I am going to introduce Thompson's groups F, T and V. They can be seen in two ways: as functions on [0,1] or as isomorphisms acting on trees.
Modern differential geometry is the art of the abstract that can be pictured. Continuum mechanics is the abstract description of concrete material phenomena. Their encounter, therefore, is as inevitable and as beautiful as the proverbial chance meeting of an umbrella and a sewing machine on a dissecting table. In this rather non-technical and lighthearted talk, some of the surprising connections between the two disciplines will be explored with a view at stimulating the interest of applied mathematicians.
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli
space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
How many triangles must a graph of density d contain? This old question due to Erdos was recently answered by Razborov, after many decades of progress by numerous authors.
We will consider the analogous question for tripartite graphs. Given a tripartite graph with prescribed edges densities between each
pair of classes how many triangles must it contain?
On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.
We investigate the behaviour of free-surface waves on time-varying potential flow in the limit as the Froude number becomes small. These waves are exponentially small in the Froude number, and are therefore inaccessible to ordinary asymptotic methods. As such, we demonstrate how exponential asymptotic techniques may be applied to the complexified free surface in order to extract information about the wave behaviour on the free surface, using a Lagrangian form of the potential flow equations. We consider the specific case of time-varying flow over a step, and demonstrate that the results are consistent with the steady state case.
Let $u \in W^{1,p}(\Omega,\R^N)$, $\Omega$ a bounded domain in
$\R^n$, be a minimizer of a convex variational integral or a weak solution to
an elliptic system in divergence form. In the vectorial case, various
counterexamples to full regularity have been constructed in dimensions $n
\geq 3$, and it is well known that only a partial regularity result can be
expected, in the sense that the solution (or its gradient) is locally
continuous outside of a negligible set. In this talk, we shall investigate
the role of the space dimension $n$ on regularity: In arbitrary dimensions,
the best known result is partial regularity of the gradient $Du$ (and hence
for $u$) outside of a set of Lebesgue measure zero. Restricting ourselves to
the partial regularity of $u$ and to dimensions $n \leq p+2$, we explain why
the Hausdorff dimension of the singular set cannot exceed $n-p$. Finally, we
address the possible existence of singularities in two dimensions.
We prove that no Riemannian manifold quasiisometric to
complex hyperbolic plane can have a better curvature pinching. The proof
uses cup-products in $L^p$-cohomology.
We study a simple heat-bath type dynamic for a simple model of
polymer interacting with an interface. The polymer is a nearest neighbor path
in
Z, and the interaction is modelised by energy penalties/bonuses given when the
path touches 0. This dynamic has been studied by D. Wilson for the case without
interaction, then by Caputo et al. for the more general case. When the
interface
is repulsive, the dynamic slows down due to the appearance of a bottleneck in
the
state space, moreover, the systems exhibits a metastable behavior, and, after
time
rescaling, behaves like a two-state Markov chain.
We consider a
directed random polymer interacting with an interface
that carries random charges some of which attract while others repel
the polymer. Such a polymer can be in a localized or delocalized
phase, i.e., it stays near the interface or wanders away respectively.
The phase it chooses depends on the temperature and the average bias
of the disorder. At a given temperature, there is a critical bias
separating the two phases. A question of particular interest, and
which has been studied extensively in the Physics and Mathematics
literature, is whether the quenched critical bias differs from the
annealed critical bias. When it does, we say that the disorder is
relevant.
Using a large deviations result proved recently by Birkner, Greven,
and den Hollander, we derive a variational formula for the quenched
critical bias. This
leads to a necessary and sufficient condition for
disorder relevance that implies easily some known results as well as
new ones.
The talk is based on joint work with Frank den Hollander.
Anomalous ( non local) diffusion processes appear in many subjects: phase transition, fracture dynamics, game theory I will describe some of the issues involved, and in particular, existence and regularity for some non local versions of the p Laplacian, of non variational nature, that appear in non local tug of war.
The talk will be devoted to criteria of absence of arbitrage opportunities under small transaction costs for a family of multi-asset models of financial market.
Please note the earlier than usual start-time!
The Mathematical Institute invites you to attend the Inaugural Lecture of Professor Gui-Qiang G. Chen. Professor in the Analysis of Partial Differential Equations. Examination Schools, 75-81 High Street, Oxford, OX 4BG.
There is no charge to attend but registration is required. Please register your attendance by sending an email to @email specifying the number of people in your party. Admission will only be allowed with prior registration.
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ABSTRACT
While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables – perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.
In this lecture, Professor Gui-Qiang G. Chen will present several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world.
Efficient linear and infinitely smooth approximation of functions from equidistant samples is a fascinating problem, at least since Runge showed in 1901 that it is not delivered by the interpolating polynomial.
In 1988, I suggested to substitute linear rational for polynomial interpolation by replacing the denominator 1 with a polynomial depending on the nodes, though not on the interpolated function. Unfortunately the so-obtained interpolant converges merely as the square of the mesh size. In 2007, Floater and Hormann have given for every integer a denominator that yields convergence of that prescribed order.
In the present talk I shall present the corresponding interpolant as well as some of its applications to differentiation, integration and the solution of boundary value problems. This is joint work with Georges Klein and Michael Floater.