Past

I will explain how to compute the symplectic cohomology of a manifold $M$ conical at infinity, whose Reeb flow at infinity arises from a Hamiltonian circle-action on $M$. For example, this allows one to compute the symplectic cohomology of negative line bundles in terms of the quantum cohomology, and (in joint work with Ivan Smith) via the open-closed string map one can determine the wrapped Fukaya category of negative line bundles over projective space. In this talk, I will show that one can explicitly compute the quantum cohomology and symplectic cohomology of Fano toric negative line bundles, which are in fact different cohomology groups, and surprisingly it is actually the symplectic cohomology which recovers the Jacobian ring of the Landau-Ginzburg superpotential.

We give a new proof of the Frankl-Rödl theorem on set systems with a forbidden intersection. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and Rödl. One consequence of our result is a Frankl-Rödl type theorem for permutations with a forbidden distance. Joint work with Peter Keevash.

Networks arise pervasively in biology, physics, technology, social science, and myriad other areas. An ordinary network consists of a collection of entities (called nodes) that interact via edges. "Multilayer networks" are a more general representation that can be used when nodes are connected to each other via multiple types of edges or a network changes in time.  In this talk, I will discuss how to find dense sets of nodes called "communities" in multilayer networks and some applications of community structure to problems in neuroscience and finance.

Sobolev spaces are the standard framework in which to analyse weak (variational) formulations of PDEs or integral equations and their numerical solution (e.g. using the Finite Element Method or the Boundary Element Method). There are many different ways to define Sobolev spaces on a given domain, for example via integrability of weak derivatives, completions of spaces of smooth functions with respect to certain norms, or restriction from spaces defined on a larger domain. For smooth (e.g. Lipschitz) domains things many of these definitions coincide. But on rough (e.g. fractal) domains the picture is much more complicated. In this talk I'll try to give a flavour of the sort of interesting behaviour that can arise, and what implications this behaviour has for a "practical" example, namely acoustic wave scattering by fractal screens. 

Special Lagranigian submanifolds are area-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. One can define the moduli space of compact special Lagrangian submanifolds of a (fixed) Calabi--Yau manifold. Mclean proves it has a structure of manifold (of dimension finite). It isn't compact in general, but one can compactify it by using geometric measure theory.

Kontsevich conjectured a mirror symmetry, and special Lagrangians should be "mirror" to holomorphic vector bundles. By using algebraic geometry one can compactify the moduli space of holomorphic vector bundles. By "counting" holomorphic vector bundles in Calabi--Yau 3-folds Richard Thomas defined holomorphic Casson invariants (Donaldson-Thomas invariants).

So far as I know it's an open question (probably very difficult) whether one can "count" special Lagrangians, or define a nice structure on the (compactified) moduli space of special Lagrangians.

To do it one has to study singularities of special Lagrangians.

One can smooth singularities in suitable situations: given a singular special Lagrangian, one can construct smooth special Lagrangians tending to it (by the gluing technique). I've proved a uniqueness theorem in a "symmetric" situation: given a symmetric singularity, there's only one way to smooth it (the point of the proof is that the symmetry reduces the problem to an ordinary differential equation).

More recently I've studied a non-symmetric situation together with Dominic Joyce and Joana Oliveira dos Santos Amorim. Our method is based on Lagrangian Floer theory, and is effective at least for pairs of two (special) Lagrangian planes intersecting transversely.

I'll give the details in the talk.

We consider equations with the simplest hysteresis operator at

the right-hand side. Such equations describe the so-called processes "with

memory" in which various substances interact according to the hysteresis

law. The main feature of this problem is that the operator at the

right-hand side is a multivalued.

We present some results concerning the optimal regularity of solutions.

Our arguments are based on quadratic growth estimates for solutions near

the free boundary. The talk is based on joint work with Darya

Apushkinskaya.

A finite set of elements in a connected real Lie group is "Diophantine" if non-identity short words in the set all lie far away from the identity. It has long been understood that in abelian groups, such sets are abundant. In this talk I will discuss recent work of Aka; Breuillard; Rosenzweig and de Saxce concerning this phenomenon (and its limitations) in the more general setting of nilpotent groups. 

We will consider infinite translation surfaces which are abelian covers of

compact surfaces with a (singular) flat metric and focus on the dynamical

properties of their flat geodesics. A motivation come from mathematical

physics, since flat geodesics on these surfaces can be obtained by unfolding

certain mathematical billiards. A notable example of such billiards is  the

Ehrenfest model, which consists of a particle bouncing off the walls of a

periodic planar array of rectangular scatterers.

