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.
The maximum principle is one of the main tools use to understand the behaviour of solutions to the Ricci flow. It is a very powerful tool that can be used to show that pointwise inequalities on the initial data of parabolic PDE are preserved by the evolution. A particular weak maximum principle for vector bundles will be discussed with references to Hamilton's seminal work [J. Differential Geom. 17 (1982), no. 2, 255–306; MR664497] on 3-manifolds with positive Ricci curvature and his follow up paper [J. Differential Geom. 24 (1986), no. 2, 153–179; MR0862046] that extends to 4-manifolds with various curvature assumptions.
This talk highlights the role of Gaussian Process models in sequential
data analysis. Issues of active data selection, global optimisation,
sensor selection and prediction in the presence of changepoints are
discussed.
Abstract. In this talk, I will present joint work with Uri Onn, Mark Berman, and Pirita Paajanen.
Let G be a linear algebraic group defined over the integers. Let O be a compact discrete valuation ring with a finite residue field of cardinality q and characteristic p. The group
G(O) has a filtration by congruence subgroups
G_m(O) (which is by definition the kernel of reduction map modulo P^m where P is the maximal ideal of O).
Let c_m=c_m(G(O)) denote the number of conjugacy classes in the finite quotient group G(O)/G_m(O) (which is called the mth congruence quotient of G(O)). The conjugacy class zeta function of
G(O) is defined to be the Dirichlet series Z_{G(O)}(s)=\sum_{m=0,1,...} c_m q^_{-ms}, where s is a complex number with Re(s)>0. This zeta function was defined by du Sautoy when G is a p-adic analytic group and O=Z_p, the ring of p-adic integers, and he proved that in this case it is a rational function in p^{-s}. We consider the question of dependence of this zeta function on p and more generally on the ring O.
We prove that for certain algebraic groups, for all compact discrete valuation rings with finite residue field of cardinality q and sufficiently large residue characteristic p, the conjugacy class zeta function is a rational function in q^{-s} which depends only on q and not on the structure of the ring. Note that this applies also to positive characteristic local fields.
A key in the proof is a transfer principle. Let \psi(x) and f(x) be resp.
definable sets and functions in Denef-Pas language.
For a local field K, consider the local integral Z(K,s)=\int_\psi(K)
|f(x)|^s dx, where | | is norm on K and dx normalized absolute value
giving the integers O of K volume 1. Then there is some constant
c=c(f,\psi) such that for all local fields K of residue characteristic larger than c and residue field of cardinality q, the integral Z(K,s) gives the same rational function in q^{-s} and takes the same value as a complex function of s.
This transfer principle is more general than the specialization to local fields of the special case when there is no additive characters of the motivic transfer principle of Cluckers and Loeser since their result is the case when the integral is zero.
The conjugacy class zeta function is related to the representation zeta function which counts number of irreducible complex representations in each degree (provided there are finitely many or finitely many natural classes) as was shown in the work of Lubotzky and Larsen, and gives information on analytic properties of latter zeta function.
We will investigate what one can detect about a discrete group from its profinite completion, with an emphasis on considering geometric properties.
Cryopreservation (using temperatures down to that of liquid nitrogen at
–196 °C) is the only way to preserve viability and function of mammalian cells for research and transplantation and is integral to the quickly evolving field of regenerative medicine. To cryopreserve tissues, cryoprotective agents (CPAs) must be loaded into the tissue. The loading is critical because of the high concentrations required and the toxicity of the CPAs. Our mathematical model of CPA transport in cartilage describes multi-component, multi-directional, non-dilute transport coupled to mechanics of elastic porous media in a shrinking and swelling domain.
Parameters are obtained by fitting experimental data. We show that predictions agree with independent spatially and temporally resolved MRI experimental measurements. This research has contributed significantly to our interdisciplinary group’s ability to cryopreserve human articular cartilage.
A family of graphs F on a fixed set of n vertices is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. Simonovits and Sos conjectured that such a family has size at most (1/8)2^{\binom{n}{2}}, and that equality holds only if F
consists of all graphs containing some fixed triangle. Recently, the author, Yuval Filmus and Ehud Friedgut proved a strengthening of this conjecture, namely that if F is an odd-cycle-intersecting family of graphs, then |F| \leq (1/8) 2^{\binom{n}{2}}. Equality holds only if F consists of all graphs containing some fixed triangle. A stability result also holds: an odd-cycle-intersecting family with size close to the maximum must be close to a family of the above form. We will outline proofs of these results, which use Fourier analysis, together with an analysis of the properties of random cuts in graphs, and some results in the theory of Boolean functions. We will then discuss some related open questions.
All will be based on joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Hebrew University of Jerusalem).