Past

We are increasingly better at predicting things about our environment. Modern weather forecasts are a lot better than they used to be, and our ability to predict climate change illustrates our better understanding of our effect on our environment. But what about predicting our collective effect on ourselves?  We now use tools like Google maps to predict how long it will take us to drive to work, and other small things, but we fail miserably when it comes to many of the big things. For example, the recent financial crisis cost the world tens of trillions of pounds, yet our ability to forecast, understand and mitigate the next economic crisis is very low. Is this inherently impossible? Or perhaps we are just not going about it the right way? The complex systems approach to economics, which brings in insights from the physical and natural sciences, presents an alternative to standard methods. Doyne will explain what this new approach is and give a few examples of its successes so far. He will then present a vision of the economics of the future which will need to confront the serious problems that the world will soon face.
 

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In 2000, Vojta solved the n-squares problem under the Bombieri-Lang conjecture, by explicitly finding all the curves of genus 0 or 1 on the surfaces related to this problem. The fundamental notion used by him is $\omega$-integrality of curves. 
In this talk, I will show a generalization of Vojta's method to find all curves of low genus in some surfaces, with arithmetic applications.
I will also explain how to use $\omega$-integrality to obtain a bound of the height of a non-constant morphism from a curve to $\mathbb{P}^2$ in terms of the number of intersections (without multiplicities) of its image with a divisor of a particular kind. This proves some new special cases of Vojta's conjecture for function fields.
 

One of the fundamental tools in numerical analysis and PDE
is the finite element method (FEM). A main ingredient in
FEM are splines: piecewise polynomial functions on a
mesh. Even for a fixed mesh in the plane, there are many open
questions about splines: for a triangular mesh T and
smoothness order one, the dimension of the vector space
  C^1_3(T) of splines of polynomial degree at most three
is unknown. In 1973, Gil Strang conjectured a formula
for the dimension of the space C^1_2(T) in terms of the
combinatorics and geometry of the mesh T, and in 1987 Lou
Billera used algebraic topology to prove the conjecture
(and win the Fulkerson prize). I'll describe recent progress
on the study of spline spaces, including a quick and self
contained introduction to some basic but quite useful tools
from topology.

In 2000, Vojta solved the n-squares problem under the Bombieri-Lang conjecture, by explicitly finding all the curves of genus 0 or 1 on the surfaces related to this problem. The fundamental notion used by him is $\omega$-integrality of curves. 

In this talk, I will show a generalization of Vojta's method to find all curves of low genus in some surfaces, with arithmetic applications.

I will also explain how to use $\omega$-integrality to obtain a bound of the height of a non-constant morphism from a curve to $\mathbb{P}^2$ in terms of the number of intersections (without multiplicities) of its image with a divisor of a particular kind.
This proves some new special cases of Vojta's conjecture for function fields.
 

We consider the problem of computing a nonnegative low rank factorization to a given nonnegative input matrix under the so-called "separabilty condition".  This assumption makes this otherwise NP hard problem polynomial time solvable, and we will use first order optimization techniques to compute such a factorization. The optimization model use is based on sparse regression with a self-dictionary, in which the low rank constraint is relaxed to the minimization of an l1-norm objective function.  We apply these techniques to endmember detection and classification in hyperspecral imaging data.

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

Not considering classified work, the first person to have asked and solved the problem of secure communication over insecure communication channels was Ralph Merkle, in a project for a Computer securitjohn y course at UC Berkeley in 1974. In this work, he gave a protocol that allow two legitimate parties to establish a secret key with an effort of the order of N, but such that an eavesdropper can not discover the secret key with non-vanishing probability if he is not willing to spend an effort of at least the order of N^2.
In this talk, we will consider key exchange protocols in the presence of a quantum eavesdropper. Unfortunately, it is easy to see that in this case, breaking Merkle’s original protocol only requires an effort of the order of N, similar to the one of the legitimate parties. We will show how to restore the security by presenting two sequences of protocols with the following properties:
- In the first sequence, the legitimate parties have access to a quantum computer, and the eavesdropper's effort is arbitrarily close to N^2.
- In the second sequence, the protocols are classical, but the eavesdropper’s effort is arbitrarily close to N^{3/2}.
We will show the key exchange protocols, the quantum attacks with the proof of their optimality. We will focus mostly on the techniques from quantum algorithms and complexity theory used to devise quantum algorithms and to prove lower bounds. The underlying tools are the quantum walk formalism, and the quantum adversary lower bound method, respectively. Finally, we will introduce a new method to prove average-case quantum query complexity lower bounds.

