Simple groups of Lie type have a purely group theoretic characterization in terms of subgroup configurations. We here show that for certain Fraisse limits, the automorphism group is a simple group.
{\bf Keble Workshop on Partial Differential Equations
in Science and Engineering}
\\
\\Place: Roy Griffiths Room in the ARCO Building, Keble College
\\Time: 1:00pm-5:10pm, Thursday, June 10.
\\
\\
Program:\\
\\ 1:00-1:20pm: Coffee and Tea
\\
\\ 1:20-2:10pm: Prof. Walter Craig (Joint with OxPDE Lunchtime Seminar)
\\
\\ 2:20-2:40pm Prof. Mikhail Feldman
\\
\\ 2:50-3:10pm Prof. Paul Taylor
\\
\\ 3:20-3:40pm Coffee and Biscuits
\\
\\ 3:40-4:00pm: Prof. Sir John Ball
\\
\\ 4:10-4:30pm: Dr. Apala Majumdar
\\
\\ 4:40-5:00pm: Prof. Robert Pego
\\
\\ 5:10-6:00pm: Free Discussion
\\
\\{\bf Titles and Abstracts:}
\\
1.{\bf Title: On the singular set of the Navier-Stokes equations
\\ Speaker: Prof. Walter Craig, McMaster University, Canada}
\\ Abstract:\\
The Navier-Stokes equations are important in
fluid dynamics, and a famous mathematics problem is the
question as to whether solutions can form singularities.
I will describe these equations and this problem, present
three inequalities that have some implications as to the
question of singularity formation, and finally, give a
new result which is a lower bound on the size of the
singular set, if indeed singularities exist.
\\
\\{\bf 2. Title: Shock Analysis and Nonlinear Partial Differential Equations of Mixed Type.
\\ Speaker: Prof. Mikhail Feldman, University of Wisconsin-Madison, USA}
\\
\\ Abstract:\\ Shocks in gas or compressible fluid arise in various physical
situations, and often exhibit complex structures. One example is reflection
of shock by a wedge. The complexity of reflection-diffraction configurations
was first described by Ernst Mach in 1878. In later works, experimental and
computational studies and asymptotic analysis have shown that various patterns
of reflected shocks may occur, including regular and Mach reflection. However,
many fundamental issues related to shock reflection are not understood,
including transition between different reflection patterns. For this reason
it is important to establish mathematical theory of shock reflection,
in particular existence and stability of regular reflection solutions for PDEs
of gas dynamics. Some results in this direction were obtained recently.
\\
In this talk we start by discussing examples of shocks in supersonic and
transonic flows of gas. Then we introduce the basic equations of gas dynamics:
steady and self-similar compressible Euler system and potential flow equation.
These equations are of mixed elliptic-hyperbolic type. Subsonic and supersonic
regions in the flow correspond to elliptic and hyperbolic regions of solutions.
Shocks correspond to certain discontinuities in the solutions. We discuss some
results on existence and stability of steady and self-similar shock solutions,
in particular the recent work (joint with G.-Q. Chen) on global existence of
regular reflection solutions for potential flow. We also discuss open problems
in the area.
\\
\\{\bf 3. Title: Shallow water waves - a rich source of interesting solitary wave
solutions to PDEs
\\ Speaker: Prof. Paul H. Taylor, Keble College and Department of Engineering Science, Oxford}
\\
\\Abstract:\\ In shallow water, solitary waves are ubiquitous: even the wave crests
in a train of regular waves can be modelled as a succession of solitary waves.
When successive crests are of different size, they interact when the large ones
catch up with the smaller. Then what happens? John Scott Russell knew by experiment
in 1844, but answering this question mathematically took 120 years!
This talk will examine solitary wave interactions in a range of PDEs, starting
with the earliest from Korteweg and De Vries, then moving onto Peregrine's
regularized long wave equation and finally the recently introduced Camassa-Holm
equation, where solitary waves can be cartoon-like with sharp corners at the crests.
For each case the interactions can be described using the conserved quantities,
in two cases remarkably accurately and in the third exactly, without actually
solving any of the PDEs.
The methodology can be extended to other equations such as the various versions
of the Boussinesq equations popular with coastal engineers, and perhaps even
the full Euler equations.
