Young Researchers in Mathematics

Astrophysics

C.J. Donnelly: Accretion Discs: Classifications and Difficulties

The disc as an astrophysical structure is well known to be ubiquitous, from galaxies to protoplanetary discs. My talk will give an overview of the different disc structures we see in the universe and their basic properties and points of interest, before focusing on the canonical thin-disc MHD model of an accretion disc and its development over the past 40 years.

Evan Keane: Using Pulsars to Detect Gravitational Waves

There are now in excess of 1800 neutron stars known which manifest themselves as pulsars. These compact astrophysical sources emit narrow beam of coherent radio emission. As the star spins the beam of radiation cuts into and out of our line of sight like a cosmic lighthouse. Thus astronomers on Earth can detect a pulse once per rotation. The spin-evolution of the fastest pulsars (with periods of a few milliseconds) is very stable which makes pulsars highly reliable cosmic clocks. In this talk I will discuss how regular monitoring of an array of pulsars can be used to detect the stochastic gravitaional wave background produced by inspiralling compact object binaries or more exiotic phenomena such as cosmic strings.

Christina Davies: A model for the solar dynamo driven by the magnetic buoyancy instability

I will use the magnetohydrodynamic (MHD) approximation to study the generation of large-scale magnetic fields in the sun via the magnetic buoyancy instability. The magnetic buoyancy instability can destabilise a stratified layer of magnetised fluid, even in the case where the layer is stable to thermal convection, and is responsible for the escape of magnetic field from the base of the convection zone to the solar surface. I will discuss the physical motivation behind the problem, the MHD equations themselves, the mean-field approximation to the MHD equations and finally the reasoning behind our approach to the problem. I will then present some results, and discuss their relevance to the generation of large-scale magnetic field in the sun.

General Relativity

Mark Durkee: The Geroch-Held-Penrose formalism in higher-dimensional gravity

There is currently great interest in studying gravity in higher dimensions. However, many useful approaches to doing computations in general relativity are specific to four dimensions. In this talk, I will give an introduction to the generalization of some such tools to arbitrary dimension, in particular frame basis methods based on the 4D Newman-Penrose and Geroch-Held-Penrose formalisms. These techniques were key to various important developments in 4D GR, for example the discovery of the Kerr metric, and the study of its stability; I will discuss likely applications of the higher dimensional versions. (see arXiv:0908.2771 and 1002.4826)

Steffen Gielen: Field theories on the space of connections

I will outline a tentative description of quantum gravity as a quantum field theory on the space of connections on a given 3-manifold, which arises as the "third quantised" version of Wheeler-DeWitt geometrodynamics (i.e. the attempt to canonically quantise GR). While the description of such a theory can in many be ways only formal, some aspects can be made mathematically rigorous using mathematical techniques from loop quantum gravity. Furthermore, it might be a promising framework to address conceptual issues arising in different approaches to quantum gravity, and to bridge gaps between them.

Norman Metzner: A Twistor Construction for Stationary and Axisymmetric Vacuum Spacetimes

This talk will give a characterisation of stationary and axisymmetric solutions of Einstein's equations (vacuum) in terms of twistor theory. The correspondence is based on the Ernst form of Einstein's equations which relate them to an anti-self-dual Yang-Mills field which in turn has a twistor characterisation by the Penrose-Ward transform. Eventually, vacuum spacetimes are described by complex vector bundles over an auxiliary Riemann sphere. The method was established for four-dimensions, and in my thesis I look how it can be generalised for problems in higher-dimensionsal relativity.

Sergio Riviera: General classical spacetime

Metric spacetime is a particular and simple case of a more general structure which leaves our main classical notions untouched. These generalized spacetimes are at the boundary of what we understand as classical physics and of what can be possible and thinkable concerning a generalization of classical spacetime. The introduction of these general spacetimes requires the development of an adequate gravity theory and a refined way to introduce matter, the latter point with important consequences for the standard model of particle physics. In my talk I will introduce these generalized spacetimes and I will discuss how matter has to be introduced.

