This talk introduces the topic of random walks on a finitely generated group and asks what properties of such a group can be detected through knowledge of such walks.
Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.
The problem of checking existence for an Euler tour of a graph is trivial (are all vertex degrees even?). The problem of counting (or even approximate counting) Euler tours seems to be very difficult. I will describe two simple classes of graphs where the problem can be
solved exactly in polynomial time. And also talk about the many many classes of graphs where no positive results are known.
We introduce and study ergodic multidimensional diffusion processes interacting through their ranks; these interactions lead to invariant measures which are in broad agreement with stability properties of large equity markets over long time-periods.
The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset.
Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.
The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of intersection local times.
The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics, such as the Sherrington-Kirkpatrick model for spin-glasses.
"We present a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points.
For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations. We give some concrete examples, and in particular we study the case of mean variance optimal portfolios with wealth dependent risk aversion"
Both parts will deal with spherical objects in the bounded derived
category of coherent sheaves on K3 surfaces. In the first talk I will
focus on cycle theoretic aspects. For this we think of the Grothendieck
group of the derived category as the Chow group of the K3 surface (which
over the complex numbers is infinite-dimensional due to a result of
Mumford). The Bloch-Beilinson conjecture predicts that over number
fields the Chow group is small and I will show that this is equivalent to
the derived category being generated by spherical objects (which
I do not know how to prove). In the second talk I will turn to stability
conditions and show that a stability condition is determined by its
behavior with respect to the discrete collections of spherical objects.
The fundamental models for lipid bilayers are curvature based and neglect the internal structure of the lipid layers. In this talk, we explore models with an additional order parameter which describes the orientation of the lipid molecules in the membrane and compare their predictions based on numerical simulations. This is joint work with Soeren Bartels (Bonn) and Ricardo Nochetto (College Park).
(Note that the talk will be in L2 and not the usual SR1)
Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain. For instance, a smooth cubic surface in complex projective three-dimensional space contains exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational) parameterization of the points of the cubic by the points in the complex projective plane.
After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x^5+y^5+z^5+u^5+v^5=0. The space of lines on X is well-known. I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.
This talk is based on joint work with R. Heath-Brown.
We review the problem of determining the high frequency asymptotics of the spectrum of the Laplacian and its relationship to the geometry of a domain. We then establish these asymptotics for some continuum random trees as well as the scaling limit of the critical random graph.
For a superprocess in a random environment in one dimensional space, a nonlinear stochastic partial differential equation is derived for its density by Dawson-Vaillancourt-Wang (2000). The joint continuity was left as an open problem. In this talk, we will give an affirmative answer to this problem.
We present an O(N logN) algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing.
Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.
Presentations by:
09.30 am Bernhard Langwallner Continuum limits of atomistic energies and new computational models of fracture
09.50 am Yasemin Sengul Well-posedness of dynamics
10.10 am Kostas Koumatos X-interfaces and nonclassical austenite-martensite interfaces
10.30 am Tim Squires Models for breast cancer and heart tissue
Abstract: Quantales are ordered algebras which can be thought of as pointfree noncommutative topologies. In recent years, their connections have been studied with fundamental notions in noncommutative geometry such as groupoids and C*-algebras. In particular, the setting of quantales corresponding to étale groupoids has been very well understood: a bijective correspondence has been defined between localic étale groupoids and inverse quantale frames. We present an equivalent but independent way of defining this correspondence for topological étale groupoids and we extend this correspondence to a non-étale setting.
Wound coils or rolls accumulate essentially flat strip compactly without folding or cutting and typically, strip is wound and unwound a number of times before its end use. The variety of material that is wound into coils or rolls is very extensive and includes magnetic tape, paper, cellophane, plastics, fabric and metals such as aluminium and steel.
Stresses wound into a coil provide its structural integrity via the frictional forces between the wraps. For a coil with inadequate inter-wrap pressure, the wraps may slip or telescope (causing surface scuffing) or the coil may slump and collapse. On the other hand, large internal stresses can cause increased creep and stress relaxation, collapse at the bore, stress wrinkling and rupture of the material in the coil.
Given the range of applications, it is not surprising that the literature on calculating stresses in wound coils is large and has a long history, which goes back at least to the wire winding of gun barrels. However the basic approach of the resulting accretion models, where the residual stress is recalculated each time a layer is added, has remained essentially the same. In this talk, we take a radically different approach in analysing the winding stresses in coils. Instead of the traditional method, we seek to deduce a winding policy that will achieve a target distribution of residual stresses within a coil. In this way, optimising the coiling tension profile is much more straight-forward, by
* Specifying the residue stresses required to avoid operational problems, tight-bore collapses, and other issues such as scuffing, then
* Determining the winding tension profile to produce the required residue stresses.
