iterated integrals over holomorphic  Eisenstein series in the low-energy expansion. Moreover, infinite families of Laplace equations can be generated for the modular forms in closed-string  low-energy expansions.
 

Tue, 22 Oct 2019

12:00 - 13:00
C1

Learning from signals on graphs with unobserved edges

Micheal Schaub
(Department of Engineering)
Abstract

In many applications we are confronted with the following scenario: we observe snapshots of data describing the state of a system at particular times, and based on these observations we want to infer the (dynamical) interactions between the entities we observe. However, often the number of samples we can obtain from such a process are far too few to identify the network exactly. Can we still reliable infer some aspects of the underlying system?
Motivated by this question we consider the following alternative system identification problem: instead of trying to infer the exact network, we aim to recover a (low-dimensional) statistical model of the network based on the observed signals on the nodes.  More concretely, here we focus on observations that consist of snapshots of a diffusive process that evolves over the unknown network. We model the (unobserved) network as generated from an independent draw from a latent stochastic block model (SBM), and our goal is to infer both the partition of the nodes into blocks, as well as the parameters of this SBM. We present simple spectral algorithms that provably solve the partition and parameter inference problems with high-accuracy.

Mon, 21 Oct 2019

16:00 - 17:00
C1

Relative decidability via the tilting correspondence

Konstantinos Kartas
(Oxford University)
Abstract

The goal of the talk is to present a proof of the following statement:
Let (K,v) be an algebraic extension of (Q_p,v_p) whose completion is perfectoid. We show that K is relatively decidable to its tilt K^♭, i.e. if K^♭ is decidable in the language of valued fields, then so is K. 
In the first part [of the talk], I will try to cover the necessary background needed from model theory and the theory of perfectoid fields.

Mon, 21 Oct 2019

16:00 - 17:00
L4

Quantitative geometric inequalities

Fabio Cavalletti
(SISSA)
Abstract

Localization technique permits to reduce full dimensional problems to possibly easier lower dimensional ones. During the last years a new approach to localization has been obtained using the powerful tools of optimal transport. Following this approach, we obtain quantitative versions of two relevant geometric inequalities  in comparison geometry as Levy-Gromov isoperimetric inequality (joint with F. Maggi and A. Mondino) and the spectral gap inequality (joint with A. Mondino and D. Semola). Both results are also valid in the more general setting of metric measure spaces verifying the so-called curvature dimension condition.

Mon, 21 Oct 2019
15:45
L6

Lower bounds on the tunnel number of composite spatial theta graphs

Scott Taylor
(Colby College)
Abstract

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots. The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs, using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

Mon, 21 Oct 2019

15:45 - 16:45
L3

Fatou's Lemmas for Varying Probabilities and their Applications to Sequential Decision Making

EUGENE FEINBERG
(Stony Brook University)
Abstract

The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma. We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time. This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Michael Zgurovsky, and Nina Zadoianchuk.

Mon, 21 Oct 2019

14:15 - 15:15
L3

Variational Inference in Gaussian processes

JAMES HENSMAN
(Prowler.io)
Abstract

 Gaussian processes are well studied object in statistics and mathematics. In Machine Learning, we think of Gaussian processes as prior distributions over functions, which map from the index set to the realised path. To make Gaussian processes a practical tool for machine learning, we have developed tools around variational inference that allow for approximate computation in GPs leveraging the same hardware and software stacks that support deep learning. In this talk I'll give an overview of variational inference in GPs, show some successes of the method, and outline some exciting direction of potential future work.

Mon, 21 Oct 2019

14:15 - 15:15
L4

The pure cohomology of multiplicative quiver varieties

Kevin McGerty
(Oxford)
Further Information

Multiplicative quiver varieties are a variant of Nakajima's "additive" quiver varieties which were introduced by Crawley-Boevey and Shaw.
They arise naturally in the study of various moduli spaces, in particular in Boalch's work on irregular connections. In this talk we will discuss joint work with Tom Nevins which shows that the tautological classes for these varieties generate the largest possible subalgebra of the cohomology ring, namely the pure part.

 

Mon, 21 Oct 2019

12:45 - 13:45
L3

The Higgs Mechanism and Hasse diagrams

Antoine Bourget
(Imperial College London)
Abstract

I will explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are hyperkahler singularities, and as such they can be described by a Hasse diagram built from a family of elementary transitions. This corresponds physically to the partial Higgs mechanism. Using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagrams.

Fri, 18 Oct 2019

16:00 - 17:00
L1

Geometry as a key to the virosphere: Mathematics as a driver of discovery in virology and anti-viral therapy

Reidun Twarock
(University of York)
Further Information

The Oxford Mathematics Colloquia are generously sponsored by Oxford University Press.