Cosmology

Mikolaj Korzynski: A geometric way to coarse-grain the flow of a fluid in general relativity

I will discuss a covariant, geometric way to assign large-scale expansion, shear and vorticity to finite, co-moving domains of a fluid in a general, no-flat sapcetime. The approach is based on the isometric embedding theorem for two-surfaces and allows for deriving the effective Raychaudhuri equation with a back reaction term explicitly present. I will show that the procedure, while non-trivial and mathematically more involved than, for example, Buchert's averaging scheme, yields the expected answer in a number of special cases and has potential applications in the study of cosmological inhomogeneities.

Algebra and Number Theory

Linda Gruendken: Different Notions of Tameness in Arithmetic Algebraic Geometry

In algebraic number theory, a (Galois) extension of number fields L/K is called tamely ramified at a prime ideal P if the ramification index e(Q/P) is relatively prime to the characteristic of the residue field for all prime ideals Q lying above P in the ring of integers O_L. The field extension itself is tamely ramified if it is tamely ramified at all primes P, and this notion of tameness is stable under composition of fields. This makes it possible to consider the maximal tamely ramified field extension of a given number field K. The purpose of my talk is to investigate and compare analogous concepts in arithmetic algebraic geometry: Given a cover of arithmetic schemes Y --> X, there are several possible generalisations of the concept of tameness of field extensions, with different functorial properties. Following and expanding on ideas introduced by G. Wiesend, M. Kerz and F. Pop, I shall present some of the properties of the resulting tame fundamental groups and show how they are related.

Jason Semeraro: p-Local Finite Groups of Rank Two

After defining the notion of a p-Local Finite Group, we will discuss the classification (due to Diaz, Ruiz and Viruel) of all such objects in the case where p is odd and the underlying p-group has rank 2. Some examples of exotic p-Local Finite Groups arise - that is those for which there is no finite group inducing all morphisms of the underlying fusion system. Time permitting we will discuss the case where p=2.

James Griffin: Wired Groups

I’ll introduce families of groups with nice algebraic and homological properties. I refer to them as “wired” because there are nice pictorial presentations in terms of blocks sliding along taut wires. These have a strong resemblance to pure braid diagrams and indeed the pure braid groups are motivating examples. When we take the homology the blocks reappear as labels on vertices of trees. I’ll give an explanation. For the majority of the talk only the basics of group theory will be required so it should be accessible.

Gary Greaves: Small-span integer symmetric matrices

Let f(x) be the characteristic polynomial of an integer symmetric matrix. Then all roots of f(x) are real, and its span is defined to be the difference between the largest root and the smallest root. I shall describe some recent classifications of cases where the span is less than or equal to 4 and talk about possible generalisations to hermitian matrices.

Maurice Chiodo: Finitely Annihilated Groups

A group is said to be Finitely Annihilated (FA) if, for each element, there is some non-trivial homomorphism to a finite group in which that element is sent to the identity. This is somewhat similar in flavour to residually finite groups; the most striking difference being that this property is preserved under reverse quotients rather than subgroups. I will introduce some equivalent definitions of FA groups, and show that in several classes a group is FA if and only if it has a non-cyclic abelianisation. I will also show that this characterisation does not hold for all groups, and finish with some comments on the decidability of being FA.

Joel Cohen: Harmonic Analysis on p-adic groups and Plancherel's Theorem

In essence, Harmonic Analysis is interested in the decomposition of functions as the superposition (sum, series, integral...) of basic ones, called harmonics, by means of inverse Fourier transforms. For example, important problems of the theory include characterizing the Fourier transform of functions (Paley-Wiener's theorem) and finding a formula for the inverse transform (Plancherel formula). This theory can be viewed in the setting of representation theory, which allows us to extend its reach beyond real analysis and the usual Fourier transform. It is of particular interest in Number Theory, where it allows us to get precious information about p-adic groups.

The talk aims at giving a brief overview of how harmonic analysis is generalized to the Representation Theory of p-adic groups and presenting Plancherel's theorem in that case. It will start with a brief introduction on Representation Theory of p-adic groups (basic theory, Hecke algebras, parabolic induction and Bernstein's center), in order to set the stage for p-adic Harmonic Analysis and Plancherel's theorem on a second part. We will specifically discuss the known versions of Plancherel's theorem (on connected reductive groups) and possible extensions (to some non-connected groups).

