Past

With the general election looming upon I will discuss the various different kinds of voting system that one could implement in such an election. I will show that these can give very different answers to the same set of voters. I will then discuss Arrow's Impossibility Theorem which shows that no voting system is compatible with 4 simple axioms which may be desireable.

Tropicalization replaces a variety by a polyhedral complex that is a "combinatorial shadow" of the original variety.  This allows algebraic geometric problems to be attacked using combinatorial and
polyhedral techniques.  While this idea has proved surprisingly effective over the last decade, it has so far been restricted to the study of varieties and algebraic cycles.  I will discuss joint work with Felipe Rincon, building on work of Jeff and Noah Giansiracusa, to understand tropicalizing schemes, and more generally the concept of a tropical scheme.

Graphons and permutons are analytic objects associated with convergent sequences of graphs and permutations, respectively. Problems from extremal combinatorics and theoretical computer science led to a study of graphons and permutons determined by finitely many substructure densities, which are referred to as finitely forcible. The talk will contain several results on finite forcibility, focusing on the relation between finite forcibility of graphons and permutons. We also disprove a conjecture of Lovasz and Szegedy about the dimension of the space of typical vertices of finitely forcible graphons. The talk is based on joint work with Roman Glebov, Andrzej Grzesik and Dan Kral.

We will begin by discussing the risk parity portfolio selection problem, which aims to find  portfolios for which the contributions of risk from all assets are equally weighted. The risk parity may be satisfied over either individual assets or groups of assets. We show how convex optimization techniques can find a risk parity solution in the nonnegative  orthant, however, in general cases the number of such solutions can be anywhere between zero and  exponential in the dimension. We then propose a nonconvex least-squares formulation which allows us to consider and possibly solve the general case. 

Motivated by this problem we present several alternating direction schemes for specially structured nonlinear nonconvex problems. The problem structure allows convenient 2-block variable splitting.  Our methods rely on solving convex subproblems at each iteration and converge to a local stationary point. Specifically, discuss approach  alternating directions method of multipliers and the alternating linearization method and we provide convergence rate results for both classes of methods. Moreover, global optimization techniques from polynomial optimization literature are applied to complement our local methods and to provide lower bounds.

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until recently.

There is a tight fit between type theories à la Martin-Löf and constructive set theories such as Constructive Zermelo-Fraenkel set theory, CZF, and its extension as well as classical Kripke-Platek set theory and extensions thereof. The technology for determining their (exact) proof-theoretic strength was developed in the 1990s. The situation is rather different when it comes to type theories (with universes) having the impredicative type of propositions Prop from the Calculus of Constructions that features in some powerful proof assistants. Aczel's sets-as-types interpretation into these type theories gives rise to  rather unusual set-theoretic axioms: negative power set and negative separation. But it is not known how to determine the proof-theoretic strengths of intuitionistic set theories with such axioms via familiar classical set theories (though it is not difficult to see that ZFC plus infinitely many inaccessibles provides an upper bound). The first part of the talk will be a survey of known results from this area. The second part will be concerned with the rather special computational and proof-theoretic behavior of such theories.

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).

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Francesco Cellarosi.

It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane. 

We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.

by Jonathan Mestel and Florencia Boshier

Numerical simulation tools for fluid and solid mechanics are often based on the discretisation of coupled systems of partial differential equations, which can easily be identified in terms of physical
conservation laws.  In contrast, much physical insight is often gained from the equivalent formulation of the relevant energy or free-energy functional, possibly subject to constraints.  Motivated by the
nonlinear static and dynamic behaviour of nematic liquid crystals and of magnetosensitive elastomers, we propose a finite-element framework for minimising these free-energy functionals, using Lagrange multipliers to enforce additional constraints.  This talk will highlight challenges, limitations, and successes, both in the formulation of these models and their use in numerical simulation.
This is joint work with PhD students Thomas Benson, David Emerson, and Dong Han, and with James Adler, Timothy Atherton, and Luis Dorfmann.

We describe the cohomology of moduli spaces of points on schemes over Abelian varieties and give explicit calculations for schemes in dimensions less that three. The construction of Gulbrandsen allows one to consider virtual motives in dimension three. In particular we see a new proof of his conjectures on the Euler numbers of generalized Kummer schemes recently proven by Shen. Joint work in progress with Junliang Shen.

A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here $G$ has an $F$-decomposition if the edges of $G$ can be covered by edge-disjoint copies of $F$ (and $F$-divisibility is a trivial necessary condition for this). We extend Wilson's theorem to graphs which are allowed to be far from complete (joint work with B. Barber, D. Kuhn, A. Lo).

I will also discuss some results and open problems on decompositions of dense graphs and hypergraphs into Hamilton cycles and perfect matchings.

I will talk about recent joint work with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold.

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.