We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.
In this talk we describe some recent work on shock
reflection problems for the potential flow equation. We will
start with discussion of shock reflection phenomena. Then we
will describe the results on existence, structure and
regularity of global solutions to regular shock reflection. The
approach is to reduce the shock reflection problem to a free
boundary problem for a nonlinear elliptic equation, with
ellipticity degenerate near a part of the boundary (the sonic
arc). We will discuss techniques to handle such free boundary
problems and degenerate elliptic equations. This talk is based
on joint works with Gui-Qiang Chen, and with Myoungjean Ba
Whether as the sudoku puzzles of popular culture or as
restricted coloring problems on graphs or hypergraphs, completing partial
Latin squares and cubes present a framework for a variety of intriguing
problems. In this talk we will present several recent results on
completing partial Latin squares and cubes.
In addition to existence, the excess-demand approach allows us to establish uniqueness and provide efficient computational algorithms for various complete- and incomplete-market stochastic financial equilibria.
A particular attention will be paid to the case when the agents exhibit constant absolute risk aversion. An overview of recent results (including those jointly obtained with M. Anthropelos and with Y. Zhao) will be given.
This will discuss the paper of Ricci, Tseytlin & Wolf from 2007.
'Compressive sampling' is a topic of current interest. It relies on data being sparse in some domain, which allows what is apparently 'sub Nyquist' sampling so that the quantities of data which must be handled become more closely related to the information rate. This principal would appear to have (at least) three applications for radar and electronic warfare: \\
The most modest application is to reduce the amount of data which we must handle: radar and electronic warfare receivers generate vast amounts of data (up to 1Gbit/second or even 10Gbit.sec). It is desirable to be able to store this data for future analysis and it is also becoming increasingly important to be able to share it between different sensors, which, prima facie, requires vast communication bandwidths and it would be valuable to be able to find ways to handle this more efficiently. \\
The second advantage is that if suitable data domains can be identified, it may also be possible to pre-process the data before the analogue to digital converters in the receivers, to reduce the demands on these critical components. \\
The most ambitious use of compressive sensing would be to find ways of modifying the radar waveforms, and the electronic warfare receiver sampling strategies, to change the domain in which the information is represented to reduce the data rates at the receiver 'front ends', i.e. make the data at the front end better match the information we really want to acquire.\\
The aim of the presentation will be to describe the issues with which we are faced, and to discuss how compressive sampling might be able to help. A particular issue which will be raised is how we might find domains in which the data is sparse.
Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.
Bloch Floquet waves are considered in structured media. Such waves are dispersive and the dispersion diagrams contain stop bands. For an example of a harmonic lattice, we discuss dynamic band gap Green’s functions characterised by exponential localisation. This is followed by simple models of exponentially localised defect modes. Asymptotic models involving uniform asymptotic approximations of physical fields in structured media are compared with homogenisation approximations.
I will discuss what is known about the cohomology of several moduli spaces coming from algebraic and differential geometry. These are: moduli spaces of non-singular curves (= Riemann surfaces) $M_g$, moduli spaces of nodal curves $\overline{M}_g$, moduli spaces of holomorphic line bundles on curves $Hol_g^k \to M_g$, and the universal Picard varieties $Pic^k_g \to M_g$. I will construct characteristic classes on these spaces, talk about their homological stability, and try to explain why the constructed classes are the only stable ones. If there is time I will also talk about the Picard groups of these moduli spaces.
Much of this work is due to other people, but some is joint with J. Ebert.
\ \ The cluster category of Dynkin type $A_\infty$ is a ubiquitous object with interesting properties, some of which will be explained in this talk.
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\ \ Let us denote the category by $\mathcal{D}$. Then $\mathcal{D}$ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $D^c(C^{∗}(S^2;k))$ of the singular cochain algebra $C^*(S^2;k)$ of the 2-sphere $S^{2}$. There is also a “universal” definition: $\mathcal{D}$ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.
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\ \ Just like cluster categories of finite quivers, $\mathcal{D}$ has many cluster tilting subcategories, with the crucial difference that in $\mathcal{D}$, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects.
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\ \ The talk will show how the cluster tilting subcategories have a rich combinatorial
structure: They can be parametrised by “triangulations of the $\infty$-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.
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\ \ This will be used to show how to obtain a subcategory of $\mathcal{D}$ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $\mathcal{D}$ generalises the situation from Dynkin type $A_n$ , and how triangulations of the $\infty$-gon are new and interesting combinatorial objects.
A categorification of cycle class maps consists to define
realization functors from constructible motivic sheaves to other
categories of coefficients (e.g. constructible $l$-adic sheaves), which are compatible with the six operations. Given a field $k$, we
will describe a systematic construction, which associates,
to any cohomology theory $E$, represented in $DM(k)$, a
triangulated category of constructible $E$-modules $D(X,E)$, for $X$
of finite type over $k$, endowed with a realization functor from
the triangulated category of constructible motivic sheaves over $X$.
In the case $E$ is either algebraic de Rham cohomology (with $char(k)=0$), or $E$ is $l$-adic cohomology, one recovers in this way the triangulated categories of $D$-modules or of $l$-adic sheaves. In the case $E$ is rigid cohomology (with $char(k)=p>0$), this construction provides a nice system of $p$-adic coefficients which is closed under the six operations.
Suppose that $A \subset \mathbb F_2^n$ has density $\Omega(1)$. How
large a subspace is $A+A:=\{a+a’:a,a’ \in A\}$ guaranteed to contain? We
discuss this problem and how the the result changes as the density
approaches $1/2$.
Starting from Morel and Voevodsky's stable homotopy theory of schemes, one defines, for each noetherian scheme of finite dimension $X$, the triangulated category $DM(X)$ of motives over $X$ (with rational coefficients). These categories satisfy all the the expected functorialities (Grothendieck's six operations), from
which one deduces that $DM$ also satisfies cohomological proper
descent. Together with Gabber's weak local uniformisation theorem,
this allows to prove other expected properties (e.g. finiteness
theorems, duality theorems), at least for motivic sheaves over
excellent schemes.