Past

We study upper bounds on topological complexity of sets definable in o-minimal structures over the reals. We suggest a new construction for approximating a large class of definable sets, including the sets defined by arbitrary Boolean combinations of equations and inequalities, by compact sets.

Those compact sets bound from above the homotopies and homologies of the approximated sets.

The construction is applicable to images under definable maps.

Based on this construction we refine the previously known upper bounds on Betti numbers of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae, and prove similar new upper bounds, individual for different Betti numbers, for their images under arbitrary continuous definable maps.

Joint work with A. Gabrielov.

If an ideal elastic spring is greatly stretched, it will develop large stresses. However, solid biological tissues are able to grow without developing such large stresses. This is because the cells within such tissues are able to lay down new fibres and remove old ones, fundamentally changing the mechanical structure of the tissue. In many ways, this is analogous to classical plasticity, where materials stretched beyond their yield point begin to flow and the unloaded state of the material changes. Unfortunately, biological tissues are not closed systems and so we are not able to use standard plasticity techniques where we require the flow to be mass conserving and energetically passive.

In this talk, a general framework will be presented for modelling the changing zero stress state of a biological tissue (or any other material). Working from the multiplicative decomposition of the deformation gradient, we show that the rate of 'desired' growth can represented using a tensor that describes both the total rate of growth and any directional biases. This can be used to give an evolution equation for the effective strain (a measure of the difference between the current state and the zero stress state). We conclude by looking at a perhaps surprising application for this theory as a method for deriving the constitutive laws of a viscoelastic fluid.

Saddle-point problems occur frequently in liquid crystal modelling. For example, they arise whenever Lagrange multipliers are used for the pointwise-unit-vector constraints in director modelling, or in both general director and order tensor models when an electric field is present that stems from a constant voltage. Furthermore, in a director model with associated constraints and Lagrange multipliers, together with a coupled electric-field interaction, a particular ''double'' saddle-point structure arises. This talk will focus on a simple example of this type and discuss appropriate numerical solution schemes.

This is joint work with Eugene C. Gartland, Jr., Department of Mathematical Sciences, Kent State University.

TBA

We will consider the monodromy action on mod 2 cohomology for SL(2) Hitchin systems. We will study Copeland's approach to the subject and use his results to compute the monodromy action on mod 2 cohomology. An interpretation of our results in terms of geometric properties of fixed points of a natural involution on the moduli space is given.

The background for the multitarget tracking problem is presented

along with a new framework for solution using the theory of random

finite sets. A range of applications are presented including

submarine tracking with active SONAR, classifying underwater entities

from audio signals and extracting cell trajectories from biological

data.

\ \ In 1939 Rademacher derived a conditionally convergent series expression for the modular j-invariant, and used this expression---the first Rademacher sum---to verify its modular invariance. We may attach Rademacher sums to other discrete groups of isometries of the hyperbolic plane, and we may ask how the automorphy of the resulting functions reflects the geometry of the group in question.

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\ \ In the case of a group that defines a genus zero quotient of the hyperbolic plane the relationship is particularly striking. On the other hand, of the common features of the groups that arise in monstrous moonshine, the genus zero property is perhaps the most elusive. We will illustrate how Rademacher sums elucidate this phenomena by using them to formulate a characterization of the discrete groups of monstrous moonshine.

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\ \ A physical interpretation of the Rademacher sums comes into view when we consider black holes in the context of three dimensional quantum gravity. This observation, together with the application of Rademacher sums to moonshine, amounts to a new connection between moonshine, number theory and physics, and furnishes applications in all three fields.

Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and use this description to construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. As an application, we compute generating functions of Euler characteristics of M in case X is a toric surface. In the torsion free case, one finds examples of new as well as known generating functions. In the pure dimension 1 case using a conjecture of Sheldon Katz, one obtains examples of genus zero Gopakumar-Vafa invariants of the canonical bundle of X.

