Past

As everyone knows, one popular notion of a (Scott-Ersov) domain is defined as a bounded complete algebraic cpo. These are closely related to algebraic lattices: (i) A domain becomes an algebraic lattice with the adjunction of an (isolated) top element. (ii) Every non-empty Scott-closed subset of an algebraic lattice is a domain. Moreover, the isolated (= compact) elements of an algebraic lattice form a semilattice (under join). This semilattice has a zero element, and, provided the top element is isolated, it also has a unit element. The algebraic lattice itself may be regarded as the ideal completion of the semilattice of isolated elements. This is all well known. What is not so clear is that there is an easy-to-construct domain of countable semilattices giving isomorphic copies of all countably based domains. This approach seems to have advantages over both the so-called "information systems" or more abstract lattice formulations, and it makes definitions of solutions to domain equations very elementary to justify. The "domain of domains" also has a natural computable structure

An emerging set of methods enables an experimental dialogue with biological systems composed of many interacting cell types---in particular, with neural circuits in the brain. These methods are sometimes called “optogenetic” because they employ light-responsive proteins (“opto-“) encoded in DNA (“-genetic”). Optogenetic devices can be introduced into tissues or whole organisms by genetic manipulation and be expressed in anatomically or functionally defined groups of cells. Two kinds of devices perform complementary functions: light-driven actuators control electrochemical signals; light-emitting sensors report them. Actuators pose questions by delivering targeted perturbations; sensors (and other measurements) signal answers. These catechisms are beginning to yield previously unattainable insight into the organization of neural circuits, the regulation of their collective dynamics, and the causal relationships between cellular activity patterns and behavior.

In the eighteenth century Gaspard Monge considered the problem of finding the best way of moving a pile of material from one site to another. This optimal transport problem has many applications such as mesh generation, moving mesh methods, image registration, image morphing, optical design, cartograms, probability theory, etc. The solution to an optimal transport problem can be found by solving the Monge-Amp\`{e}re equation, a highly nonlinear second order elliptic partial differential equation. Leonid Kantorovich, however, showed that it is possible to analyse optimal transport problems in a framework that naturally leads to a linear programming formulation. In recent years several efficient methods have been proposed for solving the Monge-Amp\`{e}re equation. For the linear programming problem, standard methods do not exploit the special properties of the solution and require a number of operations that is quadratic or even cubic in the number of points in the discretisation. In this talk I will discuss techniques that can be used to obtain more efficient methods.

Joint work with Chris Budd (University of Bath).

In this talk, we describe new profile decompositions for bounded sequences in Banach spaces of functions defined on $\mathbb{R}^d$. In particular, for "critical spaces" of initial data for the Navier-Stokes equations, we show how these can give rise to new proofs of recent regularity theorems such as those found in the works of Escauriaza-Seregin-Sverak and Rusin-Sverak. We give an update on the state of the former and a new proof plus new results in the spirit of the latter. The new profile decompositions are constructed using wavelet theory following a method of Jaffard.

\paragraph{} Poisson quasi-Nijenhuis structures with background (PqNb structures) were recently defined and are one of the most general structures within Poisson geometry. On one hand they generalize the structures of Poisson-Nijenhuis type, which in particular contain the Poisson structures themselves. On the other hand they generalize the (twisted) generalized complex structures defined some years ago by Hitchin and Gualtieri. Moreover, PqNb manifolds were found to be appropriate target manifolds for sigma models if one wishes to incorporate certain physical features in the model. All these three reasons put the PqNb structures as a new and general object that deserves to be studied in its own right.

\paragraph{} I will start the talk by introducing all the concepts necessary for defining PqNb structures, making this talk completely self-contained. After a brief recall on Poisson structures, I will define Poisson-Nijenhuis and Poisson quasi-Nijenhuis manifolds and then move on to a brief presentation on the basics of generalized complex geometry. The PqNb structures then arise as the general structure which incorporates all the structures referred above. In the second part of the talk, I will define gauge transformations of PqNb structures and show how one can use this concept to construct examples of such structures. This material corresponds to part of the article arXiv:0912.0688v1 [math.DG].\\

\paragraph{} Also, if time permits, I will shortly discuss the appearing of PqNb manifolds as target manifolds of sigma models.

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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.