Past

Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this talk we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions (this is a joint work with A.Ferraguti and R.Schnyder). Time permitting, we will also describe how to use character sum techniques to bound the size of the graph determined by the generating set (this is a joint work with D.R. Heath-Brown).

A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen. 

We characterize cutting arcs on ber surfaces that produce new ber surfaces,
and the changes in monodromy resulting from such cuts. As a corollary, we
characterize band surgeries between bered links and introduce an operation called
generalized Hopf banding. We further characterize generalized crossing changes between
bered links, and the resulting changes in monodromy.

This is joint work with Matt Rathbun, Kai Ishihara and Koya Shimokawa

It is well-known that the stochastic heat equation on R^n has a Hölder continuous function-valued solution in the case n=1, and that in dimensions 2 and above the solution is not function-valued but is forced to take values in some wider space of distributions. So what happens if the space has, in some sense, a dimension in between 1 and 2? We turn to the theory of fractals in order to answer this question. It has been shown (Kigami, 2001) that there exists a class of self-similar sets on which natural Laplacians can be defined, and so an analogue to the stochastic heat equation can be posed. In this talk we cover the following questions: Is the solution to this equation function-valued? If so, is it Hölder continuous? To answer the latter we must first prove an analogue of Kolmogorov's celebrated continuity theorem for the self-similar sets that we are working on. Joint work with Ben Hambly.

I will define the notion of "sheaf of categories with a local action of Hochschild cochains" over a stack. (This notion is analogous to D-modules, in the same way as the notion of "sheaf of categories" is analogous to quasi-coherent sheaves.) I will prove that both categories appearing in geometric Langlands carry this structure over the stack of de Rham {\check{G}}-local systems. Using this, I will explain how to glue D-mod(Bun_G) out of *tempered* D-modules associated to smaller Levi subgroups of G.

We give an explicit scheme to reconstruct any C^1 curve from its signature. It is implementable and comes with detailed stability properties. The key of the inversion scheme is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales. Joint work with Terry Lyons.

M5-branes on 4-manifolds M_4 realized as co-associatives in G_2 give rise to 2d (0,2) superconformal theories. In this talk I will propose a duality between these 2d (0,2) theories and 4d topological theories, which are sigma-models from M_4 into the Nahm moduli space. 

How might you prepare talks for different audiences (specialised seminar, colloquium-style talk, talk to a non-mathematical audience, job interview)?  Join us for advice on this, and on how to connect with your audience and get them to feel involved.

In practice, stochastic decision problems are often based on statistical estimates of probabilities. We all know that statistical error may be significant, but it is often not so clear how to incorporate it into our decision making. In this informal talk, we will look at one approach to this problem, based on the theory of nonlinear expectations. We will consider the large-sample theory of these estimators, and also connections to `robust statistics' in the sense of Huber.

Abstract: We will recall some analogies between structures arising from three-manifold topology and rings of integers in number fields. This can be used to define a Chern-Simons functional on spaces of Galois representations.  Some sample computations and elementary applications will be shown.

We show how to adapt methods originally developed in
model-independent finance / martingale optimal transport to give a
geometric description of optimal stopping times tau of Brownian Motion
subject to the constraint that the distribution of tau is a given
distribution. The methods work for a large class of cost processes.
(At a minimum we need the cost process to be adapted. Continuity
assumptions can be used to guarantee existence of solutions.) We find
that for many of the cost processes one can come up with, the solution
is given by the first hitting time of a barrier in a suitable phase
space. As a by-product we thus recover Anulova's classical solution of
the inverse first passage time problem.

Modelling catalytic kinetics is indispensable for the design of reactors and chemical processes. However, developing accurate and computationally efficient kinetic models remains challenging. Empirical kinetic models incorporate assumptions about rate-limiting steps and may thus not be applicable to operating regimes far from those where they were derived. Detailed microkinetic modelling approaches overcome this issue by accounting for all elementary steps of a reaction mechanism. However, the majority of such kinetic models employ mean-field approximations and are formulated as ordinary differential equations, which neglect spatial correlations. On the other hand, kinetic Monte Carlo (KMC) approaches provide a discrete-space continuous-time stochastic formulation that enables a detailed treatment of spatial correlations in the adlayer (resulting for instance from adsorbate-adsorbate lateral interactions), but at a significant computation expense.1,2

Motivated by these challenges, we discuss the necessity of KMC descriptions that incorporate detailed models of lateral interactions. Focusing on a titration experiment involving the oxidation of pre-adsorbed O by CO gas on Pd(111), we discuss experimental findings that show first order kinetics at low temperature (190 K) and half order kinetics at high temperature (320 K), the latter previously attributed to island formation.3 We perform KMC simulations whereby coverage effects on reaction barriers are captured by cluster expansion Hamiltonians and Brønsted-Evans-Polanyi (BEP) relations.4 By quantifying the effect of adlayer structure versus coverage effects on the observed kinetics, we rationalise the experimentally observed kinetics. We show that coverage effects lead to the half order kinetics at 320 K, rather than O-island formation as previously thought.5,6

Subsequently, we discuss our ongoing work in the development of approximations that capture such coverage effects but are much more computationally efficient than KMC, making it possible to use such models in reactor design. We focus on a model for NO oxidation incorporating first nearest neighbour lateral interactions and construct a sequence of approximations of progressively higher accuracy, starting from the mean-field treatment and continuing with a sequence of Bethe-Peierls models with increasing cluster sizes. By comparing the turnover frequencies of these models with those obtained from KMC simulation, we show that the mean-field predictions deviate by several orders of magnitude from the KMC results, whereas the Bethe-Peierls models exhibit progressively higher accuracy as the size of the explicitly treated cluster increases. While more computationally intensive than mean-field, these models still enable significant computational savings compared to a KMC simulation, thereby paving the road for employing them in multiscale modelling frameworks.

