Past

In this talk, we use Carleman type techniques to address uniqueness of self-shrinkers of mean curvature flow with given asymptotic behaviors.

In this talk I will describe two instances of Langlands functoriality concerning the group $\mathrm{Sp}_{2n}$. I will then very briefly explain how this enables one to attach Galois representations to automorphic representations of (inner forms of) $\mathrm{Sp}_{2n}$. 

Multiplicative chaos theory originated from the study of turbulence by Kolmogorov in the 1940s and it was mathematically founded by Kahane in the 1980s. Recently the theory has drawn much of attention due to its connection to SLEs and statistical physics.  In this talk I shall present some recent development of multiplicative chaos theory, as well as its applications to Liouville quantum gravity.

In this talk I will show how given a finitely generated relatively hyperbolic group G, one can construct a finite generating set X of G for which (G,X) has a number of metric properties, provided that the parabolic subgroups have these properties. I will discuss the applications of these properties to the growth series, language of geodesics, biautomatic structures and conjugacy problem. This is joint work with Yago Antolin.

We consider the problem of large-scale regression where both the number of predictors, p, and the number of observations, n, may be in the order of millions or more. Computing a simple OLS or ridge regression estimator for such data, though potentially sensible from a purely statistical perspective (if n is large enough), can be a real computational challenge. One recent approach to tackling this problem in the common situation where the matrix of predictors is sparse, is to first compress the data by mapping it to an n by L matrix with L << p, using a scheme called b-bit min-wise hashing (Li and König, 2011). We study this technique from a theoretical perspective and obtain finite-sample bounds on the prediction error of regression following such data compression, showing how it exploits the sparsity of the data matrix to achieve good statistical performance. Surprisingly, we also find that a main effects model in the compressed data is able to approximate an interaction model in the original data. Fitting interactions requires no modification of the compression scheme, but only a higher-dimensional mapping with a larger L.
This is joint work with Nicolai Meinshausen (ETH Zürich).

From a geometric viewpoint the irregular Riemann-Hilbert correspondence can be viewed as a machine that takes as input a simple
`additive' symplectic/Poisson manifold and it outputs a more complicated `multiplicative' symplectic/Poisson manifold. In the
simplest nontrivial example it converts the linear Poisson manifold Lie(G)^* into the dual Poisson Lie group G^* (which is the Poisson
manifold underlying the Drinfeld-Jimbo quantum group). This talk will firstly describe some more recent (and more complicated) examples of
such `nonperturbative symplectic/Poisson manifolds', i.e. symplectic spaces of Stokes/monodromy data or `wild character varieties'. Then
the natural generalisations (`fission algebras') of the deformed multiplicative preprojective algebras that occur will be discussed, some
of which are known to be related to Cherednik algebras.

Recently, as part of a project to find CY manifolds for which both the Hodge numbers (h^{11}, h^{21}) are small, manifolds have been found with Hodge numbers (4,1) and (1,1). The one-dimensional special geometries of their complex structures are more complicated than those previously studied. I will review these, emphasising the role of the fundamental period and Picard-Fuchs equation. Two arithmetic aspects arise: the first is the role of \zeta(3) in the monodromy matrices and the second is the fact, perhaps natural to a number theorist, that through a study of the CY manifolds over finite fields, modular functions can be associated to the singular manifolds of the family. This is a report on joint work with Volker Braun, Xenia de la Ossa and Duco van Straten.

Contraction cracks form captivating patterns such as those seen in dried mud or the polygonal networks that cover the polar regions of Earth and Mars. These patterns can be controlled, for example in the artistic craquelure sometimes found in pottery glazes. More practically, a growing zoo of patterns, including parallel arrays of cracks, spiral cracks, wavy cracks, lenticular or en-passant cracks, etc., are known from simple experiments in thin films – essentially drying paint – and are finding application in surfaces with engineered properties. Through such work we are also learning how natural crack patterns can be interpreted, for example in the use of dried blood droplets for medical or forensic diagnosis, or to understand how scales develop on the heads of crocodiles.

