Past

 I will give a short survey on classical inequalities for Laplace eigenvalues, tell about related history and questions. I will then discuss the so-called Korevaar method, and new results generalising to higher eigenvalues a number of classical inequalities known for the first Laplace eigenvalue only. 

In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof. 

In the sequel we will apply this to extend local deformations of closed $G_2$ structures, and to construct 
$C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds. 

This is joint work with Christian Bär (Potsdam).

We consider Brownian motion with Kato class measure-valued drift.   A small time large deviation principle and a Varadhan type asymptotics for the Brownian motion with singular drift are established. We also study the existence and uniqueness of the associated Dirichlet boundary value problems.

We first give a short introduction to analysis and stochastic processes on fractal state spaces and the typical difficulties involved.

We then discuss gradient operators and semilinear PDE. They are used to formulate the main result which establishes the hydrodynamic limit of the weakly asymmetric exclusion process on the Sierpinski gasket in the form of a law of large numbers for the particle density. We will explain some details and, if time permits, also sketch a corresponding large deviations principle for the symmetric case.

Roughly speaking, a commutative-by-finite Hopf algebra is a Hopf
algebra which is an extension of a commutative Hopf algebra by a
finite dimensional Hopf algebra.
There are many big and significant classes of such algebras
(beyond of course the commutative ones and the finite dimensional ones!).
I'll make the definition precise, discuss examples
and review results, some old and some new.
No previous knowledge of Hopf algebras is necessary.
 

For a polynomial $h(x)$ in $F[x]$, where $F$ is any field, let $A$ be the
$F$-algebra given by generators $x$ and $y$ and relation $[y, x]=h$.
This family of algebras include the Weyl algebra, enveloping algebras of
$2$-dimensional Lie algebras, the Jordan plane and several other
interesting subalgebras of the Weyl algebra.

In a joint work in progress with Samuel Lopes, we computed the Hochschild
cohomology $HH^*(A)$ of $A$ and determined explicitly the Gerstenhaber
structure of $HH^*(A)$, as a Lie module over the Lie algebra $HH^1(A)$.
In case $F$ has characteristic $0$, this study has revealed that $HH^*(A)$
has finite length as a Lie module over $HH^1(A)$ with pairwise
non-isomorphic composition factors and the latter can be naturally
extended into irreducible representations of the Virasoro algebra.
Moreover, the whole action can be understood in terms of the partition
formed by the multiplicities of the irreducible factors of the polynomial
$h$.
 

Ice streams are fast flowing regions of ice that generally slide over a layer of unconsolidated, water-saturated subglacial sediment known as till.  A striking feature that has been observed geophysically is that subglacial till has been found to accumulate distinctively into sedimentary wedges at the grounding zones (regions where ice sheets begin to detach from the bedrock to form freely floating ice shelves) of both past and present-day ice sheets. These grounding-zone wedges have important implications for ice-sheet stability against grounding zone retreat in response to rising sea levels, and their origins have remained a long-standing question. Using a combination of mathematical modelling, a series of laboratory experiments, field data and numerical simulations, we develop a fluid-mechanical model that explains the mechanism of the formation of these sedimentary wedges in terms of the loading and unloading of deformable till in three dynamical regimes. We also undertake a series of analogue laboratory experiments, which reveal that a similar wedge of underlying fluid accumulates spontaneously in experimental grounding zones, we formulate local conditions relating wedge slopes in each of the scenarios and compare them to available geophysical radargram data at the well lubricated, fast-flowing Whillans Ice Stream.

Mathematical biology has grown enormously over the past 40 years and has changed considerably. At first, biology inspired mathematicians to come up with models that could, at an abstract level, "explain" biological phenomena - one of the most famous being Alan Turing's model for biological pattern formation. However, with the enormous recent advances in biotechnology and computation, the field is now truly inter- and multi-disciplinary. We shall discuss the changing role mathematics is playing in applications to biology and medicine.

Genome replication is a stochastic process whereby each cell exhibits different patterns of origin activation and replication fork movement.  Despite this heterogeneity, replication is a remarkably stable process that works quickly and correctly over hundreds of thousands of iterations. Existing methods for measuring replication dynamics largely focus on how a population of cells behave on average, which precludes the detection of low probability errors that may have occurred in individual cells.  These errors can have a severe impact on genome integrity, yet existing single-molecule methods, such as DNA combing, are too costly, low-throughput, and low-resolution to effectively detect them.  We have created a method that uses Oxford Nanopore sequencing to create high-throughput genome-wide maps of DNA replication dynamics in single molecules.  I will discuss the informatics approach that our software uses, our use of mathematical modelling to explain the patterns that we observe, and questions in DNA replication and genome stability that our method is uniquely positioned to answer.

Given a stability condition defined over a category, every object in this category
is filtered by some distinguished objects called semistables. This
filtration, that is unique up-to-isomorphism, is know as the
 Harder-Narasimhan filtration.
One less studied property of stability conditions, when defined over an
 abelian category, is the fact that each of them induce a chain of torsion
classes that is naturally indexed.
 In this talk we will study arbitrary indexed chain of torsion classes. Our
first result states that every indexed chain of torsion classes induce a
 Harder-Narasimhan filtration. Following ideas from Bridgeland we
 show that the set of all indexed chains of torsion classes satisfying a mild 
 technical condition forms a topological space. If time we
 will characterise the neighbourhood or some distinguished points. 