References

1    M. Stamatakis and D. G. Vlachos, ACS Catal. 2 (12), 2648 (2012).

2    M. Stamatakis, J Phys-Condens Mat 27 (1), 013001 (2015).

3    I. Nakai, H. Kondoh, T. Shimada, A. Resta, J. N. Andersen, and T. Ohta, J. Chem. Phys. 124 (22), 224712 (2006).

4    J. Nielsen, M. d’Avezac, J. Hetherington, and M. Stamatakis, J. Chem. Phys. 139 (22), 224706 (2013).

5    M. Stamatakis and S. Piccinin, ACS Catal. 6 (3), 2105 (2016).

6    S. Piccinin and M. Stamatakis, ACS Catal. 4, 2143 (2014).

Rational Krylov methods are applicable to a wide range of scientific computing problems, and ​the rational Arnoldi algorithm is a commonly used procedure for computing an ​orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this​ ​algorithm is the solution of a large linear system of equations at each iteration. We explore the​ ​option of solving several linear systems simultaneously, thus constructing the rational Krylov​ ​basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly​ ​conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the​ ​new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that ​allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm.
​ ​
The computational benefits are illustrated using several numerical examples from different application areas.
​ ​
This ​talk is based on joint work with Mario Berljafa  available as an Eprint at http://eprints.ma.man.ac.uk/2503/
 

I'll discuss the problem of controlling energy concentration in YM flow over a four-manifold. Based on a study of the rotationally symmetric case, it was conjectured in 1997 that bubbling can only occur at infinite time. My thesis contained some strong elementary results on this problem, which I've now solved in full generality by a more involved method.

A Kähler group is a group which can be realised as the fundamental group of a close Kähler manifold. We will prove that for a Kähler group $G$ we have that $G$ is residually free if and only if $G$ is a full subdirect product of a free abelian group and finitely many closed hyperbolic surface groups. We will then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler: We explain how to construct subgroups of direct products of surface groups which have even first Betti number but are not Kähler. All relevant notions will be explained in the talk.

The Algebraic Eraser is a cryptosystem (more precisely, a class of key
agreement schemes) introduced by Anshel, Anshel, Goldfeld and Lemieux
about 10 years ago. There is a concrete instantiation of the Algebraic
Eraser called the Colored Burau Key Agreement Protocol (CBKAP), which
uses a blend of techniques from permutation groups, matrix groups and
braid groups. SecureRF, the company owning the trademark to the
Algebraic Eraser, is marketing this system for lightweight
environments such as RFID tags and other Internet of Things
applications; they have proposed making this scheme the basis for an
ISO RFID standard.

This talk gives an introduction to the Algebraic Eraser, a brief
history of the attacks on this scheme using ideas from group-theoretic
cryptography, and describes the countermeasures that have been
proposed. I would not recommend the scheme for the proposed
applications: the talk ends with a brief sketch of a recent convincing
cryptanalysis of this scheme due to Ben-Zvi, Blackburn and Tsaban
(which appeared at CRYPTO this summer), and significant attacks
on the protocol in the proposed ISO standard due to Blackburn and
Robshaw (which appeared at ACNS earlier this year).

The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action.

(Joint with Jack Jeffries)

The threshold for existence of a giant component in a random graph with given vertex degrees was found by Molloy and Reed (1995), and several authors have since studied the size of the largest and other components in various cases. The critical window was found by Hatami and Molloy (2012), and has a width that depends on whether the asymptotic degree distribution has a finite third moment or not. I will describe some new results (joint work with Remco van der Hofstad and Malwina Luczak) on the barely supercritical case, where this difference between finite and infinite third moment also is seen.

Multi-index methods are a generalization of multilevel methods in high dimensional problem and are based on taking mixed first-order differences along all dimensions. With these methods, we can accurately and efficiently compute a quadrature or construct an interpolation where the integrand requires some form of high dimensional discretization. Multi-index methods are related to Sparse Grid methods and the Combination Technique and have been applied to multiple sampling methods, i.e., Monte Carlo, Stochastic Collocation and, more recently, Quasi Monte Carlo.

In this talk, we describe and analyse the Multi-Index Monte Carlo (MIMC) and Multi-Index Stochastic Collocation (MISC) methods for computing statistics of the solution of a PDE with random data. Provided sufficient mixed regularity, MIMC and MISC achieve better complexity than their corresponding multilevel methods. We propose optimization procedures to select the most effective mixed differences to include in these multi-index methods. We also observe that in the optimal case, the convergence rate of MIMC and MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We finally show the effectiveness of MIMC and MISC in some computational tests, including PDEs with random coefficients and Stochastic Particle Systems.
 

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional. 
In this talk, I report on recent developments in the 3-dimensional setting of this theory.

I will discuss some of the relationships between ODE IVPs, usually solved by marching, and ODE BVPs, usually solved by global discretizations.