Past

Computer simulation of the propagation and interaction of linear waves
is a core task in computational science and engineering.
The finite element method represents one of the most common
discretisation techniques for Helmholtz and Maxwell's equations, which
model time-harmonic acoustic and electromagnetic wave scattering.
At medium and high frequencies, resolution requirements and the
so-called pollution effect entail an excessive computational effort
and prevent standard finite element schemes from an effective use.
The wave-based Trefftz methods offer a possible way to deal with this
problem: trial and test functions are special solutions of the
underlying PDE inside each element, thus the information about the
frequency is directly incorporated in the discrete spaces.

This talk is concerned with a family of those methods: the so-called
Trefftz-discontinuous Galerkin (TDG) methods, which include the
well-known ultraweak variational formulation (UWVF).
We derive a general formulation of the TDG method for Helmholtz
impedance boundary value problems and we discuss its well-posedness
and quasi-optimality.
A complete theory for the (a priori) h- and p-convergence for plane
and circular/spherical wave finite element spaces has been developed,
relying on new best approximation estimates for the considered
discrete spaces.
In the two-dimensional case, on meshes with very general element
shapes geometrically graded towards domain corners, we prove
exponential convergence of the discrete solution in terms of number of
unknowns.

This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH Zurich,
Switzerland) and Ilaria Perugia (Vienna, Austria).

As the title says, in this talk I will be giving a casual introduction to higher categories. I will begin by introducing strict n-categories and look closely at the resulting structure for n=2. After discussing why this turns out to be an unsatisfying definition I will discuss in what ways it can be weakened. Broadly there are two main classes of models for weak n-categories: algebraic and geometric. The differences between these two classes will be demonstrated by looking at bicategories on the algebraic side and quasicategories on the geometric.

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein 
measure.

In this talk we aim to introduce the key ideas of homotopy type theory and show how it draws on and has applications to the areas of logic, higher category theory, and homotopy theory. We will discuss how types can be viewed both as propositions (statements about mathematics) as well as spaces (mathematical objects themselves). In particular we will define identity types and explore their groupoid-like structure; we will also discuss the notion of equivalence of types, introduce the Univalence Axiom, and consider some of its implications. Time permitting, we will discuss inductive types and show how they can be used to define types corresponding to specific topological spaces (e.g. spheres or more generally CW complexes).\\

This talk will assume no prior knowledge of type theory; however, some very basic background in category theory (e.g. the definition of a category) and homotopy theory (e.g. the definition of a homotopy) will be assumed.

Let $A$ and $B$ be boundaries of CAT(0) spaces. A function $f:A \to B$ is called a {\em boundary isomorphism} if $f$ is a homeomorphism in the visual topology and

$f$ is an isometry in the Tits metric. A compact metrizable space $Y$ is said to be {\em Tits rigid}, if for any two CAT(0) group boundaries $Z_1$ and $Z_2$ homeomorphic to $Y$, $Z_1$ is boundary isomorphic to $Z_2$.

We prove that the join of two Cantor sets and its suspension are Tits rigid.

Mirror Symmetry predicts a surprising relationship between the virtual numbers of degree-d rational curves in a target space X and variations of Hodge structure on a different space X’, called the mirror to X.  Concretely, it predicts that one can compute genus-zero Gromov–Witten invariants (which are the virtual numbers of rational curves) in terms of hypergeometric functions (which are the solutions to a differential equation that controls the variation of Hodge structure).  Existing proofs of this rely on beautiful but fearsomely complicated localization calculations in equivariant cohomology.  I will describe a new proof of the Mirror Theorem, for a broad range of target spaces X, which is much simpler and more conceptual. This is joint work with Cristina Manolache.

Partial differential equations defined on surfaces appear in various applications, for example image processing and reconstruction of non-planar images. In this talk, I will present a penalty method for evolution equations, based on an implicit representation of the surface. I will derive a simple equation in the surrounding space, formulated with an extension operator, and then show some analysis and applications of the method.

Several problems lead to the question of how well can a fine grid function be approximated by a coarse grid function, such as preconditioning in finite element methods or data compression and image processing. Particular challenges in answering this question arise when the functions may be only piecewise-continuous, or when the coarse space is not nested in the fine space. In this talk, we solve the problem by using a stable approximation from a space of globally smooth functions as an intermediate step, thereby allowing the use of known approximation results to establish the approximability by a coarse space. We outline the proof, which relies on techniques from the theory of discontinuous Galerkin methods and on the theory of Helmholtz decompositions. Finally, we present an application of our to nonoverlapping domain decomposition preconditioners for hp-version DGFEM.

Cox processes arise as a natural extension of inhomogeneous Poisson Processes, when the intensity function itself is taken to be stochastic. In multiple applications one is often concerned with characterizing the posterior distribution over the intensity process (given some observed data). Markov Chain Monte Carlo methods have historically been successful at such tasks. However, direct methods are doubly intractable, especially when the intensity process takes values in a space of continuous functions.

In this talk I'll be presenting a method to overcome this intractability that is based on the idea of "thinning" and that does not resort to approximations.

I will show how families of concentrating stationary vortices for the shallow

water equations can be constructed and studied asymptotically. The main tool

is the study of asymptotics of solutions to a family of semilinear elliptic

problems. The same method also yields results for axisymmetric vortices for

the Euler equation of incompressible fluids.

Abstract: The signature of a path characterizes the non-commutative evolvements along the path trajectory. Nevertheless, one can extract local commutativities from the signature, thus leading to an inversion scheme.

It is well known that several solutions to the Skorokhod problem

optimize certain ``cost''- or ``payoff''-functionals. We use the

theory of Monge-Kantorovich transport to study the corresponding

optimization problem. We formulate a dual problem and establish

duality based on the duality theory of optimal transport. Notably

the primal as well as the dual problem have a natural interpretation

in terms of model-independent no arbitrage theory.

In optimal transport the notion of c-monotonicity is used to

characterize the geometry of optimal transport plans. We derive a

similar optimality principle that provides a geometric

characterization of optimal stopping times. We then use this

principle to derive several known solutions to the Skorokhod

embedding problem and also new ones.

This is joint work with Mathias Beiglböck and Alex Cox.

This talk is based on a joint work with B. Rémy (Lyon) in which we study some subgroups of topological Kac–Moody groups and the implications of this study on the subgroup structure of the ambient Kac–Moody group.

In this talk we will discuss when one right-angled Artin group is a subgroup of another one and explain how this basic algebraic problem may provide answers to questions in geometric group theory and model theory such as classification of right-angled Artin groups up to quasi-isometries and universal equivalence.

Numerical models provide valuable tools for integrating understanding of riverine processes and morphology. Moreover, they have considerable potential for use in investigating river responses to environmental change and catchment management, and for aiding the interpretation of alluvial deposits and landforms. For this potential to be realised fully, such models must be capable of representing diverse river styles, and the spatial and temporal transitions between styles that can be driven by environmental forcing. However, while numerical modelling of rivers has advanced significantly over the past few decades, this has been accomplished largely by developing separate approaches to modelling different styles of river (e.g., meanders and braided networks). In addition, there has been considerable debate about what should constitute the ‘basic ingredients’ of river models, and the degree to which the environmental processes governing river evolution can be simplified in such models. This seminar aims to examine these unresolved issues, with particular reference to the simulation of large rivers and their floodplains.