Abstract

Viruses encapsulate their genetic material into protein containers that act akin to molecular Trojan horses, protecting viral genomes between rounds of infection and facilitating their release into the host cell environment. In the majority of viruses, including major human pathogens, these containers have icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to better understand how viruses form, evolve and infect their hosts, and point the way to novel antiviral solutions.

In this talk, I will present an overarching theory of virus architecture, that contains the seminal Caspar Klug theory as a special case and solves long-standing open problems in structural virology. Combining insights into virus structure with a range of different mathematical modelling techniques, such as Gillespie algorithms, I will show how viral life cycles can be better understood through the lens of viral geometry. In particular, I will discuss a recent model for genome release from the viral capsid. I will also demonstrate the instrumental role of the Hamiltonian path concept in the discovery of a virus assembly mechanism that occurs in many human pathogens, such as Picornaviruses – a family that includes the common cold virus– and Hepatitis B and C virus. I will use multi-scale models of a viral infection and implicit fitness landscapes in order to demonstrate that therapeutic interventions directed against this mechanism have advantages over conventional forms of anti-viral therapy. The talk will finish with a discussion of how the new mathematical and mechanistic insights can be exploited in bio-nanotechnology for applications in vaccination and gene therapy.

Fri, 18 Oct 2019

14:00 - 15:00
L1

Making the most of the intercollegiate classes

Dr Vicky Neale, Dr Richard Earl, Dr Neil Laws and George Cooper
Abstract

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.  

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)

Fri, 18 Oct 2019

14:00 - 15:00
L3

Cell polarity formation and the dynamics of small G proteins; or, why your Turing bifurcations should always be subcritical

Professor Alan Champneys
(Dept of Engineering Maths University of Bristol)
Abstract

In this talk I shall describe recent work inspired by problems in cell biology, namely how the dynamics of small G-proteins underlies polarity formation. Their dynamics is such that their active membrane bound form diffuses more slowly. Hence you might expect Turing patterns. Yet how do cells form backs and fronts or single isolated patches. In understanding these questions we shall show that the key is to identify the parameter region where Turing bifurcations are sub-critical. What emerges is a unified 2-parameter bifurcation diagram containing pinned fronts, localised spots, localised patterns. This diagram appears in many canonical models such as Schnakenberg and Brusselator, as well as biologically more realistic systems. A link is also found between theories of semi-string interaction asymptotics and so-called homoclinic snaking. I will close with some remarks about relevance to root hair formation and to the importance of subcriticality in biology. 

Fri, 18 Oct 2019

12:00 - 13:00
L4

DPM: A deep learning algorithm for estimating PDE models from data

Justin Sirignano
(The University of Illinois at Urbana-Champaign)
Abstract

Machine learning for scientific applications faces the challenge of limited data. To reduce overfitting, we propose a framework to leverage as much as possible a priori-known physics for a problem. Our approach embeds a deep neural network in a partial differential equation (PDE) system, where the pre-specified terms in the PDE capture the known physics and the neural network will learn to describe the unknown physics. The neural network is estimated from experimental and/or high-fidelity numerical datasets. We call this approach a “deep learning PDE model” (DPM). Once trained, the DPM can be used to make out-of-sample predictions for new physical coefficients, geometries, and boundary conditions. We implement our approach on a classic problem of the Navier-Stokes equation for turbulent flows. The numerical solution of the Navier-Stokes equation (with turbulence) is computationally expensive and requires a supercomputer. We show that our approach can estimate a (computationally fast) DPM for the filtered velocity of the Navier-Stokes equations. 

Thu, 17 Oct 2019
16:00
L6

One-level density of Dirichlet L-functions

Kyle Pratt
(Oxford)
Abstract

I will discuss work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.
 

Thu, 17 Oct 2019

16:00 - 17:30
C5

A biased view of two-row Springer theory

Filip Zivanovic
Abstract

Springer theory is an important branch of geometric representation theory. It is a beautiful interplay between combinatorics, geometry and representation theory.
It started with Springer correspondence, which yields geometric construction of irreducible representations of symmetric groups, and Ginzburg's construction of universal enveloping algebra U(sl_n).

Here I will present a view of two-row Springer theory of type A (thus looking at nilpotent elements with two Jordan blocks) from a scope of a symplectic topologist (hence the title), that yields connections between symplectic-topological invariants and link invariants (Floer homology and Khovanov homology) and connections to representation theory (Fukaya category and parabolic category O), thus summarising results by Abouzaid,
Seidel, Smith and Mak on the subject.