Robert Kurinczuk: Representations of p-adic groups

In recent years the representation theory of p-adic groups has advanced rapidly, inspired by the conjectures of Langlands. In this talk we sketch the local picture of Langlands and then focus on exploring p-adic groups via their representations.

Edward L. Sanders: p-adic Langlands Theory for algebraic tori

Langlands generalised the proofs of Class Field Theory to work for general algebraic tori. For an algebraic torus, this relates representations of the global "Weil group" in an associated complex group to representations of the adele group that occur as p-adic automorphic forms. In the talk I will then like to focus on certain classes of these automorphic forms and then giving them some p-adic geometric interpretation.

Daniel Loughran: Counting integral solutions to Diophantine equations

Given a diophantine equation with infinitely many solutions, one is interested in quantifying the "density" of said solutions. One approach is to count the solutions of bounded height. Quite often the geometry of the underlying variety gives you clues as to which methods to use, and in line with this Manin has made a conjecture on the expected formulae attained for a certain class of varieties. I shall discuss some of my work in this area for del Pezzo surfaces.

Rishi Vyas: Global Dimension and Completion

Given a ring R, one can define an homological invariant known as the global dimension of R, gdim(R). We describe this notion of dimension, and investigate how it behaves when we complete our rings with regards to a filtration. We will then specialize to Auslander Regular (AR) rings, a class of rings which behave well with respect to homological properties, and show that for AR rings with 'nice enough' filtrations, the global dimension does not change when we complete.

Strings

Paul Richmond: M2-branes and background fields

We discuss the coupling of multiple M2-branes to the background 3-form and 6-form gauge fields of eleven-dimensional supergravity, including the coupling of the Fermions. In particular we show in detail how a natural generalization of the Myers flux-terms, along with the resulting curvature of the background metric, leads to mass terms in the effective field theory.

Daniel Brattan: Charged, conformal non-relativistic hydrodyanmics

We embed a holographic model of an U(1) charged fluid with Galilean invariance in string theory and calculate its specific heat capacity and Prandtl number. Such theories are generated by a R-symmetry twist along a null direction of a N=1 superconformal theory. We study the hydrodynamic properties of such systems employing ideas from the fluid-gravity correspondence.

Rak-Kyeong Seong: An Introduction to Counting Orbifolds

D3-branes probing abelian orbifolds of C3 have been studied intensively in recent years. More recently, through the works by Bagger-Lambert, Gustavsson, and Aharony-Bergman-Jafferis-Maldecena (ABJM), this idea has been extended to cover the description of M2-branes probing certain orbifolds of C4. In this talk, I will present several methods of counting the abelian orbifolds C3/\Gamma using orbifold actions, brane tilings and toric diagrams. The methods will be extended to count orbifolds of C4, and closed form formulas will be given for the counting series.

Miguel Paulos: Holography and the fate of the viscosity bound

Strongly coupled theories with a gravity dual generically satisfy the KSS bound eta/s=1/4pi. Generically this can be perturbatively violated in string theory. In this talk we will discuss recent attempts to discuss non-perturbative violations of the bound and the (im)possibility of a newbound in the context of higher curvature theories.

Ben Crampton: The braneworld scenario

In the standard theory of compactification, the extra six or seven dimensions are curled up as some tiny compact manifold or orbifold. The braneworld scenario presents us with an intruiging alternative: we are confined to a D3 brane and the extra dimensions are large or even infinite in extent! If this is the correct model, we may see evidence of string theory at the LHC. But if gravity propogates in the bulk, why would we measure an inverse square law?

I will give a crash course in Kaluza-Klein theory, emphasising the subtlety of a consistent truncation. Then I'll introduce the Randall-Sundrum model (specifically RS2) and show that under the right circumstances the bulk propagation amplitudes for gravitational waves are exponentially suppressed, and gravity looks four dimensional. Finally, I'll look at recent work constructing more realistic braneworld-like models as consistent truncations of M-Theory.

High Energy Physics

James Barnard : Strong coupling, discrete symmetry and flavour

It is entirely possible that any new physics discovered at the LHC will be strongly coupled in nature. Moreover, strong coupling could be precisely what is required to address several of the problems in contemporary particle physics. In this talk I will demonstrate how a combination of strong coupling and simple, discrete symmetry is sufficient to give a natural explanation of the observed Standard Model flavour structure. Hierarchies between both masses and CKM matrix elements are understood in terms of a different strong coupling (or confinement) scales, which have a naturally exponential hierarchy. Meanwhile, imposing a simple Z_3 permutation symmetry results in the experimentally favoured tribimaximal mixing in the neutrino sector.

