Past

I will discuss the specific long-time behaviour of kinetic solutions to stochastic conservation laws with non-linear multiplicative noises.

.. showing that a field K is isomorphic to Q if it has the same absolute Galois group and if it satisfies a very small additional condition (very similar to my talk 2 years ago).

Abstract: The Kontsevich graph weights are period integrals whose
values make Kontsevich's star-product associative for any Poisson
structure. We illustrate, by using software, to what extent these
weights are determined by their properties: the associativity
constraint for the star-product (for all Poisson structures), the
multiplicativity (decomposition into prime graphs), the cyclic
relations, and some relations due to skew-symmetry. Up to the order 4
in ℏ we express all the weights in terms of 10 parameters (6
parameters modulo gauge-equivalence), and we verify pictorially that
the star-product expansion is associative modulo ō(ℏ⁴) for every value
of the 10 parameters. This is joint work with Arthemy Kiselev.
 

In this talk, I’ll share the progress that we have made in the field of e-voting, including the proposal of a new paradigm of e-voting system called self-enforcing e-voting (SEEV). A SEEV system is End-to-End (E2E) verifiable, but it differs from all previous E2E systems in that it does not require tallying authorities. The removal of tallying authorities significantly simplifies the election management and makes the system much more practical than before. A prototype of a SEEV system based on the DRE-i protocol (Hao et al. USENIX JETS 2014) has been built and used regularly in Newcastle University for classroom voting and student prize competitions with very positive student feedback. Lessons from our experience of designing, analysing and deploying an e-voting system for real-world applications are also presented.

Let $A$ denote either the automorphism group of the free group of rank $n$ or the mapping class group of an orientable surface of genus $n$ with at most 1 boundary component, and let $G$ be either the subgroup of IA-automorphisms or the Torelli subgroup of $A$, respectively. I will discuss various finiteness properties of subgroups containing $G_N$, the $N$-th term of the lower central series of $G$, for sufficiently small $N$. In particular, I will explain why
(1) If $n \geq 4N-1$, then any subgroup of G containing $G_N$ (e.g. the $N$-th term of the Johnson filtration) is finitely generated
(2) If $n \geq 8N-3$, then any finite index subgroup of $A$ containing $G_N$ has finite abelianization.
The talk will be based on a joint work with Sue He and a joint work with Tom Church and Andrew Putman

Deficiency is a measure of how complicated the presentations of a particular group need to be; it is defined as the maximum of the number of generators minus the number of relators (over all finite presentations of the group). This talk will introduce the basics of deficiency, give a deft example of Swan which illustrates why our understanding of deficiency is deficient, and conclude with some new examples that defy this defeatism: finite $p$-groups can have any deficiency you could (reasonably) wish for.

The $r$-neighbour bootstrap process on a graph $G$ starts with an initial set of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ becomes infected during the process, then we say that the initial set percolates.

In this talk I will discuss the proof of a conjecture of Balogh and Bollobás: for fixed $r$ and $d\to\infty$, the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\frac{1+o(1)}{r}\binom{d}{r-1}$. One of the key ideas behind the proof exploits a connection between bootstrap percolation and weak saturation. This is joint work with Jonathan Noel.

Let G be a split reductive group over a finite extension k of Q_p. Reeder and Yu have given a new construction of supercuspidal representations of G(k) using geometric invariant theory. Their construction is uniform for all p but requires as input stable vectors in certain representations coming from Moy-Prasad filtrations. In joint work, Jessica Fintzen and I have classified the representations of this kind which contain stable vectors; as a corollary, the construction of Reeder-Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G(k). 

The past few years have seen a surge of interest in nonconvex reformulations of convex optimizations using nonlinear reparameterizations of the optimization variables. Compared with the convex formulations, the nonconvex ones typically involve many fewer variables, allowing them to scale to scenarios with millions of variables. However, one pays the price of solving nonconvex optimizations to global optimality, which is generally believed to be impossible. In this talk, I will characterize the nonconvex geometries of several low-rank matrix optimizations. In particular, I will argue that under reasonable assumptions, each critical point of the nonconvex problems either corresponds to the global optimum of the original convex optimizations, or is a strict saddle point where the Hessian matrix has a negative eigenvalue. Such a geometric structure ensures that many local search algorithms can converge to the global optimum with random initializations. Our analysis is based on studying how the convex geometries are transformed under nonlinear parameterizations.

I will give an overview of the complexes used in algebraic topology using the language of abstract complexes.

This is a lunch seminar, so feel free to bring your lunch along!

In this presentation, we give an overview of the numerical methods used in commercial oil and gas reservoir simulation. The models are described by flow through porous media and are solved using a series of nested numerical methods. Most of the computational effort resides in solving large linear systems resulting from Newton iterations. Therefore, we will go in greater detail about the iterative linear solvers and preconditioning techniques.

Note: This talk will cover similar topics to the InFoMM group meeting talks on Friday 28th April, but I will discuss more mathematical details for this JAMS talk.