The problem of computing the Galois group of an irreducible, rational polynomial has been studied for many years. I will discuss the methods developed over the years to approach this problem, and give some examples of them in practice. These methods mainly involve constructing and factorising resolvent polynomials, and thereby determining better upper bounds for the conjugacy class of the Galois group within the symmetric group, i.e. describe its action on the roots of the polynomial explicitly. I will describe how using approximations to the zeros of the polynomial allows us to construct resolvents, and in particular, how using p-adic approximations can be advantageous over numerical approximations, and how this can yield a direct and systematic method of determining the Galois group.

The Hall algebra can be constructed using the Waldhausen S-construction. We will give a systematic recipe for this and show how it extends naturally to give a bi-algebraic structure. As a result we obtain a more transparent proof of Green's theorem about the bi-algebra structure on the Hall algebra.

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large,
$$
R_k(C_n)=2^{k-1}(n-1)+1.
$$
This resolves a conjecture of Bondy and Erdős for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation.  This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal $k$-colourings for this problem and perfect matchings in the $k$-dimensional hypercube $Q_k$.

It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

Thin glass sheets are used in smartphone, battery and semiconductor technology, and may be manufactured by first producing a relatively thick glass slab (known as a preform) and subsequently redrawing it to a required thickness. Theoretically, if the sheet is redrawn through an infinitely long heater zone, a product with the same aspect ratio as the preform may be manufactured. However, in reality the effect of surface tension and the restriction to factories of finite size prevent this. In this talk I will present a mathematical model for a viscous sheet undergoing redraw, and use asymptotic analysis in the thin-sheet, low-Reynolds-number limit to investigate how the product shape is affected by process parameters. 

I will survey of some of the many significant connections between integrable many-body physics and mathematics. I exploit an algebraic structure called a fusion category, familiar from the study of conformal field theory, topological quantum field theory and knot invariants. Rewriting statistical-mechanical models in terms of a fusion category allows the derivation of combinatorial identities for the Tutte polynomial, the analysis of discrete ``holomorphic'' observables in probability, and to defining topological defects in lattice models. I will give a little more detail on topological defects, explaining how they allows exact computations of conformal-field-theory quantities directly on the lattice, as well as a greatly generalised set of duality transformations.
 

We consider the Beris-Edwards model of liquid crystal dynamics. We study a non-dimensionalisation and regime suited for the study of defect patterns, that amounts to a combined high Ericksen and high Reynolds  number regime. 
We identify some of the flow mechanisms responsible for the appearance of localized gradients that increase in time.
This is joint work with Hao Wu (Fudan).
 

Decorate knot cobordisms functorially induce maps on knot Floer homology.
We compute these maps for elementary cobordisms, and hence give a formula for 
the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of
concordances and present some applications to invertible concordances. 
This is joint work with Marco Marengon.
 

We revisit small-noise expansions in the spirit of Benarous, Baudoin-Ouyang, Deuschel-Friz-Jacquier-Violante for bivariate diffusions driven by fractional Brownian motions with different Hurst exponents. A particular focus is devoted to rough stochastic volatility models which have recently attracted considerable attention.
We derive suitable expansions (small-time, energy, tails) in these fractional stochastic volatility models and infer corresponding expansions for implied volatility. This sheds light (i) on the influence of the Hurst parameter in the time-decay of the smile and (ii) on the asymptotic behaviour of the tail of the smile, including higher orders.

Abstract: (This is joint work with Mark McLean, Stony Brook University N.Y.).

The classical McKay correspondence is a 1-1 correspondence between finite subgroups G of SL(2,C) and simply laced Dynkin diagrams (the ADE classification). These diagrams determine the representation theory of G, and they also describe the intersection theory between the irreducible components of the exceptional divisor of the minimal resolution Y of the simple surface singularity C^2/G. In particular those components generate the homology of Y. In the early 1990s, Miles Reid conjectured a far-reaching generalisation to higher dimensions: given a crepant resolution Y of the singularity C^n/G, where G is a finite subgroup of SL(n,C), the claim is that the conjugacy classes of G are in 1-1 correspondence with generators of the cohomology of Y. This has led to much active research in algebraic geometry in recent years, in particular Batyrev proved the conjecture in 2000 using algebro-geometric techniques (Kontsevich's motivic integration machinery). The goal of my talk is to present work in progress, jointly with Mark McLean, which proves the conjecture using symplectic topology techniques. We construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y.

This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the crepant setup.