Past

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.

In this talk, we will discuss the dimension of $J_0(N)[m]$ at an Eisenstein prime m for
square-free level N. We will also study the structure of $J_0(N)[m]$ as a Galois module.
This work generalizes Mazur’s work on Eisenstein ideals of prime level to the case of
arbitrary square-free level up to small exceptional cases.

Acto-myosin network growth and remodeling in vivo is based on a large number of chemical and mechanical processes, which are mutually coupled and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed detailed physico-chemical, stochastic models of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work sheds light on the interplay between the chemical and mechanical processes, and also will highlights the importance of diffusional and active transport phenomena. For example, we showed that molecular transport plays an important role in determining the shapes of the commonly observed force-velocity curves. We also investigated the nonlinear mechano-chemical couplings between an acto-myosin network and an external deformable substrate.

From a theoretical point of view, theta is a relatively simple quantity: the rate of change in value of a financial derivative with respect to time. In a Black-Scholes world, the theta of a delta hedged option can be viewed as `rent’ paid in exchange for gamma. This relationship is fundamental to the risk-management of a derivatives portfolio. However, in the real world, the situation becomes significantly more complicated. In practice the model is continually being recalibrated, and whereas in the Black-Scholes world volatility is not a risk factor, in the real world it is stochastic and carries an associated risk premium. With the heightened interest in automation and electronic trading, we increasingly need to attempt to capture trading, marking and risk management practice algorithmically, and this requires careful consideration of the relationship between the risk neutral and historical measures. In particular these effects need to be incorporated in order to make sense of theta and the time evolution of a derivatives portfolio in the historical measure. 

Gradient-based optimisation techniques have become a common tool in the analysis of fluid systems. They have been applied to replace and extend large-scale matrix decompositions to compute optimal amplification and optimal frequency responses in unstable and stable flows. We will show how to efficiently extract linearised and adjoint information directly from nonlinear simulation codes and how to use this information for determining common flow characteristics. We also extend this framework to deal with the optimisation of less common norms. Examples from aero-acoustics and mixing will be presented.