Past

A G\"odel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the formula is obtained. Some further examples of self-referential sentences are considered, such as sentences that \anf{say of themselves} that they are $\Sigma^0_n$-true (or $\Pi^0_n$-true), and their formal properties are investigated.

In this seminar I will discuss a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.

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.

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.

Using the fact that certain exotic spheres do not admit Lagrangian embeddings into $T^*{\mathcal S}^{n+1}$, as proven by Abouzaid and Ekholm-Smith, we produce non-trivial homotopy classes of the group of compactly supported symplectomorphisms of $T^*{\mathcal S}^n$. In particular, we show that the Hamiltonian isotopy class of the symplectic Dehn twist depends on the parametrisation used in the construction.  Related results are also obtained for $T^*({\mathcal S}^n \times {\mathcal S}^1)$.

Joint work with Jonny Evans.

We present advances in several fundamental fields of computer vision: image segmentation, object tracking, stereo reconstruction for depth map estimation and full 3D multi-view reconstruction. The basic method applied to these fields is convex relaxation. Convex relaxation methods allow for global optimization of numerous energy functionals and provide a step towards less user input and more automation. We will show how the respective computer vision problems can be formulated in this convex optimization framework. Efficient parallel implementations of the arising numerical schemes using graphics processing units allow for interactive applications.

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. (Such a graph is said to be $r$-locally-$F$.) This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional integer lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They can be viewed as quotient lattices in certain 'flat orbifolds'.) Our results are best possible in the sense that '3' cannot be replaced with '2'. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory. We will also discuss some results and open problems on the properties of a random n-vertex graph which is $r$-locally-$F$. This is all joint work with Itai Benjamini (Weizmann Institute of Science). 

We investigate an X-ray imaging system that fires multiple point sources of radiation simultaneously from close proximity to a probe. Radiation traverses the probe in a non-parallel fashion, which makes it necessary to use tomosynthesis as a preliminary step to calculating a 2D shadowgraph. The system geometry requires imaging techniques that differ substantially from planar X-rays or CT tomography. We present a proof of concept of such an imaging system, along with relevant artefact removal techniques.  This work is joint with Kishan Patel.

      I will show uniqueness result for BV solutions of scalar conservation laws with discontinuous flux in several space dimensions. The proof is based on the notion of kinetic solution and on a careful analysis of the entropy dissipation along the discontinuities of the flux.
 

The idea of A-free group, where A is a discrete ordered abelian group, has been introduced by Myasnikov, Remeslennikov and Serbin. It generalises the construction of free groups. A proof will be outlined that a group is A-free for some A if and only if it acts freely and without inversions on a \lambda-tree, where \lambda is an arbitrary ordered abelian group.

Vector bundles over a compact manifold can be defined via transition 
functions to a linear group. Often one imposes 
conditions on this structure group. For example for real vector bundles on 
may  ask that all 
transition functions lie in the special orthogonal group to encode 
orientability. Commutative K-theory arises when we impose the condition 
that the transition functions commute with each other whenever they are 
simultaneously defined.

We will introduce commutative K-theory and some natural variants of it, 
and will show that they give rise to  new generalised 
cohomology theories.

This is joint work with Adem, Gomez and Lind building on previous work by 
Adem, F. Cohen, and Gomez.

The Taylor expansion of a controlled differential equation suggests that the solution at time 1 depends on the driving path only through the latter's iterated integrals up to time 1, if the vector field is infinitely differentiable. Hambly and Lyons proved that this remains true for Lipschitz vector fields if the driving path has bounded total variation. We extend the Hambly-Lyons result for weakly geometric rough paths in finite dimension. Joint work with X. Geng, T. Lyons and D. Yang.