Past

For reasonable domains $\Omega\subseteq\mathbb{R}^{d+
1}$, and given some boundary data $f\in C(\partial\Omega)$, we can solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $\partial\Omega$. For $x_{0}\in \Omega$, the association $f\rightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $\omega_{\Omega}^{x_{0}}$ on $\partial\Omega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $E\subseteq \partial\Omega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$. In this talk, we will survey some very recent results about the relationship between the measure theoretic behavior of harmonic measure and the geometry of the boundary of its domain. In particular, we will study how absolute continuity of harmonic measure with respect to $d$-dimensional Hausdorff measure implies rectifiability of the boundary and vice versa.
 

The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to
perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).

Using tropical geometry and new methods in the theory of Fukaya categories, we explain a mirror symmetry equivalence relating the Fukaya category of a hypersurface and the category of coherent sheaves on the boundary of a toric variety.

Given a graph $G$, we can form a hypergraph $H$ whose edges correspond to the triangles in $G$. If $G$ is the standard Erdős-Rényi random graph with independent edges, then $H$ is random, but its edges are not independent, because of overlapping triangles. This is (presumably!) a major complication when proving results about triangles in random graphs.  However, it turns out that, for many purposes, we can treat the triangles as independent, in a one-sided sense (and losing something in the density): we can find an independent random hypergraph within the set of triangles. I will present two proofs, one of which generalizes to larger complete (and some non-complete) subgraphs.

In this talk, a novel discontinuous Galerkin (DG) method is introduced by utilising the principle of discrete least squares. The key idea is to build polynomial approximations by the method of  (weighted) discrete least squares instead of usual interpolation or (discrete) $L^2$ projections. The resulting method hence uses more information of the underlying function and provides a more robust alternative to common DG methods. As a result, we are able to construct high-order schemes which are conservative as well as linear stable on any set of collocation points. Several numerical tests highlight the new discontinuous Galerkin discrete least squares (DG-DLS) method to significantly outperform present-day DG methods.

High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework.

However, if generalised SBP operators are used, the treatment of boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely —interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain.

In this talk, the concept of generalised SBP operators and their application to entropy stable semidiscretisations will be presented. Several recent ideas extending the range of possible methods are discussed, presenting both advantages and several shortcomings.

In this talk I will first briefly introduce 1-parameter persistent homology, and discuss some applications and the theoretical challenges in the multiparameter case. If time remains I will explain how tools from commutative algebra give invariants suitable for the study of data. This last part is based on the preprint https://arxiv.org/abs/1708.07390.
 

Shortly after Mason & Skinner introduced the so-called ambitwistor strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In the heterotic version, however, both models give somewhat unsatisfactory descriptions of the supergravity sector.

In this talk, I will show how the original pure-spinor version of the heterotic ambitwistor string can be modified in a consistent manner that renders the supergravity sector treatable. In addition to the massless states, the spectrum of the new model --- which we call sectorized heterotic string --- contains a single massive level. In the limit in which a dimensionful parameter is taken to infinity, these massive states become the unexpected massless states (e.g. a 3-form potential) first encountered by Mason & Skinner."

Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.

A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this
talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).

Inverting the signature of a path with ideas from linear algebra with implementations.

A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the  Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole. 
 

After recalling some backgrounds and motivations, we'll report some recent results on the optimal L^2 extensions and multiplier ideal sheaves, with emphasizing the close relations between SCV and PDE.

We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e., each data point is a solution to a system of polynomial equations. In this case, the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. We study the sampling requirements for matrix completion under a variety model with a focus on a union of subspaces. We also propose an efficient matrix completion algorithm that minimizes a surrogate of the rank of the matrix of monomial features, which is able to recover synthetically generated data up to the predicted sampling complexity bounds. The proposed algorithm also outperforms standard low-rank matrix completion and subspace clustering techniques in experiments with real data.

Let \Gamma_{g,1} denote the mapping class group of a genus g surface with one parametrized boundary component.  The group homology H_i(\Gamma_{g,1}) is independent of g, as long as g is large compared to i, by a famous theorem of Harer known as homological stability, now known to hold when 2g > 3i.  Outside that range, the relative homology groups H_i(\Gamma_{g,1},\Gamma_{g-1,1}) contain interesting information about the failure of homological stability.  In this talk, I will discuss a metastability result; the relative groups depend only on the number k = 2g-3i, as long as g is large compared to k.  This is joint work with Alexander Kupers and Oscar Randal-Williams.

The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem.  It provides us with an  interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions).  My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.
 

After a brief review on the classical Prandtl system, we introduce our recent work on the well-posedness and high Reynolds numbers limit for the MHD boundary layer that shows the tangential magnetic field stabilizes the boundary layer. And then we will discuss some instability phenomena of the shear flow for both the classical Prandtl and MHD boundary layer systems. The talk includes some recent joint works with Chengjie Liu, Yaguang Wang on the classical Prandtl equation, and with Chengjie Liu and Feng Xie on the magnetohydrodynamic boundary layer.

We will discuss models for vehicular traffic flow on networks. The models include both the Lighthill-Whitham-Richards (LWR) model and Follow-the-Leader (FtL) models.
The emphasis will be on the Braess paradox in which adding a road to a traffic network can make travel times worse for all drivers. 
In addition we will present a novel proof how FtL models approximate the LWR model in case of heavy traffic.

Abstract: If A is a finite dimensional algebra, and D(A) the unbounded
derived category of the full module category Mod-A, then it is
straightforward to see that D(A) is generated (as a "localizing
subcategory") by the indecomposable projectives, and by the simple 
modules. It is not so obvious whether it is generated by the 
indecomposable injectives. In 2001, Keller gave a talk in which he 
remarked that"injectives generate" would imply several of the well-known
homological conjectures, such as the Nunke condition and hence the 
generalized Nakayama
conjecture, and asked if there was any relation to the finitistic 
dimension conjecture. I'll show that an algebra that satisfies "injectives 
generate" also satisfies the finitistic dimension conjecture and discuss 
some examples. I'll present things in a fairly concrete way, so most of 
the talk won't assume much knowledge of derived categories.

 

Abstract: In this talk I will discuss the interplay between the local and
the global invariants in modular representation theory with a focus on the
first Hochschild cohomology $\mathrm{HH}^1(B)$ of a block algebra $B$. In
particular, I will show the compatibility between $r$-integrable 
derivations
and stable equivalences of Morita type. I will also show that if
$\mathrm{HH}^1(B)$ is a simple Lie algebra such that $B$ has a unique
isomorphism class of simple modules, then $B$ is nilpotent with an
elementary abelian defect group $P$ of order at least 3. The second part 
is joint work with M. Linckelmann.

Abstract: I will describe how the ADE preprojective algebras appear in 
certain Conformal Field Theories, namely SU(2) WZW models, and explain
the generalisation to the SU(3) case, where 'almost CY3' algebras appear.

Abstract: In this talk, we will introduce new affine algebraic varieties 
for algebras given by quiver and relations. Each variety contains a 
distinguished element in the form of a monomial algebra. The properties 
and characteristics of this monomial algebra govern those of all other 
algebras in the variety. We will show how amongst other things this gives 
rise to a new way to determine whether an algebra is quasi-hereditary. 
This is a report on joint work both with Ed Green and with Ed Green and 
Lutz Hille.