\\
{\bf 4. Title: Austenite-Martensite interfaces
\\ Speaker: Prof. Sir John Ball, Queen's College and Mathematical Institute, Oxford}
\\
\\Abstract:\\ Many alloys undergo martensitic phase transformations
in which the underlying crystal lattice undergoes a change of shape
at a critical temperature. Usually the high temperature phase (austenite)
has higher symmetry than the low temperature phase (martensite).
In order to nucleate the martensite it has to somehow fit geometrically
to the austenite. The talk will describe different ways in which this
occurs and how they may be studied using nonlinear elasticity and
Young measures.
\\
\\{\bf 5. Title: Partial Differential Equations in Liquid Crystal Science and
Industrial Applications
\\ Speaker: Dr. Apala Majumdar, Keble College and Mathematical Institute, Oxford}
\\
\\Abstract:\\
Recent years have seen a growing demand for liquid crystals in modern
science, industry and nanotechnology. Liquid crystals are mesophases or
intermediate phases of matter between the solid and liquid phases of
matter, with very interesting physical and optical properties.
We briefly review the main mathematical theories for liquid crystals and
discuss their analogies with mathematical theories for other soft-matter
phases such as the Ginzburg-Landau theory for superconductors. The
governing equations for the static and dynamic behaviour are typically
given by systems of coupled elliptic and parabolic partial differential
equations. We then use this mathematical framework to model liquid crystal
devices and demonstrate how mathematical modelling can be used to make
qualitative and quantitative predictions for practical applications in
industry.
\\
\\{\bf 6. Title: Bubble bath, shock waves, and random walks --- Mathematical
models of clustering
\\Speaker: Prof. Robert Pego, Carnegie Mellon University, USA}
\\Abstract:\\ Mathematics is often about abstracting complicated phenomena into
simple models. This talk is about equations that model aggregation
or clustering phenomena --- think of how aerosols form soot particles
in the atmosphere, or how interplanetary dust forms comets, planets
and stars. Often in such complex systems one observes universal trend
toward self-similar growth. I'll describe an explanation for this
phenomenon in two simple models describing: (a) ``one-dimensional
bubble bath,'' and (b) the clustering of random shock waves.
A behavioral mean-variance portfolio selection problem in continuous time is formulated and studied. Based on the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by a probability distortion function. Then the problem is no longer a convex optimization problem. This feature distinguishes it from the conventional linear-quadratic (LQ) problems.
The stochastic optimal LQ control theory no longer applies. We take the quantile function of the terminal cash flow as the decision variable.
The corresponding optimal terminal cash flow can be recovered by the optimal quantile function. Then the efficient strategy is the hedging strategy of the optimal terminal cash flow.
On any Hermitian manifold there is a unique Hermitian connection, called the Bismut connection, which has torsion a three-form. One says that the triplet consisting of the Hermitian structure together with the Bismut connection specifies a Kähler-with-torsion structure, or briefly a KT structure. If the torsion three-form is closed, we have a strong KT structure. The first part of this talk will discuss these notions and also address the problem of classifying strong KT structures.
\paragraph{} Despite their name, KT manifolds are generally not Kähler. In particular the fundamental two-form is not closed. If the KT structure is strong, we have instead a closed three-form. Motivated by the usefulness of moment maps in geometries involving symplectic forms, one may ask whether it is possible to construct a similar type of map, when we replace the symplectic form by a closed three-form. The second part of the talk will explain the construction of such maps, which are called multi-moment maps.
Clustering phenomena occur in numerous areas of science. This series of lectures will discuss:
(i) basic kinetic models for clustering- Smoluchowski's coagulation equation, random shock clustering, ballistic aggregation, domain-wall merging;
(ii) Criteria for approach to self-similarity- role of regular variation;
(iii) The scaling attractor and its measure representation. A particular theme is the use of methods and insights from probability in tandem with dynamical systems theory. In particular there is a
close analogy of scaling dynamics with the stable laws of probability and infinite divisibility.
Homology counts components and cycles, K-theory counts vector bundles and bundles of Clifford algebra modules. What about geometric models for other generalized cohomology theories? There is a vision, introduced by Segal, Stolz, and Teichner, that certain cohomology theories should be expressible in terms of topological field theories.
