Past

In this talk, we will present results regarding the regularity and

rigidity of solutions to the Monge-Ampere equation, inspired by the role

played by this equation in the context of prestrained elasticity. We will

show how the Nash-Kuiper convex integration can be applied here to achieve

flexibility of Holder solutions, and how other techniques from fluid

dynamics (the commutator estimate, yielding the degree formula in the

present context) find their parallels in proving the rigidity. We will indicate

possible avenues for the future related research.

Let F be a fixed infinite, vertex-transitive graph. We say a graph G is `r-locally F' if for every vertex v of G, the ball of radius r and centre v in G is isometric to the ball of radius r in F. For each positive integer n, let G_n = G_n(F,r) be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r-locally F. We investigate the properties that the random graph G_n has with high probability --- i.e., how these properties depend upon the fixed graph F. 
We show that if F is a Cayley graph of a torsion-free group of polynomial growth, then there exists a positive integer r_0 such that for every integer r at least r_0, with high probability the random graph G_n = G_n(F,r) defined above has largest component of size between n^{c_1} and n^{c_2}, where 0 < c_1 < c_2  < 1 are constants depending upon F alone, and moreover that G_n has at least exp(poly(n)) automorphisms. This contrasts sharply with the random d-regular graph G_n(d) (which corresponds to the case where F is replaced by the infinite d-regular tree).
Our proofs use a mixture of results and techniques from group theory, geometry and combinatorics, including a recent and beautiful `rigidity' result of De La Salle and Tessera.
We obtain somewhat more precise results in the case where F is L^d (the standard Cayley graph of Z^d): for example, we obtain quite precise estimates on the number of n-vertex graphs that are r-locally L^d, for r at least linear in d, using classical results of Bieberbach on crystallographic groups.
Many intriguing open problems remain: concerning groups with torsion, groups with faster than polynomial growth, and what happens for more general structures than graphs.
This is joint work with Itai Benjamini (Weizmann Institute).
 

The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic).  Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.

In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1<a<2, and prove convergence to an a-stable Levy process.  This includes to the best of our knowledge the first natural examples where the M_2 Skorokhod topology is the appropriate one.



 

I will consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. A PDE (free boundary problem) approach is used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths.  An Eulerian---mass flow---formulation of the problem is introduced. Its dual is given by Hamilton-Jacobi-Bellman type variational inequalities.  Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglb\"ock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety.

 We study codimension one holomorphic distributions on projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is an integrable phenomenon. This part of the  talk  is work joint with  M. Jardim(Unicamp) and O. Calvo-Andrade(Cimat).

We also study foliations by curves via the investigation  of their  singular schemes and  conormal  sheaves and we provide a classification  of foliations of degree at most 3 with  conormal  sheaves locally free.  Foliations of degrees  1 and 2 are aways given by a global intersection of two codimension one distributions. In the classification of degree 3 appear Legendrian foliations, foliations whose  conormal sheaves are instantons and other ” exceptional”
type examples. This part of the  talk   is  work joint with  M. Jardim(Unicamp) and S. Marchesi(Unicamp).

Gathering for women and non-binary mathematicians

Supersymmetric localization is a powerful technique to evaluate a class of functional integrals in supersymmetric field theories. It reduces the functional integral over field space to ordinary integrals over the space of solutions of the off-shell BPS equations. The application of this technique to supergravity suffers from some problems, both conceptual and practical. I will discuss one of the main conceptual problems, namely how to construct the fermionic symmetry with which to localize. I will show how a deformation of the BRST technique allows us to do this. As an application I will then sketch a computation of the one-loop determinant of the super-graviton that enters the localization formula for BPS black hole entropy.
 

The moments of characteristic polynomials play a central role in Random Matrix Theory.  They appear in many applications, ranging from quantum mechanics to number theory.  The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.

Glacier flow is an example of a gravity driven non-linear viscous flow at low Reynolds numbers. As a glacier flows over an undulating bed, the surface topography is modified in response. Some information about bed conditions is therefore contained in the shape of the surface and the surface velocity field. I will present theoretical and numerical work on how basal conditions on glaciers affect ice flow, and how one can obtain information about basal conditions through surface-to-bed inversion. I’ll give an overview over inverse methodology currently used in glaciology, and how satellite data is now routinely used to invert for bed properties of the Greenland and the Antarctic Ice Sheets.

Computer vision approaches have made huge advances with deep learning research. These algorithms can be employed as a basis for phenotyping of biological traits from imaging modalities. This can be employed, for example, in the context of facial photographs of rare diseases as a means of aiding diagnostic pathways, or as means to large scale phenotyping in histological imaging. With any data set, inherent biases and problems in the data available for training can have a detrimental impact on your models. I will describe some examples of such data set problems and outline how to build models that are not confounded – despite biases in the training data. 

Let F be a totally real or CM number field.  Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for GL_n(F).  We show that the nilpotent ideal appearing in Scholze's construction can be removed when F splits completely at the relevant prime.  As a key component of the proof, we show that the compactly supported cohomology of certain unitary and symplectic Shimura varieties with level  Gamma_1(p^{\infty}) vanishes above the middle degree. This is joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih. 

J. M. Foster 1 , N. E. Courtier 2 , S. E. J. O’Kane 3 , J. M. Cave 3 , R. Niemann 4 , N. Phung 5 , A. Abate 5 , P. J. Cameron 4 , A. B. Walker 3 & G. Richardson 2 .

