Past

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.

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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.

Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using ``local’’ techniques, such as RBF generated finite differences (RBF-FD).

In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations. The method is quite flexible, only requiring a set of ``scattered’’ nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems.

This talk represents joint work with Edward Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).

A Kähler group is a finitely presented group that can be realized as fundamental group of a compact Kähler manifold. It is known that every finitely presented group can be realized as fundamental group of a compact real and even symplectic manifold of dimension greater equal than 4 and of a complex manifold of complex dimension greater equal than 2. In contrast, the question which groups are Kähler groups is surprisingly harder and there are large classes of examples for both, Kähler, and non-Kähler groups. This talk will give a brief introduction to the theory of Kähler manifolds and then discuss some basic examples and properties of Kähler groups. It is aimed at a general audience and no prior knowledge of the field will be required.

For a finitely presented group, the Word Problem asks for an algorithm

which declares whether or not words on the generators represent the

identity. The Dehn function is the time-complexity of a direct attack

on the Word Problem by applying the defining relations.

A "hydra phenomenon" gives rise to novel groups with extremely fast

growing (Ackermannian) Dehn functions. I will explain why,

nevertheless, there are efficient (polynomial time) solutions to the

Word Problems of these groups. The main innovation is a means of

computing efficiently with compressed forms of enormous integers.

This is joint work with Will Dison and Eduard Einstein.

Quantifying the uncertainty in computational simulations is one of the central challenges confronting the field of computational science and engineering today. The uncertainty quantification of inverse problems is neatly addressed in the Bayesian framework, where instead of seeking one unique minimiser of a regularised misfit functional, the entire posterior probability distribution is to be characterised. In this talk I review the deep connection between deterministic PDE-constrained optimisation techniques and Bayesian inference for inverse problems, discuss some recent advances made in the Bayesian viewpoint by adapting deterministic techniques, and mention directions for future research.

Composite dilation wavelets are affine systems which extend the notion of wavelets by incorporating a second set of dilations.  The addition of a second set of dilations allows the composite system to capture directional information in addition to time and frequency information.  We classify admissible dilation groups at two extremes: frequency localization through minimally supported frequency composite dilation wavelets and time localization through crystallographic Haar-type composite dilation wavelets. 

The instanton corrections to the hyperkähler metric on moduli spaces of meromorphic flat SL(2,C)-connections on a Riemann surface with prescribed singularities have recently been studied by Gaiotto, Moore and Neitzke. The instantons are given by certain special trajectories of the meromorphic quadratic differentials which form the base of Hitchin's integrable system structure on the moduli space. Bridgeland and Smith interpret such quadratic differentials as defining stability conditions on an associated 3-Calabi-Yau triangulated category whose stable objects correspond to these special trajectories.

The smallest non-trivial examples are provided by the moduli spaces of quaternionic dimension one. In these cases it is possible to study explicitly the periods of the Seiberg-Witten differential on the fibres of the Hitchin system which define the central charge of the stability condition and lift the period map to the space of stability conditions. This provides in particular a new categorical perspective on the original Seiberg-Witten gauge theories.

The talk is based on my paper with E. Beggs appearing in Class. Quantum

Gravity.

Working within a bimodule approach to noncommutative geometry, we show that

even a small amount of noncommutativity drastically constrains the moduli

space of

noncommutative metrics. In particular, the algebra [x,t]=x is forced to have

a geometry

corresponding to a gravitational source at x=0 so strong that even light

cannot

escape. This provides a non-trivial example of noncommutative Riemannian

geometry

and also serves as an introduction to some general results.

One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed,

transonic shock can be represented as a monotone function of exit pressure.

From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system

when exit pressure is prescribed in a proper range.

In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system,

which is formulated as a free boundary problem with mixed type PDE system.