I will discuss mud cracks, how they form, and their use as a simple laboratory analogue system. For flat mud layers I will show how sequential crack formation leads to a rectilinear crack network, with cracks meeting each other at roughly 90°. By allowing cracks to repeatedly form and heal, I will describe how this pattern evolves into a hexagonal pattern. This is the origin of several striking real-world systems: columnar joints in starch and lava; cracks in gypsum-cemented sand; and the polygonal terrain in permafrost. Finally, I will turn to look at crack patterns over uneven substrates, such as paint over the grain of wood, or on geophysical scales involving buried craters, and identify when crack patterns are expected to be dominated by what lies beneath them. In exploring all these different situations I will highlight the role of energy release in selecting the crack patterns that are seen.

Donaldson-Thomas theory for Calabi-Yau 3-folds is a complexification of Chern-Simons theory. In this talk, I will discuss joint work with Naichung Conan Leung on the complexification of Donaldson theory.

Unordered configuration spaces on (connected) manifolds are basic objects
that appear in connection with many different areas of topology. When the
manifold M is non-compact, a theorem of McDuff and Segal states that these
spaces satisfy a phenomenon known as homological stability: fixing q, the
homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)
denotes the space of k-point configurations and homology is taken with
coefficients in Z. However, this statement is in general false for closed
manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we
extend all the previously known results in this situation. One key idea is
to introduce so-called "replication maps" between configuration spaces,
which in a sense replace the "stabilisation maps" that exist only in the
case of non-compact manifolds. One corollary of our results is to recover a
"homological periodicity" theorem of Nagpal -- taking homology with field
coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is
eventually periodic in k -- and we obtain a much simpler estimate for the
period. Another result is that homological stability holds with Z[1/2]
coefficients whenever M is odd-dimensional, and in fact we improve this to
stability with Z coefficients for 3- and 7-dimensional manifolds.

The Network and Criminality Workshop will explore the capacity of mathematics and computation to extract insight on network structures relevant to crime, riots, terrorism, etc. It will include presentations on current work (both application-oriented and on methods that can be applied in the future) and active discussion on how to address existing challenges.

Invited speakers (in alphabetical order) are as follows:

Prof. Alex Arenas, Professor of Computer Science & Mathematics, URV, http://deim.urv.cat/~alexandre.arenas/

Prof. Henri Berestycki, Professor of Mathematics, EHESS, http://en.wikipedia.org/wiki/Henri_Berestycki

Prof. Andrea Bertozzi, Professor of Mathematics, UCLA, http://www.math.ucla.edu/~bertozzi/

Dr. Paolo Campana, Research Fellow, Oxford, http://www.sociology.ox.ac.uk/academic-staff/paolo-campana.html

Toby Davies, Graduate Student,  UCL, http://www.bartlett.ucl.ac.uk/casa/people/mphil-phd-students/Toby_Davies

Dr. Hannah Fry, Lecturer in the mathematics of cities, UCL, https://iris.ucl.ac.uk/iris/browse/profile?upi=HMFRY30

Dr. Yves van Gennip, Lecturer in Mathematics, Nottingham, http://www.nottingham.ac.uk/mathematics/people/y.vangennip

Prof. Sandra González-Bailón, Assistant Professor at UPenn, http://dimenet.asc.upenn.edu/people/sgonzalezbailon/

Prof. Federico Varese, Professor of Criminology, Oxford, http://www.law.ox.ac.uk/profile/federico.vareserecep

If you are interested in attending this workshop, please register by following this link: https://www.maths.ox.ac.uk/node/13764/.

We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.

We propose a non-perturbative formulation of structure constants of single trace operators in planar N=4 SYM.  We match our results with both weak and strong coupling data available in the literature. Based on work with Benjamin Basso and Pedro Vieira.