I will describe how the 0-th K-theory group can be formulated in terms of equivalence classes of 1-dimensional topological field theories. Then I will discuss what it means to twist a topological field theory, and explain that the n-th K-theory group comes from twisted 1-dimensional topological field theories.
The expectation is that 2-dimensional topological field theories should be analogously related to elliptic cohomology. I will take an extended digression to explain what elliptic cohomology is and why it is interesting. Then I will discuss 2-dimensional twisted field theory and explain how it leads us toward a notion of higher
("2-dimensional") algebra.
Based on joint work with Arthur Bartels and Andre Henriques
We will introduce both the classical Hanna Neumann Conjecture and its strengthened version, discuss Stallings' reformulation in terms of immersions of graphs, and look at some partial results. If time allows we shall also look at the new approach of Joel Friedmann.
We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.
Ordinary homology is a geometrically defined invariant of spaces: the 0-th homology group counts the number of components; the n-th homology group counts n-cycles, which correspond to an intuitive notion of 'n-dimensional holes' in a space. K-theory, or more specifically the 0-th K-theory group, is defined in terms of vector bundles, and so also has an immediate relationship to geometry. By contrast, the n-th K-theory group is typically defined homotopy-theoretically using the black box of Bott periodicity.
I will describe a more geometric perspective on K-theory, using Z/2-graded vector bundles and bundles of modules for Clifford algebras. Along the way I will explain Clifford algebras, 2-categories, and Morita equivalence, explicitly check the purely algebraic 8-fold periodicity of the Clifford algebras, and discuss how and why this periodicity implies Bott periodicity.
The talk will not presume any prior knowledge of K-theory, Clifford algebras, Bott periodicity, or the like.
Based on joint work with Arthur Bartels and Andre Henriques
Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.
Consider a configuration of points in $d$-dimensional Euclidean space
together with a set of constraints
which fix the direction or the distance between some pairs of points.
Basic questions are whether the constraints imply that the configuration
is unique or locally unique up to congruence, and whether it is bounded. I
will describe some solutions
and partial solutions to these questions.
Consider the space M of parabolas y=ax^2+bx+c, with (a, b, c) as coordinates on M. Two parabolas generically intersect at two (possibly complex) points, and we can define a conformal structure on M by declaring two points to be null separated iff the corresponding parabolas are tangent. A simple calculation of discriminant shows that this conformal structure is flat.
In this talk (based on joint works with Godlinski and Sokolov) I shall show how replacing parabolas by rational plane curves of higher degree allows constructing curved conformal structures in any odd dimension. In dimension seven one can use this "twistor" construction to find G_2 structures in a conformal class.
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.
\\
\ \ 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.
\\
\ \ 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.
\\
\ \ 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.
\\
\ \ 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.
Colorectal cancer (CRC) is one of the leading causes of cancer-related death worldwide, demanding a response from scientists and clinicians to understand its aetiology and develop effective treatment. CRC is thought to originate via genetic alterations that cause disruption to the cellular dynamics of the crypts of Lieberkűhn, test-tube shaped glands located in both the small and large intestine, which are lined with a monolayer of epithelial cells. It is believed that during colorectal carcinogenesis, dysplastic crypts accumulate mutations that destabilise cell-cell contacts, resulting in crypt buckling and fission. Once weakened, the corrupted structure allows mutated cells to migrate to neighbouring crypts, to break through to the underlying tissue and so aid the growth and malignancy of a tumour. To provide further insight into the tissue-level effects of these genetic mutations, a multi-scale model of the crypt with a realistic, deformable geometry is required. This talk concerns the progress and development of such a model, and its usefulness as a predictive tool to further the understanding of interactions across spatial scales within the context of colorectal cancer.
We have seen that L-functions of elliptic curves of conductor N coincide exactly with L-functions of weight 2 newforms of level N from the Modularity Theorem. We will show how, using modular symbols, we can explicitly compute bases of newforms of a given level, and thus investigate L-functions of an elliptic curve of given conductor. In particular, such calculations allow us to numerically test the Birch-Swinnerton-Dyer conjecture.
This will be a review of recent work that obtains amplitudes at strong coupling from certain minimal surfaces in AdS.