I describe a conjecture equating the two items appearing in the title.
In this talk we will first consider the Allen-Cahn action functional that controls the probability of rare events in an Allen-Cahn type equation with additive noise. Further we discuss a perturbation of the Allen-Cahn equation by a stochastic flow. Here we will present a tightness result in the sharp interface limit and discuss the relation to a version of stochastically perturbed mean curvature flow. (This is joint work with Luca Mugnai, Leipzig, and Hendrik Weber, Warwick.)
Coarse geometry provides a very useful organising point of view on the study
of geometry and analysis of discrete metric spaces, and has been very
successful in the context of geometric group theory and its applications. On
the other hand, the work of Carlsson, Ghrist and others on persistent
homology has paved the way for applications of topological methods to the
study of broadly understood data sets. This talk will provide an
introduction to this fascinating topic and will give an overview of possible
interactions between the two.
Topological spaces and manifolds are commonly used to model configuration
spaces of systems of various nature. However, many practical tasks, such as
those dealing with the modelling, control and design of large systems, lead
to topological problems having mixed topological-probabilistic character
when spaces and manifolds depend on many random parameters.
In my talk I will describe several models of stochastic algebraic topology.
I will also mention some open problems and results known so far.
"Rough paths of inhomogeneous degree of smoothness (Pi-rough paths) can be treated as p-rough paths (of homogeneous degree of
smoothness) for a sufficiently large p. The theory of integration with respect to p-rough paths can be applied to prove existence and uniqueness of solutions of differential equations driven by Pi-rough paths. However the required conditions on the one-form determining the differential equation are too strong and can be weakened. The talk proves the existence and uniqueness under weaker conditions and explores some applications of Pi-rough paths
The behaviour of the tail of the distribution of the first passage time over a fixed level has been known for many years, but until recently little was known about the behaviour of the probability mass function or density function. In this talk we describe recent results of Vatutin and Wachtel, Doney, and Doney and Rivero which give such information whenever the random walk or Levy process is asymptotically stable.
Many proteins cleave and reseal DNA molecules in precisely orchestrated
ways. Modelling these reactions has often relied on the axis of the DNA
double helix
being circular, so these cut-and-seal mechanisms can be
tracked by corresponding changes in the knot type of the DNA axis.
However, when the DNA molecule is linear, or the
protein action does not manifest itself as a change in knot type, or the
knots types are not 4-plats, these knot theoretic models are less germane.
We thus give a taxonomy of local DNA axis configurations. More precisely, we
characterise
all rational tangles obtained from a given rational tangle via a rational
subtangle
replacement (RSR). This builds on work of Berge and Gabai.
We further determine the sites for these RSR of distance greater than 1.
Finally, we classify all knots in lens spaces whose exteriors are
generalised Seifert fibered spaces and their lens space surgeries, extending work of
Darcy-Sumners.
Biologically then, this classification is endowed with a distance that
determines how many protein reactions
of a particular type (corresponding to steps of a specified size) are
needed to proceed from one local conformation to another.
We conclude by discussing a variety of biological applications of this
categorisation.
Joint work with Ken Baker
The central axis of the famous DNA double helix can become knotted
or linked as a result of numerous biochemical processes, most notably
site-specific recombination. Site-specific recombinases are naturally
occurring enzymes that cleave and reseal DNA molecules in very precise ways.
As a by product of their main purpose, they manipulate cellular DNA in
topologically interesting and non-trivial ways. So if the axis of the DNA
double helix is circular, these cut-and-seal mechanisms can be tracked by
corresponding changes in the knot type of the DNA axis. In this talk, I'll
explain several topological strategies to investigate these biological
situations. As a concrete example, I will disscuss my recent work, which
predics what types of DNA knots and links can arise from site-specific
recombination on DNA twist knots.
Persistent homology is a relatively new tool to analyse the topology of data sets.
We will give a brief introduction and tutorial as preparation for the third talk in the afternoon.
We present three examples of credit products whose valuation poses challenging modeling problems related to armageddon scenarios and extreme losses, analyzing their behaviour pre- and in-crisis.
The products are Credit Index Options (CIOs), Collateralized Debt Obligations (CDOs), and Credit Valuation Adjustment (CVA) related products. We show that poor mathematical treatment of possibly vanishing numeraires in CIOs and lack of modes in the tail of the loss distribution in CDOs may lead to inaccurate valuation, both pre- and especially in crisis. We also consider the limits of copula models in trying to represent systemic risk in credit intensity models. We finally enlarge the picture and comment on a number of common biases in the public perception of modeling in relationship with the crisis.
This will be on the topic of the CASE project Thales will be sponsoring from Oct '11.
It is known that the expansion of the real field by some quasianalytic algebras of functions are o-minimal and polynomially bounded. We prove that, for these structures, the preparation theorem for definable functions proved by L. van den Dries and P. Speissegger has an explicit form, from which it is easy to deduce a quantifier elimination result.
"Most drivers will recognize the scenario: you are making steady progress along the motorway when suddenly you come to a sudden halt at the tail end of a lengthy queue of traffic. When you move off again you look for the cause of the jam, but there isn't one. No accident damaged cars, no breakdown, no dead animal, and no debris strewn on the road. So what caused everyone to stop?" RAC news release (2005)
The (by now well-known) answer is that such "phantom traffic jams" exist as waves that propagate upstream (opposite to the driving direction) - so that the vast majority of individuals do not observe the instant at which the jam was created - yet what exactly goes on at that instant is still a matter of debate. In this talk I'll give an overview of empirical data and models to describe such spatiotemporal patterns. The key property we need is instability: and using the framework of car-following (CF) models, I'll show how different sorts of linear (convective and absolute) and nonlinear instability can be used to explain empirical patterns.
Polynomial Programs are ussually solved by using hierarchies of convex relaxations. This scheme rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves an initial relaxation without incurring exponential growth in size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems.
Joint work with Bissan Ghaddar and Miguel Anjos
The topology of the moduli space of stable bundles (of coprime rank and degree) on a smooth curve can be understood from different points of view. Atiyah and Bott calculated the Betti numbers by gauge-theoretic methods (using equivariant Morse theory for the Yang-Mills functional), arriving at the same inductive formula which had been obtained previously by Harder and Narasimhan using arithmetic techniques. An intermediate interpretation (algebro-geometric in nature but dealing with infinite-dimensional parameter spaces as in the gauge theory picture) comes from thinking about vector bundles in terms of matrix divisors, generalising the Abel-Jacobi map to higher rank bundles.
I'll sketch these different approaches, emphasising their parallels, and in the end I'll speculate about how (some of) these methods could be made to work when the underlying curve acquires nodal singularities.
We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information.
The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. Using stochastic flow ideas we establish properties of the value function (monotonicity, convexity in the log-stock price), conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward. From this we derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not.
We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.
This is joint work with Boris Andreianov and Nils Henrik Risebro.