In this talk we discuss a utility-risk portfolio selection problem. By considering the first order condition for the objective function, we derive a primitive static problem, called Nonlinear Moment Problem, subject to a set of constraints involving nonlinear functions of “mean-field terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems, their positive answers have long been missing in the literature. In particular, the existence result in utility-downside-risk problem is in contrast with that of mean-downside-risk problem considered in Jin-Yan-Zhou (2005) in which they prove the non-existence of optimal solution instead and we can show the same non-existence result via the corresponding Nonlinear Moment Problem. This is joint work with K.C. Wong (University of Hong Kong) and S.C.P. Yam (Chinese University of Hong Kong).

For this workshop, we have identified two subject of interest for us in the field of particle technology, one the wet granulation is a size enlargement process of converting small-diameter solid particles (typically powders) into larger-diameter agglomerates to generate a specific size, the other one the mechanical centrifugal air classifier is employed when the particle size that you need to separate is too fine to screen.

Optimization methods for large-scale machine learning must confront a number of challenges that are unique to this discipline. In addition to being scalable, parallelizable and capable of handling nonlinearity (even non-convexity), they must also be good learning algorithms. These challenges have spurred a great amount of research that I will review, paying particular attention to variance reduction methods. I will propose a new algorithm of this kind and illustrate its performance on text and image classification problems.

Ice streams are narrow bands of rapidly sliding ice within an otherwise

slowly flowing continental ice sheet. Unlike the rest of the ice sheet,

which flows as a typical viscous gravity current, ice streams experience

weak friction at their base and behave more like viscous 'free films' or

membranes. The reason for the weak friction is the presence of liquid

water at high pressure at the base of the ice; the water is in turn

generated as a result of dissipation of heat by the flow of the ice

stream. I will explain briefly how this positive feedback can explain the

observed (or inferred, as the time scales are rather long) oscillatory

behaviour of ice streams as a relaxation oscillation. A key parameter in

simple models for such ice stream 'surges' is the width of an ice stream.

Relatively little is understood about what controls how the width of an

ice stream evolves in time. I will focus on this problem for most of the

talk, showing how intense heat dissipation in the margins of an ice stream

combined with large heat fluxes associated with a switch in thermal

boundary conditions may control the rate at which the margin of an ice

stream migrates. The relevant mathematics involves a somewhat non-standard

contact problem, in which a scalar parameter must be chosen to control the

location of the contact region. I will demonstrate how the problem can be

solved using the Wiener-Hopf method, and show recent extensions of this

work to more realistic physics using a finite element discretization.

A compact space is a Rosenthal compactum if it can be embedded into the space of Baire class 1 functions on a Polish space. Those objects have been well studied in functional analysis and set theory. In this talk, I will explain the link between them and the model-theoretic notion of NIP and how they can be used to prove new results in model theory on the topology of the space of types.
 

Quiver varieties and their quantizations feature prominently in
geometric representation theory. Multiplicative quiver varieties are
group-like versions of ordinary quiver varieties whose quantizations
involve quantum groups and $q$-difference operators. In this talk, we will
define and give examples of representations of quivers, ordinary quiver
varieties, and multiplicative quiver varieties. No previous knowledge of
quivers will be assumed. If time permits, we will describe some phenomena
that occur when quantizing multiplicative quiver varieties at a root of
unity, and work-in-progress with Nicholas Cooney.

I will review some classical problems in number theory concerning the statistical distribution of the primes, square-free numbers and values of the divisor function; for example, fluctuations in the number of primes in short intervals and in arithmetic progressions.  I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals. 

The growth of the exchange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). Motivated by a number of empirical market observations, we study the relationship between the ETF and LETF implied volatility surfaces under general stochastic volatility models. Analytic approximations for prices and implied volatilities are derived for LETF ​options, along with rigorous error bounds. In these price and IV expressions, we identify their non-trivial dependence on the leverage ratio. Moreover, we introduce a "moneyness scaling" procedure to enhance the comparison of implied volatilities across leverage ratios, and test it with empirical price data.