The dynamics of flat geodesics on compact translation surfaces is now well

understood thanks to the beautiful connection with Teichmueller dynamics. We

will survey some recent advances on the study of infinite translation

surfaces and in particular focus on a joint work with K. Fraczek,  in which

we proved that the Ehrenfest model and more in general geodesic flows on

certain abelain covers have no dense orbits. We will try to convey an

heuristic idea of how Teichmueller dynamics plays a crucial role in the

proofs.

Sampling a $d$-dimensional continuous signal (say a semimartingale) $X:[0,T] \rightarrow \mathbb{R}^d$ at times $D=(t_i)$, we follow the recent papers [Gyurko-Lyons-Kontkowski-Field-2013] and [Lyons-Ni-Levin-2013] in constructing a lead-lag path; to be precise, a piecewise-linear, axis-directed process $X^D: [0,1] \rightarrow
\mathbb{R}^{2d}$ comprised of a past and future component. Lifting $X^D$ to its natural rough path enhancement, we can consider the question of convergence as
the latency of our sampling becomes finer.

tba

I will introduce the physical phenomena of transonic shocks, and review the progresses on related boundary value problems of the steady compressible Euler equations. Some Ideas/methods involved in the studies will be presented through specific examples. The talk is based upon joint works with my collaborators.

Droplet snap-off and coalescence are very rich hydrodynamic phenomena that are even richer in liquid crystals where both the bulk phase and the interface have anisotropic properties. We studied both phenomena in suspensions of colloidal platelets with isotropic-nematic phase coexistence.

We observed two different scenarios for droplet snap-off depending on the relative values of the elastic constant and anchoring strength, in both cases markedly different from Newtonian pinching.[1] Furthermore, we studied coalescence of nematic droplets with the bulk nematic phase. For small droplets this qualitatively resembles coalescence in isotropic fluids, while larger droplets act as if they are immiscible with their own bulk phase. We also observed an interesting deformation of the director field inside the droplets as they sediment towards the bulk phase, probably as a result of flow inside the droplet. Finally, we found that mutual droplet coalescence is accompanied by large droplet deformations that closely resemble coalescence of isotropic droplets.[2]

[1] A.A. Verhoeff and H.N.W. Lekkerkerker, N. J. Phys. 14, 023010 (2012)

[2] M. Manga and H.A. Stone, J. Fluid Mech. 256, 647 (1993)

Note: Change of time and (for Logic) place! Joint with Number Theory (double header)

Quivers are directed graphs which can be thought of as "space" in noncommutative geometry. In this talk, we will try to establish a link between noncommutative geometry and its commutative counterpart. We will show how one can construct (differential graded) quivers which are "equivalent" (in the sense of derived category of representations) to vector bundles on smooth varieties.

There are many financial models used in practice (CIR/Heston, Vasicek,

Stein-Stein, quadratic normal) whose popularity is due, in part, to their

analytically tractable asset pricing. In this talk we will show that it is

possible to generalise these models in various ways while maintaining

tractability. Conversely, we will also characterise the family of models

which admit this type of tractability, in the spirit of the classification

of polynomial term structure models.

Much of the mathematical modelling of urban systems revolves around the use spatial interaction models, derived from information theory and entropy-maximisation techniques and embedded in dynamic difference equations. When framed in the context of a retail system, the

dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be analysed analytically. In this contribution, we present some analysis of the continuous retail model and corresponding discrete version, which yields insights into the effect of space on the system, and an understanding of why certain retail centers are more successful than others. This class of models turns out to have wide reaching applications: from trade and migration flows to the spread of riots and the prediction of archeological sites of interest, examples of which we explore in more detail during the talk.

The goal of this lecture is describing recent joint work with Henri Darmon, in which we construct an Euler system of twisted Gross-Kudla diagonal cycles that allows us to prove, among other results, the following statement (under a mild non-vanishing hypothesis that we shall make explicit):

Let $E/\mathbb{Q}$ be an elliptic curve and $K=\mathbb{Q}(\sqrt{D})$ be a real quadratic field. Let $\psi: \mathrm{Gal}(H/K) \rightarrow \mathbb{C}^\times$ be an anticyclotomic character. If $L(E/K,\psi,1)\ne 0$ then the $\psi$-isotypic component of the Mordell-Weil group $E(H)$ is trivial.

Such a result was known to be a consequence of the conjectures on Stark-Heegner points that Darmon formulated at the turn of the century. While these conjectures still remain highly open, our proof is unconditional and makes no use of this theory.