This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao

In 1997 V. Bentkus and F. Götze introduced a technique for estimating $L^p$ norms of certain exponential sums without needing an explicit estimate for the exponential sum itself. One uses instead a kind of estimate I call a "moat lemma". I explain this term, and discuss the implications for several kinds of point-counting problem which we all know and love.

The'signature', from the theory of differential equations driven by rough paths,
provides a very efficient way of characterizing curves. From a machine learning
perspective, the elements of the signature can be used as a set of features for
consumption by a classification algorithm.

Using datasets of letters, digits, Indian characters and Chinese characters, we
see that this improves the accuracy of online character recognition---that is
the task of reading characters represented as a collection of pen strokes.

A "bordism representation" (*) is a representation of the abstract

structure formed by manifolds and bordisms between them, and hence of

fundamental interest in topology. I will give an overview of joint work

establishing a simple generators-and-relations presentation of the

3-dimensional oriented bordism bicategory, and also its "signature" central

extension. A representation of this bicategory corresponds in a 2-1 fashion

to a modular category, which must be anomaly-free in the oriented case. J/w

Chris Douglas, Chris Schommer-Pries, Jamie Vicary.

(*) These are also known as "topological quantum field theories".

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.

The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role that the surface subgroup problem plays in some important solved and unsolved problems in the theory of 3-manifolds, the geometric group theory, and the theory of arithmetic manifolds.

Jakobshavn Isbrae and many other fast flowing outlet glaciers of present

and past ice sheets lie in deep troughs which often have several

overdeepened sections. To make their fast flow possible their bed needs

to be slippery which in turn means high basal water pressures. I will

present a model of subglacial water flow and its application to

Jakobshavn. I find that, somewhat surprisingly, the reason for

Jakobshavn's fast flow might be the pressure dependence of the melting

point of ice. The model itself describes the unusual fluid dynamics occurring underneath the ice; it has an interesting mathematical structure that presents computational challenges.

We consider the pricing of American put options in a model-independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting, we are able to provide conditions on the American Put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process. (Joint with Christoph Hoeggerl).

A pseudofinite group is an infinite model of the theory of finite groups. I will discuss what can be said about pseudofinite groups under various tameness assumptions on the theory (e.g. NIP, supersimplicity), structural results on pseudofinite permutation groups, and connections to word maps and generalisations.

Ricci solitons were introduced by Richard Hamilton in the 80's and they are a generalization of the better know Einstein metrics. During this talk we will define the notion of Ricci soliton and I will try to convince you that these metrics arise "naturally" in a number of different settings. I will also present various examples and talk a bit about some symmetry properties that Ricci solitons have.

Note: This talk is meant to be introductory and no prior knowledge about Einstein metrics will be assumed (or necessary).

After a short survey of the notion of level of distribution for
arithmetic functions, and its importance in analytic number theory, we
will explain how our recent studies of twists of Fourier coefficients of
modular forms (and especially Eisenstein series) by "trace functions"
lead to an improvement of the results of Friedlander-Iwaniec and
Heath-Brown for the ternary divisor function in arithmetic progressions
to prime moduli.

This is joint work with É. Fouvry and Ph. Michel.

Many real-life complex systems arise as a network of simple interconnected individual agents. A central question is to determine how network topology and individual agent dynamics combine to create the global dynamics.

In this talk we focus on the case of continuous-time random walks on networks, with a waiting time of the walker on each node assuming arbitrary probability distributions. Such random walks are useful to model diffusion processes over complex temporal networks representing human interactions, often characterized by non-Poissonian contact patterns.

We find that the mixing time of the random walker, i.e. the relaxation time for the process to reach stationarity, is determined by a combination of three factors: the spectral gap, associated to bottlenecks in the underlying topology, burstiness, related to the second moment of the waiting time distribution, and the characteristic time of its exponential tail, which is an indicator of the tail `fatness'. We show

theoretically that a strong modular structure dampens the importance of burstiness, and empirically that either of the three factors may be dominant in real-life data.

These results are available in arXiv:1309.4155