Past

tba

The Wulff droplet arises by conditioning a spin system in a dominant

phase to have an excess of signs of opposite type. These gather

together to form a droplet, with a macroscopic Wulff profile, a

solution to an isoperimetric problem.

I will discuss recent work proving that the phase boundary that

delimits the signs of opposite type has a characteristic scale, both

at the level of exponents and their logarithmic corrections.

This behaviour is expected to be shared by a broad class of stochastic

interface models in the Kardar-Parisi-Zhang class. Universal

distributions such as Tracy-Widom arise in this class, for example, as

the maximum behaviour of repulsive particle systems. time permitting,

I will explain how probabilistic resampling ideas employed in spin

systems may help to develop a qualitative understanding of the random

mechanisms at work in the KPZ class.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.

Given a graph we want to draw it in the plane; well we *want* to draw it in the plane, but sometimes we just can't. So we resort to various compromises. Sometimes we add crossings and try to minimize the crossings. Sometimes we add handles and try to minimize the number of handles. Sometimes we add crosscaps and try to minimize the number of crosscaps.

Sometimes we mix these parameters: add a given number of handles (or crosscaps) and try to minimize the number of crossings on that surface. What if we are willing to trade: say adding a handle to reduce the number of crossings? What can be said about the relative value of such a trade? Can we then add a second handle to get an even greater reduction in crossings? If so, why didn't we trade the second handle in the first place? What about a third handle?

The crossing sequence cr_1, cr_2, ... , cr_i, ... has terms the minimum number of crossings over all drawings of G on a sphere with i handles attached. The non-orientable crossing sequence is defined similarly. In this talk we discuss these crossing sequences.

By Dan Archdeacon, Paul Bonnington, Jozef Siran, and citing works of others.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.

This talk is the second in a series of an elementary introduction to the ideas unifying elliptic curves, modular forms and Galois representations. I will discuss what it means for an elliptic curve to be modular and what type of representations one associates to such objects.

In this talk, we introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover all homotopy groups of any dimensional spheres can be obtained from the geometric Massey products on certain links.

Since the work of Feigenbaum and Coullet-Tresser on universality in the period doubling bifurcation, it is been understood that crucial features of unimodal (one-dimensional) dynamics depend on the behavior of a renormalization (and infinite dimensional) dynamical system. While the initial analysis of renormalization was mostly focused on the proof of existence of hyperbolic fixed points, Sullivan was the first to address more global aspects, starting a program to prove that the renormalization operator has a uniformly hyperbolic (hence chaotic) attractor. Key to this program is the proof of exponential convergence of renormalization along suitable ``deformation classes'' of the complexified dynamical system. Subsequent works of McMullen and Lyubich have addressed many important cases, mostly by showing that some fine geometric characteristics of the complex dynamics imply exponential convergence.

We will describe recent work (joint with Lyubich) which moves the focus to the abstract analysis of holomorphic iteration in deformation spaces. It shows that exponential convergence does follow from rougher aspects of the complex dynamics (corresponding to precompactness features of the renormalization dynamics), which enables us to conclude exponential convergence in all cases.

In this talk, we report some of our recent work on hybrid switching diffusions in which continuous dynamics and discrete events coexist. Motivational examples in singular perturbed Markovian systems, manufacturing, and financial engineering will be mentioned. After presenting criteria for recurrence and ergodicity, we consider numerical methods for controlled switching diffusions and related game problems. Rates of convergence of Markov chain approximation methods will also be studied.

Andrew Stewart -

The role of the complete Coriolis force in ocean currents that cross the equator

Large scale motions in the atmosphere and ocean are dominated by the Coriolis force due to the Earth's rotation. This tends to prevent fluid crossing the equator from one hemisphere to the other. We investigate the flow of a deep ocean current, the Antarctic Bottom Water, across the equator using a shallow water model that includes the Earth's complete Coriolis force. By contrast, most theoretical models of the atmosphere and ocean use the so-called traditional approximation that neglects the component of the Coriolis force associated with the locally horizontal component of the Earth's rotation vector. Using a combination of analytical and numerical techniques, we show that the cross-equatorial transport of the Antarctic Bottom Water may be substantially influenced by the interaction of the complete Coriolis force with bottom topography.

In the first part of my presentation, I plan to review several applications modelled by delay differential equations (DDEs) starting from familiar examples such as traffic flow problems to physiology and industrial problems. Although delay differential equations have the reputation to be difficult mathematical problems, there is a renewed interest for both old and new problems modelled by DDEs. In the second part of my talk, I’ll emphasize the need of developing asymptotic tools for DDEs in order to guide our numerical simulations and help our physical understanding. I illustrate these ideas by considering the response of optical optoelectronic oscillators that have been studied both experimentally and numerically.

Jointly with Number Theory

Consider a family of abelian varieties whose base is an algebraic variety. The union of all torsion groups over all fibers of the family will be called the set of torsion points of the family. If the base variety is a point then the family is just an abelian variety.

In this case the Manin-Mumford Conjecture, a theorem of Raynaud, implies that a subvariety of the abelian variety contains a Zariski dense set of torsion points if and only if it is itself essentially an abelian subvariety. This talk is on possible extensions to certain families where the base is a curve. Conjectures of André and Pink suggest considering "special points": these are torsion points whose corresponding fibers satisfy an additional arithmetic property. One possible property is for the fiber to have complex multiplication; another is for the fiber to be isogenous to an abelian variety fixed in advance.

We discuss some new results on the distribution of such "special points"

on the subvarieties of certain families of abelian varieties. One important aspect of the proof is the interplay of two height functions.

I will give a brief introduction to the theory of heights in the talk.

Meshless (or meshfree) methods are a relatively new numerical approach for the solution of ordinary- and partial differential equations. They offer the geometrical flexibility of finite elements but without requiring connectivity from the discretization support (ie a mesh). Meshless methods based on the collocation of radial basis functions (RBF methods) are particularly easy to code, and have a number of theoretical advantages as well as practical drawbacks. In this talk, an adaptive RBF scheme is presented for a novel application, namely the solution of (a rather broad class of) delayed- and neutral differential equations.