Chris Monahan : Lattice perturbation theory and the b-quark mass mess

The search for Beyond the Standard Model physics requires precise theoretical predictions of Standard Model (SM) processes to allow meaningful comparison with experimental results. Of particular interest is the study of heavy quark flavour physics, which will be explored at new experiments such as LHCb and offers the possibility of insight into new physics through loop-suppressed rare B-meson decays. Lattice QCD is the only available tool for precise calculations of nonperturbative contributions to such processes, but current lattices are too coarse to simulate heavy quarks directly. An effective field theory, NonRelativistic Quantum ChromoDynamics (NRQCD), can be used to factor out the effects of high energy scales. These effects are then absorbed into renormalised coefficients in the effective Lagrangian and the resulting theory modelled in lattice simulations. Current lattice predictions of the b-quark mass from NRQCD are constrained by the uncertainty in the heavy quark mass renormalisation, which can be calculated in lattice perturbation theory. This talk will introduce lattice perturbation theory as a tool for predicting the heavy quark mass from NRQCD and discuss some of the conceptual difficulties associated with defining and measuring quark mass.

Stephen Goatham : Fermions and Skyrmions

The Skyrme model is a nonlinear theory of pions in three spatial dimensions. After a brief introduction to the model I will describe an investigation of a background charge one Skyrme field chirally coupled to light fermions on the 3-sphere. The Dirac equation for the system commutes with a generalised angular momentum or grand spin. It can be solved explicitly for a Skyrme configuration given by the hedgehog form. I will present the energy spectrum and corresponding eigenfunctions for this system.

Olga Goulko : The Fermi Gas at Unitarity

Fermionic matter is ubiquitous in nature, from the electrons in metals and semiconductors or the neutrons in the inner crust of neutron stars, to gases of fermionic atoms that can be created and studied under laboratory conditions. Due to Fermi-Dirac statistics, a dilute system of spin-polarised fermions exhibits no interactions and can be viewed as an ideal gas. However, interactions become crucial for fermions of several spin species. An especially intriguing case is the Fermi gas interacting with divergent scattering length -- the unitary regime, in which the gas is dilute and strongly interacting at the same time.

In this talk I will explain how the unitary Fermi gas can be studied non-perturbatively with Quantum Monte Carlo methods. I will focus on calculations of the critical temperature, which is of order of the Fermi temperature and thus much higher than in the weakly interacting limit. This is not only relevant for experimental study but also promises valuable insights into high-temperature superfluidity. I will present and discuss results for equal, as well as unequal number of fermions in the two spin components. So far, most numerical studies were limited to the former case. In the latter, the so-called imbalanced case, a much richer structure of the phase diagram can be observed and new interesting effects arise.

Geometry and Topology

Oyku Yurttas: Dynnikov coordinates and pseudo-Anosov braids

Isotopy classes of orientation preserving homeomorphims on a finitely punctured disk are represented by braids. In this talk, I will present a method for computing the topological entropy of pseudo-Anosov braids using the Dynnikov coordinate system which is computationally much more efficient than the usual Thurston's train track approach. If time permits, I will talk about the relation between Dynnikov matrices and the train track transition matrix with an illustrative example that also shows the local dynamics around the unstable foliation on Teichmuller boundary.

Irida Altman: Homology Cylinders

A homology cylinder is a homology cobordism between two copies of an orientable surface. Introduced by Geroufalidis and Levine, the group of homology cobordisms of homology cylinders over a surface can be viewed as an enlargement of the mapping class group. There is a natural surjective map from the group of smooth homology cobordisms onto the group of topological homology cobordisms. It has been shown that the surjection has a ``large'' kernel: in the special case, when the surface is a disk or a sphere, this follows from the work of Fintushel, Stern, Furuta and Freedman; in the general case, this follows from the recent work of Cha, Friedl and Kim. My goal is to discuss the latter.

