Past

I will introduce classical results on finiteness theorem with a way of connecting them to idea of covering spaces. I will talk about the proof of FLT under this connection.

There is a simple way to “glue together” a coupled pair of continuum random trees to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). We present an explicit and canonical way to embed the sphere into the Riemann sphere. In this embedding, the measure is Liouville quantum gravity with parameter gamma in (0,2), and the curve is space-filling version of SLE with kappa=16/gamma^2. Based on joint work with Bertrand Duplantier and Scott Sheffield

TBC

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.
This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.
I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.
This is joint work with Daniel Woodhouse.

We consider a controlled system, in which an input $X: [0, T] \rightarrow E:= \mathbb{R}^{d}$ is a continuous but potentially highly oscillatory path and the corresponding output $Y$ is the line integral along $X$, for some unknown function $f: E \rightarrow E$. The rough paths theory provides a general framework to answer the question on which mild condition of $X$ and $f$, the integral $I(X)$ is well defined. It is robust enough to allow to treat stochastic integrals in a deterministic way. In this paper we are interested in identification of controlled systems of this type. The difficulty comes from the high dimensionality caused by the input of a function type. We propose novel adaptive and non-parametric algorithms to learn the functional relationship between the  input and the output from the data by carefully choosing the feature set of paths based on the rough paths theory and applying linear regression techniques. The algorithms is demonstrated on a financial application where the task is to predict the P$\&$L of the unknown trading strategy.

From a complex analytic perspective, both Teichmüller spaces and
symmetric spaces can be realised as contractible bounded domains, that
have several features in common but also exhibit many differences. In
this talk we will study isometric maps between these two important
classes of bounded domains equipped with their intrinsic Kobayashi metric.

Calabi-Yau manifolds without flux are perhaps the best-known
supergravity backgrounds that leave some supersymmetry unbroken. The
supersymmetry conditions on such spaces can be rephrased as the
existence and integrability of a particular geometric structure. When
fluxes are allowed, the conditions are more complicated and the
analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for
generic D=4, N=2 backgrounds with flux in both type II and
eleven-dimensional supergravity. The geometry is characterised by a
pair of G-structures in 'exceptional generalised geometry' that
interpolate between complex, symplectic and hyper-Kahler geometry.
Supersymmetry is then equivalent to integrability of the structures,
which appears as moment maps for diffeomorphisms and gauge
transformations. Similar structures also appear in D=5 and D=6
backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds
in type IIB, where deformations of these geometric structures give
exactly marginal deformations of the dual field theories.

In the course of work with Jamshid Derakhshan on definability in adele rings, we came upon various problems about definability and model completeness for possibly infinite dimensional algebraic extensions of p-adic fields (sometimes involving uniformity across p). In some cases these problems had been closely approached in the literature but never  explicitly considered.I will explain what we have proved, and try to bring out many big gaps in our understanding of these matters. This  seems appropriate just over 50 years after the breakthroughs of Ax-Kochen and Ershov.

Abstract: Four manifolds are some of the most intriguing objects in topology. So far, they have eluded any attempt of classification and their behaviour is very different from what one encounters in other dimensions. On the other hand, Einstein metrics are among the canonical types of metrics one can find on a manifold. In this talk I will discuss many of the peculiarities that make dimension four so special and see how Einstein metrics could potentially help us understand more about four manifolds.

Solving systems of linear equations $Ax=b$ is easy, but how can we solve such a system when given a "noisy" version of $b$? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum. 

We propose the use of a constraint preconditioner for the iterative solution of the linear system arising from the finite element discretization of the coupled Stokes-Darcy system. The Stokes-Darcy system is a set of coupled PDEs that can be used to model a freely flowing fluid over porous media flow. The fully coupled system matrix is large, sparse, non-symmetric, and of saddle point form. We provide for exact versions of the constraint preconditioner spectral and field-of-values bounds that are independent of the underlying mesh width. We present several numerical experiments, using the deal.II finite element library, that illustrate our results in both two and three dimensions. We compare exact and inexact versions of the constraint preconditioner against standard block diagonal and block lower triangular preconditioners to illustrate its favorable properties.

I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.

For a finitely generated group $G$ with subgroup $H$ we define $e(G,H)$, the relative ends of the pair $(G,H)$, to be the number of ends of the Cayley graph of G quotiented out by the left action of H. We will examine some basic properties of relative ends and will outline the theorem of Sageev showing that $e(G,H)>1$ if and only if $G$ acts essentially on a simply connected CAT(0) cube complex. If time permits, we will outline Niblo's proof of Stallings' theorem using Sageev's construction.

I will give a historical overview of some of the theorems proved by the
Ancient Greeks, which are now taken for granted but were, and are,
landmarks in the history of mathematics. Particular attention will be
given to the calculation of areas, including theorems of Hippocrates,
Euclid and Archimedes.

Let $G$ be a compact Lie group and $k$ be a field of characteristic $p\ge 0$ such that $H^*(G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category over $k$ if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian $G$-action on $X$. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians. 

A graph is quasirandom if its edge distribution is similar (in a well defined quantitative way) to that of a random graph with the same edge density. Classical results of Thomason and Chung-Graham-Wilson show that a variety of graph properties are equivalent to quasirandomness. On the other hand, in some known proofs the error terms which measure quasirandomness can change quite dramatically when going from one property to another which might be problematic in some applications.

Simonovits and Sós proved that the property that all induced subgraphs have about the expected number of copies of a fixed graph $H$ is quasirandom. However, their proof relies on the regularity lemma and gives a very weak estimate. They asked to find a new proof for this result with a better estimate. The purpose of this talk is to accomplish this.

Joint work with D. Conlon and J. Fox

We provide a simple convergence analysis unified for TR and a new ARC algorithms, which we name ARCq and which is very close in spirit to trust region methods, closer than the original ARC is. We prove global convergence to second order points. We also obtain as a corollary the convergence of the original ARC method. Since one of our aims is to achieve a simple presentation, we sacrifice some generality which we discuss at the end of our developments. In this simplified setting, we prove the optimal complexity property for the ARCq and identify the key elements which allow it. We then propose efficient implementations using a Cholesky like factorization as well as a scalable version based on conjugate gradients.