The notion of a boundary graph property is a relaxation of that of a

minimal property. Several fundamental results in graph theory have been obtained in

terms of identifying minimal properties. For instance, Robertson and Seymour showed that

there is a unique minimal minor-closed property with unbounded tree-width (the planar

graphs), while Balogh, Bollobás and Weinreich identified nine minimal hereditary

properties of labeled graphs with the factorial speed of growth. However, there are

situations where the notion of minimal property is not applicable. A typical example of this type

is given by graphs of large girth. It is known that for each particular value of k, the

graphs of girth at least k are of unbounded tree-width and their speed of growth is

superfactorial, while the limit property of this sequence (i.e., the acyclic graphs) has bounded

tree-width and its speed of growth is factorial. To overcome this difficulty, the notion of

boundary properties of graphs has been recently introduced. In the present talk, we use this

notion in order to identify some classes of graphs which are well-quasi-ordered with

respect to the induced subgraph relation.

Let $\phi$ be a convex, $C^1$-function and consider the functional: $$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set and $\bf u: \Omega \to \mathbb{R}^N$. The associated Euler Lagrange system is $$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$ In a fundamental paper K.~Uhlenbeck proved everywhere $C^{1,\alpha}$-regularity for local minimizers of the $p$-growth functional with $p\ge 2$. Later on a large number of generalizations have been made. The case $1

{\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$-Hölder continuous for some $\alpha>0$.

We present a unified approach to the superquadratic and subquadratic $p$-growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.

We consider the time average of the (renormalized) current fluctuation field in one-dimensional weakly asymmetric simple exclusion.

The asymmetry is chosen to be weak enough such that the density fluctuation field still converges in law with respect to diffusive scaling. Remark that the density fluctuation field would evolve on a slower time scale if the asymmetry is too strong and that then the current fluctuations would have something to do with the Tracy-Widom distribution. However, the asymmetry is also chosen to be strong enough such that the density fluctuation field does not converge in law to an infinite-dimensional Ornstein-Uhlenbeck process, that is something non-trivial is happening.

We will, at first, motivate why studying the time average of the current fluctuation field helps to understand the structure of this non-trivial scaling limit of the density fluctuation field and, second, show how one can replace the current fluctuation field by a certain functional of the density fluctuation field under the time average. The latter result provides further evidence for the common belief that the scaling limit of the density fluctuation field approximates the solution of a Burgers-type equation

Abstract: The goal of this talk is to discuss threeproblems on fractional and related stochastic fields, in which wavelet methodshave turned out to be quite useful.

  The first problemconsists in constructing optimal random series representations of Lévyfractional Brownian field; by optimal we mean that the tails of the seriesconverge to zero as fast as possible i.e. at the same rate as the l-numbers.Note in passing that there are close connections between the l-numbers of aGaussian field and its small balls probabilities behavior.

  The secondproblem concerns a uniform result on the local Hölder regularity (the pointwiseHölder exponent) of multifractional Brownian motion; by uniform we mean thatthe result is satisfied on an event with probability 1 which does not depend onthe location.

  The third problemconsists in showing that multivariate multifractional Brownian motion satisfiesthe local nondeterminism property. Roughly speaking, this property, which wasintroduced by Berman, means that the increments are asymtotically independentand it allows to extend to general Gaussian fields many results on the localtimes of Brownian motion.

TBA

The viability of a market impact model is usually considered to be equivalent to the absence of price manipulation strategies in the sense of Huberman & Stanzl (2004). By analyzing a model with linear instantaneous, transient, and permanent impact components, we discover a new class of irregularities, which we call transaction-triggered price manipulation strategies. Transaction-triggered price manipulation is closely related to the non-existence of measure-valued solutions to a Fredholm integral equation of the first kind. We prove that price impact must decay as a convex decreasing function of time to exclude these market irregularities along with standard price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest. Joint work with Aurélien Alfonsi, Jim Gatheral, and Alla Slynko.