In 1997, Maxim Kontsevich gave a universal formula for the
quantization of Poisson brackets.  It can be viewed as a perturbative
expansion in a certain two-dimensional topological field theory.  While the
formula is explicit, it is currently impossible to compute in all but the
simplest cases, not least because the values of the relevant Feynman
integrals are unknown.  In forthcoming joint work with Peter Banks and Erik
Panzer, we use Francis Brown's approach to the periods of the moduli space
of genus zero curves to give an algorithm for the computation of these
integrals in terms of multiple zeta values.  It allows us to calculate the
terms in the expansion on a computer for the first time, giving tantalizing
evidence for several open conjectures concerning the convergence and sum of
the series, and the action of the Grothendieck-Teichmuller group by gauge
transformations.

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics and engineering. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three main advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available.

Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag (1999) is incorrect. We will also use the algorithm to calculate distinct local minimisers of a topology optimisation problem via the combination of deflated continuation and a semismooth Newton method.

Character varieties have been studied largely by means of their correspondence to the moduli space of Higgs bundles. In this talk we will report on a method to study their Hodge structure, in particular to compute their E- polynomials. Moreover, we will explain some applications of the given method such as, the study of the topology of the moduli space of doubly periodic instantons. This is joint work with A. González, V.Muñoz and P. Newstead.

Permutations of finitely many elements are often drawn as permutation diagrams. We take this point of view as a motivation to construct and describe more complicated algebras arising for instance from differential operators, from operators acting on (co)homologies, from invariant theory, or from Hecke algebras. The surprising fact is that these diagrams are elementary and simple to describe, but at the same time describe relations between cobordisms as well as categories of represenetations of p-adic groups. The goal of the talk is to give some glimpses of these phenomena and indicate which role categorification plays here.
 

The management of natural resources, from fisheries and climate change to gut bacteria colonies, all require the development of ecological models that represent the full spectrum of population interactions, from competition, through mixotrophy and mutualism, to predation.

Mixotrophic plankton, that both photosynthesise and eat other plankton, underpin all marine food webs and help regulate climate by facilitating gas exchange between the ocean and atmosphere. We show the recent discovery that their feeding preferences change with increasing temperature implies climate change could dramatically alter the structure of marine food webs.

We describe a theoretical framework that reveals the key role of mixotrophy in facilitating transitions between trophic interactions. Mixotrophy smoothly and stably links competition to predation, and extends this linkage to include mutualism in both facultative and obligate forms. Such smooth stable transitions further allow the development of eco-evolutionary theory at the population level through quantitative trait modelling.

Transseries arise naturally when solving differential equations around essential singularities. Just like most Taylor series are not convergent, most transseries do not converge to real functions, even when using advanced summation techniques.

On the other hand, we can show that all classical transseries induce analytic functions on the surreal line. In fact, this holds for an even larger (proper) class of series which we call "omega-series".

Omega-series can be composed and differentiated, like LE-series, and they form a differential subfield of surreal numbers equipped with the simplest derivation. This raises once again the question whether all surreal numbers can be also interpreted as functions. Unfortunately, it turns out that the simplest derivation is in fact incompatible with this goal.

This is joint work with A. Berarducci.

In this talk, I will give a brief overview of the Langlands program and Langlands functoriality with reference to the examples of Galois representations attached to cusp forms and the Jacquet-Langlands correspondence for $\mathrm{GL}_2$. I will then explain how one can generalise this idea, sketching a proof of a Jacquet-Langlands type correspondence from $\mathrm{U}_n(B)$, where $B$ is a quaternion algebra, to $\mathrm{Sp}_{2n}$ and showing that one can attach Galois representations to regular algebraic cuspidal automorphic representations of $\mathrm{Sp}_{2n}$.
 

In this lecture I intend to review a few selected recent results on numerical approximations for high-dimensional nonlinear parabolic partial differential equations (PDEs), nonlinear stochastic ordinary differential equations (SDEs), and high-dimensional nonlinear forward-backward stochastic ordinary differential equations (FBSDEs). Such equations are key ingredients in a number of pricing models that are day after day used in the financial engineering industry to estimate prices of financial derivatives. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on the Heston model, as well as on nonlinear pricing models for financial derivatives. We illustrate our results by several numerical simulations and we also calibrate some of the considered derivative pricing models to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.

Many systems in nature consist of stochastically interacting agents or particles. Stochastic processes have been widely used to model such systems, yet they are notoriously difficult to analyse. In this talk I will show how ideas from statistics can be used to tackle some challenging problems in the field of stochastic processes.

In the first part, I will consider the problem of inference from experimental data for stochastic reaction-diffusion processes. I will show that multi-time distributions of such processes can be approximated by spatio-temporal Cox processes, a well-studied class of models from computational statistics. The resulting approximation allows us to naturally define an approximate likelihood, which can be efficiently optimised with respect to the kinetic parameters of the model. 

In the second part, we consider more general path properties of a certain class of stochastic processes. Specifically, we consider the problem of computing first-passage times for Markov jump processes, which are used to describe systems where the spatial locations of particles can be ignored.  I will show that this important class of generally intractable problems can be exactly recast in terms of a Bayesian inference problem by introducing auxiliary observations. This leads us to derive an efficient approximation scheme to compute first-passage time distributions by solving a small, closed set of ordinary differential equations.