Past

Historically, the study of positively curved manifolds has always been challenging. There are many reasons for this, but among them is the fact that the existence of a metric of positive curvature on a manifold imposes strong topological restrictions. In this talk we will discuss some of these topological implications and we will introduce the main results in this area. We will also present some recent results that relate positive curvature to the smooth structure of the manifold.

Let $G$ be a connected reductive group and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $Bun_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called $coeff_{G}$, from the DG category of D-modules on $Bun_G$ to a certain DG category $Wh(G, ext)$, called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor  $$\mathbb{L}_G : \operatorname{IndCoh}_{N}(LocSys_{\check{G}} ) \to D(Bun_G)$$ with the Whittaker model. For $G = GL_n$ and $G = PGL_n$, we prove that $coeff_G$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.

In numerical analysis the design and analysis of computational methods is often based on, and closely linked to, a well-posedness result for the underlying continuous problem. In particular the continuous dependence of the continuous model is inherited by the computational method when such an approach is used. In this talk our aim is to design a stabilised finite element method that can exploit continuous dependence of the underlying physical problem without making use of a standard well-posedness result such as Lax-Milgram's Lemma or The Babuska-Brezzi theorem. This is of particular interest for inverse problems or data assimilation problems which may not enter the framework of the above mentioned well-posedness results, but can nevertheless satisfy some weak continuous dependence properties. First we will discuss non-coercive elliptic and hyperbolic equations where the discrete problem can be ill-posed even for well posed continuous problems and then we will discuss the linear elliptic Cauchy problem as an example of an ill-posed problem where there are continuous dependence results available that are suitable for the framework that we propose.

After motivating why we would like to find examples of simple totally disconnected locally compact groups, I will describe a construction due to Banks, Elder and Willis which yields infinitely many such examples when given certain groups acting on a tree.

Given any family of varieties over a number field, if we have that the existence of local points everywhere is equivalent to the existence of a global point (for each member of the family), then we say that the family satisfies the Hasse principle. Of more interest, in this talk, is the case when the Hasse principle fails: we will give an overview of the "geography" of the currently known obstructions.

I will explain an approach to Hamiltonian reduction using derived
symplectic geometry. Roughly speaking, the reduced space can be
presented as an intersection of two Lagrangians in a shifted symplectic
space, which therefore carries a natural symplectic structure. A slight
modification of the construction gives rise to quasi-Hamiltonian
reduction. This talk will also serve as an introduction to the wonderful
world of derived symplectic geometry where statements that morally ought
to be true are indeed true.

This brief lecture will highlight several best-practice observations and
research for writing software for mathematical research, drawn from a
number of sources, including; Best Practices for Scientific Computing
[BestPractices], Code Complete [CodeComplete], and personal observation
from the presenter.  Specific focus will be given to providing the
definition of important concepts, then describing how to apply them
successfully in day-to-day research settings.  Following the outline from
Best Practices, we will cover the following topics:

* Write Programs for People, Not Computers
* Let the Computer Do the Work
* Make Incremental Changes
* Don't Repeat Yourself (or Others)
* Plan for Mistakes
* Optimize Software Only after It Works Correctly
* Document Design and Purpose, Not Mechanics
* Collaborate

[BestPractices]
http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001745
[CodeComplete] http://www.cc2e.com/Default.aspx

 We describe a framework for defining and classifying TQFTs via
surgery. Given a functor 
from the category of smooth manifolds and diffeomorphisms to
finite-dimensional vector spaces, 
and maps induced by surgery along framed spheres, we give a set of axioms
that allows one to assemble functorial coboridsm maps. 
Using this, we can reprove the correspondence between (1+1)-dimensional
TQFTs and commutative Frobenius algebras, 
and classify (2+1)-dimensional TQFTs in terms of a new structure, namely
split graded involutive nearly Frobenius algebras 
endowed with a certain mapping class group representation. The latter has
not appeared in the literature even in conjectural form. 
This framework is also well-suited to defining natural cobordism maps in
Heegaard Floer homology.

I will discuss several recent developments regarding the construction of fluxes for F-theory on Calabi-Yau fourfolds. Of particular importance to the effective physics is the structure of the middle (co)homology groups, on which new results are presented. Fluxes dynamically drive the fourfold to Noether-Lefschetz loci in moduli space. While the structure of such loci is generally unknown for Calabi-Yau fourfolds, this problem can be answered in terms of arithmetic for K3 x K3 and a classification is possible.

The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimals sheaves, which might be useful on it own. 

The Calabi conjecture, posed in 1954 and proved by Yau in 1976, guaranties the existence of Ricci-flat Kahler metrics on compact Kahler manifolds with vanishing first Chern class, providing examples of the so called Calabi-Yau manifolds. The latter are of great importance to the fields of Riemannian Holonomy Groups, having Hol0 as a subgroup of SU; Calibrated Geometry, more precisely Special Lagrangian Geometry; and to String theory with the discovery of the phenomenon of Mirror Symmetry (to mention a few!). In the talk, we will discuss the necessary background to formulate the Calabi conjecture and explain some of the main ideas behind its proof by Yau, which itself is a jewel from the point of view of non-linear PDEs.