Past

In April 2010 Eguchi--Ooguri--Tachikawa observed a fascinating connection between the elliptic genus of a K3 surface and the largest Mathieu group. We will report on joint work with Miranda Cheng and Jeff Harvey that identifies this connection as one component of a system of surprising relationships between a family of finite groups, their representation theory, and automorphic forms of various kinds Mock modular forms, and particularly their shadows, play a key role in the analysis, and we find several of Ramanujan's mock theta functions appearing as McKay--Thompson series arising from the umbral groups.

In this talk I review the use of the spectral decomposition for understanding the solution of ill-posed inverse problems. It is immediate to see that regularization is needed in order to find stable solutions. These solutions, however, do not typically allow reconstruction of signal features such as edges. Generalized regularization assists but is still insufficient and methods of total variation are commonly suggested as an alternative. In the talk I consider application of standard approaches from Tikhonov regularization for finding appropriate regularization parameters in the total variation augmented Lagrangian implementations. Areas for future research will be considered.

Twistor theory is a technology that can be used to translate analytical problems on Euclidean space $\mathbb R^4$ into problems in complex algebraic geometry, where one can use the powerful methods of complex analysis to solve them. In the first half of the talk we will explain the geometry of the Twistor correspondence, which realises $\mathbb R^4$ , or $S^4$, as the space of certain "real" lines in the (projective) Twistor space $\mathbb{CP}^3$. Our discussion will start from scratch and will assume very little background knowledge. As an application, we will discuss the Twistor description of instantons on $S^4$ as certain holomorphic vector bundles on $\mathbb{CP}^3$ due to Ward.

In this talk, I will present reflected backward stochastic differential equations (reflected BSDEs) and their connection with the pricing of American options. Then I will present a simple perturbative method for studying them. Under the appropriate assumptions on the coefficient, the terminal condition and the lower obstacle, similar to those used by Kobylankski, this method allows to prove the existence of a solution. I will also provide the usual comparison theorem and a new proof for a refined comparison theorem, specific to RBSDEs.

We study the mean-field models describing the evolution of distributions of particle radii obtained by taking the small volume fraction limit of the free boundary problem describing the micro phase separation of diblock copolymer melts, where micro phase separation consists of an ensemble of small balls of one component. In the dilute case, we identify all the steady states and show the convergence of solutions.

Next we study the dynamics for a free boundary problem in two dimension, obtained as a gradient flow of Ohta- Kawasaki free energy, in the case that one component is a distorted disk with a small volume fraction. We show the existence of solutions that a small, almost circular interface moves along a curve determined via a Green’s function of the domain. This talk is partly based on a joint work with Xiaofeng Ren.

Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. We introduce the ASAP algorithm, for the graph realization problem in R^d, given a sparse and noisy set of distance measurements associated to the edges of a globally rigid graph. ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in R^3, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The reflections and rotations are estimated using a recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. Furthermore, the algorithm successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose SP-ASAP, a faster version of ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that ASAP and SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms. Time permitting, we briefly discuss the analogy between the graph realization and the low-rank matrix completion problems, as well as an application of synchronization over Z_2 and its variations to bipartite multislice networks.

By recent work of Voevodsky and others, type theories are now considered as a candidate

for a homotopical foundations of mathematics. I will explain what are type theories using the language

of (essentially) algebraic theories. This shows that type theories are in the same "family" of algebraic

concepts such as groups and categories. I will also explain what is homotopic in (intensional) type theories.

Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk.

Random geometric graphs have been well studied over the last 50 years or so. These are graphs that

are formed between points randomly allocated on a Euclidean space and any two of them are joined if

they are close enough. However, all this theory has been developed when the underlying space is

equipped with the Euclidean metric. But, what if the underlying space is curved?

The aim of this talk is to initiate the study of such random graphs and lead to the development of

their theory. Our focus will be on the case where the underlying space is a hyperbolic space. We

will discuss some typical structural features of these random graphs as well as some applications,

related to their potential as a model for networks that emerge in social life or in biological

sciences.

Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the

first talk.

Large-scale zonal jets are observed in a wide range of geophysical and astrophysical flows; most strikingly in the atmospheres of the Jovian gas giant planets. Jupiter's upper atmosphere is highly turbulent, with many small vortices, and strong westerly winds at the equator. We consider the thermal shallow water equations as a model for Jupiter's upper atmosphere. Originally proposed for the terrestrial atmosphere and tropical oceans, this model extends the conventional shallow water equations by allowing horizontal temperature variations with a modified Newtonian cooling for the temperature field. We perform numerical simulations that reproduce many of the key features of Jupiter’s upper atmosphere. However, the simulations take a long time to run because their time step is severely constrained by the inertia-gravity wave speed. We filter out the inertia-gravity waves by forming the quasigeostrophic limit, which describes the rapidly rotating (small Rossby number) regime. We also show that the quasigeostrophic energy equation is the quasigeostrophic limit of the thermal shallow water pseudo-energy equation, analogous to the derivation of the acoustic energy equation from gas dynamics. We perform numerical simulations of the quasigeostrophic equations, which again reproduce many of the key features of Jupiter’s upper atmosphere. We gain substantial performance increases by running these simulations on graphical processing units (GPUs).

We show how the reduction procedure for generalized Kahler  
structures can be used to recover Hitchin's results about the  
existence of a generalized Kahler structure on the moduli space of  
instantons on bundle over a generalized Kahler manifold. In this setup  
the proof follows closely the proof of the same claim for the Kahler  
case and clarifies some of the stranger considerations from Hitchin's  
proof.

We consider the stationary flow of Prandtl-Eyring fluids in two

dimensions. This model is a good approximation of perfect plasticity.

The corresponding potential is only slightly super linear. Thus, many

severe problems arise in the existence theory of weak solutions. These

problems are overcome by use of a divergence free Lipschitz

truncation. As a second application of this technique, we generalize

the concept of almost harmonic functions to the Stokes system.

The classical associahedra are cell complexes, in fact polytopes,

introduced by Stasheff to parametrize the multivariate operations

naturally occurring on loop spaces of connected spaces.

They form a topological operad $ Ass_\infty $ (which provides a resolution

of the operad $ Ass $ governing spaces-with-associative-multiplication)

and the complexes of cellular chains on the associahedra form a dg

operad governing $A_\infty$-algebras (that is, a resolution of the

operad governing associative algebras).

In classical applications it was not necessary to consider units for

multiplication, or it was assumed units were strict. The introduction

of non-strict units into the picture was considerably harder:

Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in

their work on Lagrangian intersection Floer theory, and equivalent

descriptions of the dg operad for homotopy unital $A_\infty$-algebras

have now been given, for example, by Lyubashenko and by Milles-Hirsch.

In this talk we present the "missing link": a cellular topological

operad $uAss_\infty$ of "unital associahedra", providing a resolution

for the operad governing topological monoids, such that the cellular

chains on $uAss_\infty$ is precisely the dg operad of

Fukaya-Ono-Oh-Ohta.

(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)

This is joint work with P. Caputo and D. Chafai. In this talk, we
will consider various probability distributions on the set of stochastic
 matrices with n states and on the set of Laplacian/Kirchhoff
matrices on n states. They will arise naturally from the conductance model on
n states with i.i.d conductances. With the help of random matrix
theory, we will study the spectrum of these processes.