1 School of Mathematics & Physics, University of Portsmouth, UK. {@email}

2 School of Mathematics, University of Southampton, UK.

3 School of Physics, University of Bath, UK.

4 School of Chemistry, University of Bath, UK.

5 Helmholtz-Zentrum Berlin, Germany.

Metal halide perovskite has emerged as a highly promising photovoltaic material. Perovskite-based solar cells now exhibit power conversion efficiencies exceeding 22%; higher than that of market-leading multi-crystalline silicon, and comparable to the Shockley-Queisser limit of around 33% (the maximum obtainable efficiency for a single junction solar cell). In addition to fast electronic phenomena, occurring on timescales of nanoseconds, they also exhibit much slower dynamics on the timescales of several seconds and up to a day. One well-documented example of this is the ‘anomalous’ hysteresis observed in current-voltage scans where the applied voltage is varied whilst the output current is measured. There is now a consensus that this is caused by the motion of ions in the perovskite material affecting the internal electric field and in turn the electronic transport.

We will discuss the formulation of a drift-diffusion model for the coupled electronic and ionic transport in a perovskite solar cell as well as its systematic simplification via the method of matched asymptotic expansions. We will use the resulting reduced model to give a cogent explanation for some experimental observations including, (i) the apparent disappearance of current-voltage hysteresis for certain device architectures, and (ii) the slow fading of performance under illumination during the day and subsequent recovery in the dark overnight. Finally, we suggest ways in which materials and geometry can be chosen to reduce charge carrier recombination and improve device performance.

The FEniCS system [1] allows the description of finite element discretisations of partial differential equations using a high-level syntax, and the automated conversion of these representations to working code via automated code generation. In previous work described in [2] the high-level representation is processed automatically to derive discrete tangent-linear and adjoint models. The processing of the model code at a high level eases the technical difficulty associated with management of data in adjoint calculations, allowing the use of optimal data management strategies [3].

This previous methodology is extended to enable the calculation of higher order partial differential equation constrained derivative information. The key additional step is to treat tangent-linear
equations on an equal footing with originating forward equations, and in particular to treat these in a manner which can themselves be further processed to enable the derivation of associated adjoint information, and the derivation of higher order tangent-linear equations, to arbitrary order. This enables the calculation of higher order derivative information -- specifically the contraction of a Kth order derivative against (K - 1) directions -- while still making use of optimal data management strategies. Specific applications making use of Hessian information associated with models written using the FEniCS system are presented.

[1] "Automated solution of differential equations by the finite element method: The FEniCS book", A. Logg, K.-A. Mardal, and  G. N. Wells (editors), Springer, 2012
[2] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, "Automated derivation of the adjoint of high-level transient finite element programs", SIAM Journal on Scientific Computing 35(4), C369--C393, 2013
[3] A. Griewank, and A. Walther, "Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation", ACM Transactions on Mathematical Software 26(1), 19--45, 2000

For the North Analysis British Seminar, see: https://www.maths.ox.ac.uk/node/29687

A core problem in the study of manifolds and their topology is that of telling them apart. That is, when can we say whether or not two manifolds are homeomorphic? In two dimensions, the situation is simple, the Classification Theorem for Surfaces allows us to differentiate between any two closed surfaces. In three dimensions, the problem is a lot harder, as the century long search for a proof of the Poincaré Conjecture demonstrates, and is still an active area of study today.
As an early pioneer in the area of 3-manifolds Seifert carved out his own corner of the landscape instead of attempting to tackle the entire problem. By reducing his scope to the subclass of 3-manifolds which are today known as Seifert fibred spaces, Seifert was able to use our knowledge of 2-manifolds and produce a classification theorem of his own.
In this talk I will define Seifert fibred spaces, explain what makes them so much easier to understand than the rest of the pack, and give some insight on why we still care about them today.
 

The aim of this talk is to tell the story of Non-Abelian Hodge Theory for curves. The starting point is the space of representations of the fundamental group of a compact Riemann surface. This space can be endowed with the structure of a complex algebraic variety in three different ways, giving rise to three non-algebraically isomorphic moduli spaces called the Betti, de Rham and Dolbeault moduli spaces respectively. 

After defining and outlining the construction of these three moduli spaces, I will describe the (non-algebraic) correspondences between them, collectively known as Non-Abelian Hodge Theory. Finally, we will see how the rich structure of the Dolbeault moduli space can be used to shed light on the topology of the space of representations.

The core idea behind metric spaces is the triangular inequality. Metrics have been generalized in many ways, but the most tempting way to alter them would be to "flip" the triangular inequality, obtaining an "anti-metric". This, however, only allows for trivial spaces where the distance between any two points is 0. However, if we intertwine the concept of antimetrics with the structures of partial linear--and cyclic--orders, we can define a structure where the anti-triangular inequality holds conditionally. We define this structure, give examples, and show an interesting result involving metrics and antimetrics.

Roger Penrose is the ultimate scientific all-rounder.  He started out in algebraic geometry but within a few years had laid the foundations of the modern theory of black holes with his celebrated paper on gravitational collapse. His exploration of foundational questions in relativistic quantum field theory and quantum gravity, based on his twistor theory, had a huge impact on differential geometry. His work has influenced both scientists and artists, notably Dutch graphic artist M. C. Escher.

Roger Penrose is one of the great ambassadors for science. In this lecture and in conversation with mathematician and broadcaster Hannah Fry he will talk about work and career.

This lecture is in partnership with the Science Museum in London where it will take place. Please email @email to register.

You can also watch online:

https://www.facebook.com/OxfordMathematics

https://livestream.com/oxuni/Penrose-Fry

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

In 1985, Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free set of size at least cn, where a product-free set of a group is a subset that does not contain three elements x,y and z  satisfying xy=z. Gowers showed that the answer is no in the early 2000s, by linking the existence of product-free sets of large density to the existence of low dimensional unitary representations.

In this talk, I will provide an answer to the aforementioned question by model theoretic means. Furthermore, I will relate some of Gowers' results to the existence of nontrivial definable compactifications of nonstandard finite groups.