Landmark-based human action recognition in videos is a challenging task in computer vision. One crucial step is to design discriminative features for spatial structure and temporal dynamics. To this end, we use and refine the path signature as an expressive, robust, nonlinear, and interpretable representation for landmark-based streamed data. Instead of extracting signature features from raw sequences, we propose path disintegrations and transformations as preprocessing to improve the efficiency and effectiveness of signature features. The path disintegrations spatially localize a pose into a collection of m-node paths from which the signatures encode non-local and non-linear geometrical dependencies, while temporally transform the evolutions of spatial features into hierarchical spatio-temporal paths from which the signatures encode long short-term dynamical dependencies. The path transformations allow the signatures to further explore correlations among different informative clues. Finally, all features are concatenated to constitute the input vector of a linear fully-connected network for action recognition. Experimental results on four benchmark datasets demonstrated that the proposed feature sets with only linear network achieves comparable state-of-the-art result to the cutting-edge deep learning methods. 

Starting with a countable transitive model of V=L, we show that by 
adding a single Cohen real, c, most intermediate models do no satisfy choice. In 
fact, most intermediate models to L[c] are not even definable.

The key part of the proof is the Bristol model, which is intermediate to L[c], 
but is not constructible from a set. We will give a broad explanation of the 
construction of the Bristol model within the constraints of time.

Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. I'll discuss some recent refinements of Gallagher's theorem, one of which is joint work with Niclas Technau. A key new ingredient is the correspondence between Bohr sets and generalised arithmetic progressions. It is hoped that these are the first steps towards a metric theory of multiplicative diophantine approximation on manifolds. 

I will give an overview of the Nielsen-Thurston theory of the mapping class group and its connection to hyperbolic geometry and dynamics. Time permitting, I will discuss the surface entropy conjecture and a theorem of Hamenstadt on entropies of `generic' elements of the mapping class group. No prior knowledge of the concepts involved is required.

Injection of a gas into a gas/liquid foam is known to give rise to instability phenomena on a variety of time and length scales. Macroscopically, one observes a propagating gas-filled structure that can display properties of liquid finger propagation as well as of fracture in solids. Using a discrete model, which incorporates the underlying film instability as well as viscous resistance from the moving liquid structures, we describe brittle cleavage phenomena in line with experimental observations. We find that  the dimensions of the foam sample significantly affect the speed of the  cracks as well as the pressure necessary to sustain them: cracks in wider samples travel faster at a given driving stress, but are able to avoid arrest and maintain propagation at a lower pressure (the  velocity gap becomes smaller). The system thus becomes a study case for stress concentration and the transition between discrete and continuum systems in dynamical fracture; taking into account the finite dimensions of the system improves agreement with experiment.

Driven by its numerous applications in computational science, the approximation of smooth, high-dimensional functions via sparse polynomial expansions has received significant attention in the last five to ten years.  In the first part of this talk, I will give a brief survey of recent progress in this area.  In particular, I will demonstrate how the proper use of compressed sensing tools leads to new techniques for high-dimensional approximation which can mitigate the curse of dimensionality to a substantial extent.  The rest of the talk is devoted to approximating functions defined on irregular domains.  The vast majority of works on high-dimensional approximation assume the function in question is defined over a tensor-product domain.  Yet this assumption is often unrealistic.  I will introduce a method, known as polynomial frame approximation, suitable for broad classes of irregular domains and present theoretical guarantees for its approximation error, stability, and sample complexity.  These results show the suitability of this approach for high-dimensional approximation through the independence (or weak dependence) of the various guarantees on the ambient dimension d.  Time permitting, I will also discuss several extensions.

Stein ($1981$) proved the borderline Sobolev embedding result which states that for $n \geq 2,$ $u \in L^{1}(\mathbb{R}^{n})$ and $\nabla u \in L^{(n,1)}(\mathbb{R}^{n}; \mathbb{R}^{n})$ implies $u$ is continuous. Coupled with standard Calderon-Zygmund estimates for Lorentz spaces, this implies $u \in C^{1}(\mathbb{R}^{n})$ if $\Delta u \in L^{(n,1)}(\mathbb{R}^{n}).$ The search for a nonlinear generalization of this result culminated in the work of Kuusi-Mingione ($2014$), which proves the same result for $p$-Laplacian type systems. \paragraph{} In this talk, we shall discuss how these results can be extended to differential forms. In particular, we can prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ $k$-differential form with $\delta \left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k \leq n-1, $ $1 < p < \infty, $ with uniformly positive, bounded, Dini continuous scalar function $a$, then $du$ is continuous.

I will give a self-contained introduction to the theory of cross ratios on boundaries of Gromov hyperbolic and CAT(-1) spaces, focussing on the connections to the following two questions. When are two spaces with the 'same' Gromov boundary isometric/quasi-isometric? Are closed Riemannian manifolds completely determined (up to isometry) by the lengths of their closed geodesics?

Young diagrams frequently appear in the study of partitions and representations of the symmetric group. By filling these diagrams with numbers, we obtain Young tableau, combinatorial objects onto which we can define the structure of a monoid via insertion algorithms. We will explore this structure and it's connection to a basis of the ring of symmetric polynomials. If we have time, we will mention alternative monoid structures and their corresponding bases.