We also state some connections to some open problems.
Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. We give a brief history of point vortices. We then obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.
The geometric Langlands correspondence relates rank n integrable connections
on a complex Riemann surface $X$ to regular holonomic D-modules on
$Bun_n(X)$, the moduli stack of rank n vector bundles on $X$. If we replace
$X$ by a smooth irreducible curve over a finite field of characteristic p
then there is a version of the geometric Langlands correspondence involving
$l$-adic perverse sheaves for $l\neq p$. In this lecture we consider the
case $l=p$, proposing a $p$-adic version of the geometric Langlands
correspondence relating convergent $F$-isocrystals on $X$ to arithmetic
$D$-modules on $Bun_n(X)$.
The geometric Langlands correspondence relates rank n integrable connections on a complex Riemann surface $X$ to regular holonomic D-modules on $Bun_n(X)$, the moduli stack of rank n vector bundles on $X$. If we replace $X$ by a smooth irreducible curve over a finite field of characteristic p then there is a version of the geometric Langlands correspondence involving $l$-adic perverse sheaves for $l\neq p$. In this lecture we consider the case $l=p$, proposing a $p$-adic version of the geometric Langlands correspondence relating convergent $F$-isocrystals on $X$ to arithmetic $D$-modules on $Bun_n(X)$.
We present recent numerical techniques for the treatment of integral formulations of Helmholtz boundary value problems in the case of high frequencies. The combination of $H^2$-matrices with further developments of the adaptive cross approximation allows to solve such problems with logarithmic-linear complexity independent of the frequency. An advantage of this new approach over existing techniques such as fast multipole methods is its stability over the whole range of frequencies, whereas other methods are efficient either for low or high frequencies.
We find and describe four futures markets where the bid-ask spread is bid down to the fixed price tick size practically all the time, and which match coun- terparties using a pro-rata rule. These four markets’ offered depths at the quotes on average exceed mean market order size by two orders of magnitude, and their order cancellation rates (the probability of any given offered lot being cancelled) are significantly over 96 per cent. We develop a simple theoretical model to explain these facts, where strategic complementarities in the choice of limit order size cause traders to risk overtrading by submitting over-sized limit orders, most of which they expect to cancel.
Joint work with Jonathan Field.
We shall dedicate the first half of the talk to introduce
classical Higgs bundles and describe the fibres of the corresponding
Hitchin fibration in terms of spectral data. Then, we shall define
principal Higgs bundles and look at some examples. Finally, we
consider the particular case of $SL(2,R)$, $U(p,p)$ and $Sp(2p,2p)$ Higgs
bundles and study their spectral data. Time permitting, we shall look
at different applications of our new methods.
In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.
Graphs and digraphs behave quite differently, and many classical results for graphs are often trivially false when extended to general digraphs. Therefore it is usually necessary to restrict to a smaller family of digraphs to obtain meaningful results. One such very natural family is Eulerian digraphs, in which the in-degree equals out-degree at every vertex.
In this talk, we discuss several natural parameters for Eulerian digraphs and study their connections. In particular, we show that for any Eulerian digraph G with n vertices and m arcs, the minimum feedback arc set (the smallest set of arcs whose removal makes G acyclic) has size at least $m^2/2n^2+m/2n$, and this bound is tight. Using this result, we show how to find subgraphs of high minimum degrees, and also long cycles in Eulerian digraphs. These results were motivated by a conjecture of Bollob\'as and Scott.
Joint work with Ma, Shapira, Sudakov and Yuster
Stability plays an important role in engineering, for it limits the load carrying capacity of all kinds of structures. Many failure mechanisms in advanced engineering materials are stability-related, such as localized deformation zones occurring in fiber-reinforced composites and cellular materials, used in aerospace and packaging applications. Moreover, modern biomedical applications, such as vascular stents, orthodontic wire etc., are based on shape memory alloys (SMA’s) that exploit the displacive phase transformations in these solids, which are macroscopic manifestations of lattice-level instabilities.
The presentation starts with the introduction of the concepts of stability and bifurcation for conservative elastic systems with a particular emphasis on solids with periodic microstructures. The concept of Bloch wave analysis is introduced, which allows one to find the lowest load instability mode of an infinite, perfect structure, based solely on unit cell considerations. The relation between instability at the microscopic level and macroscopic properties of the solid is studied for several types of applications involving different scales: composites (fiber-reinforced), cellular solids (hexagonal honeycomb) and finally SMA's, where temperature- or stress-induced instabilities at the atomic level have macroscopic manifestations visible to the naked eye.
This talk will attempt to say something about the p-adic zeta function, a p-adic analytic object which encodes information about Galois cohomology of Tate twists in its special values. We first explain the construction of the p-adic zeta function, via p-adic Fourier theory. Then, after saying something about Coleman integration, we will explain the interpretation of special values of the p-adic zeta function as limiting values of p-adic polylogarithms, in analogy with the Archimedean case. Finally, we will explore the consequences for the de Rham and etale fundamental groupoids of the projective line minus three points.
In this talk, I will focus on an infinite class of 3d N=4 gauge theories
which can be constructed from a certain set of ordered pairs of integer
partitions. These theories can be elegantly realised on brane intervals in
string theory. I will give an elementary review on such brane constructions
and introduce to the audience a symmetry, known as mirror symmetry, which
exchanges two different phases (namely the Higgs and Coulomb phases) of such
theories. Using mirror symmetry as a tool, I will discuss a certain
geometrical aspect of the vacuum moduli spaces of such theories in the
Coulomb phase. It turns out that there are certain infinite subclasses of
such spaces which are special and rather simple to study; they are complete intersections. I will mention some details and many interesting features of these spaces.
Consider a fully-connected social network of people, companies,
or countries, modeled as an undirected complete graph with real numbers on
its edges. Positive edges link friends; negative edges link enemies.
I'll discuss two simple models of how the edge weights of such networks
might evolve over time, as they seek a balanced state in which "the enemy of
my enemy is my friend." The mathematical techniques involve elementary
ideas from linear algebra, random graphs, statistical physics, and
differential equations. Some motivating examples from international
relations and social psychology will also be discussed. This is joint work
with Seth Marvel, Jon Kleinberg, and Bobby Kleinberg.
Data assimilation in highly nonlinear and high dimensional systems is a hard
problem. We do have efficient data-assimilation methods for high-dimensional
weakly nonlinear systems, exploited in e.g. numerical weather forecasting.
And we have good methods for low-dimensional (
In this talk, we elaborate on the notions of no-free-lunch that have proved essential in the theory of financial mathematics---most notably, arbitrage of the first kind. Focus will be given in most recent developments. The precise connections with existence of deflators, numeraires and pricing measures are explained, as well as the consequences that these notions have in the existence of bubbles and the valuation of illiquid assets in the market.