Will Merry: On the Rabinowitz Floer homology of twisted cotangent bundles

Rabinowitz Floer homology is the semi-infinite dimensional Morse homology associated to the Rabinowitz action functional, and can be viewed as a Floer type Lagrange multiplier problem. In this talk I will describe the construction of Rabinowitz Floer homology and its applications to the study of the symplectic topology of energy hypersurfaces ${H=k}$ of mechanical Hamiltonians $H$ on twisted cotangent bundles.

In particular, I will show how Rabinowitz Floer homology can be used to prove that when the energy value k is greater than the Man\'e critical value, such an energy hypersurface ${H=k}$ is never displaceable.

Richard Harris: Exotic Symplectic Manifolds

I shall begin by introducing the theory of Lefschetz fibrations, a very useful tool for studying symplectic manifolds, before outlining Maydanskiy's recent construction of certain exotic symplectic manifolds using these techniques, and if time permits my attempts to extend his results. The talk should be reasonably pictorial with most of the necessary technical machinery from symplectic topology being treated as a black box.

David Witt Nystrom: Test configurations and Okounkov bodies

In the mid-nineties Andrei Okounkov found a way of associating convex bodies, called Okounkov bodies, to holomorphic line bundles on Kähler manifolds, which generalizes the association of a toric variety with its moment polytope. Okounkov bodies has since then mainly been used to investigate the fundamental birational invariant of a line bundle called its volume.

A big open problem in Kähler geometry is the characterization of all line bundles whose first Chern class contains a a constant scalar curvature metric. The Yau-Tian-Donaldson conjecture says that. that a line bundle has a constant scalar curvature metric iff it is K-stable. In formulating the condition of K-stability, one uses the concept of a test configuration, which was introduced by Simon Donaldson. Test configurations can be thought of as generalizing holomorphic C^*-actions of the line bundle. Donaldson has showed that certain symmetric test configurations of a toric line bundle can be encoded by a concave functions on the associated moment polytope, thereby translating the K-stability condition to convex geometry/analysis.

I will descibe how a test configuration on an arbitrary line bundle gives rise to a concave function on the Okounkov body, thus generalizing the toric case.

Chris Braun: Topological Quantum Field Theories and Moduli Spaces of Klein Surfaces

The ribbon graph decomposition of moduli space gives orbi-cell complexes homotopy equivalent to moduli spaces of Riemann surfaces. Ribbon graphs are related to the A-infinity operad which gives a dual interpretation of this in terms of topological conformal field theory. This is an enlightening point of view and I will explain how it works, starting from the (abstract) definition of a TQFT, with lots of pictures. I will show how considering this point of view for unoriented manifolds leads naturally to a Möbius graph decomposition for moduli spaces of Klein surfaces (real algebraic curves). In the unoriented case we are also led to consider a partial compactification given by certain spaces of stable symmetric Riemann surfaces (which are closer in spirit to the Deligne-Mumford moduli spaces) having another quite different graph decomposition.

Ana Lucia Gardia Pulido: String Topology and Graded Complete Intersection Algebras

In this talk I will briefly introduce the main concepts of the String Topology, its properties and relations with algebra. I'll explain what a Graded Complete Intersection (GCI) algebra is and for manifolds whose cohomology ring is a GCI algebra I'll essentially show how to compute its String Topology.

Thomas Wieber: A result on vector valued modular forms derived from differential geometry

We want to estimate the dimension of a certain subspace of vector valued modular forms for Igusa's congruence group $\Gamma_2[2,4]$. We will first discuss modular forms to see that the above modular forms can be viewed as $\Gamma_2[2,4]$-invariant tensors. By a tricky construction they are isomorphic to rational tensors on the projective plane with some additional properties.

Michael Walter: Geometric K-homology

K-homology, the dual of K-theory, is the generalised homology theory of elliptic operators (in an abstract sense) over (not necessarily commutative) spaces. In my talk, I will introduce a geometric picture due to Baum and Douglas, inspired by singular bordism, and sketch a recent proof of its equivalence.

Along the way, we will meet Kasparov's bivariant KK-theory, which provides a powerful unified language for the above, and illuminate the role of Clifford algebras in connecting geometry and analysis. I will attempt to describe how most geometrical and topological constructions have analytical counterparts that can be elegantly expressed in Kasparov's theory and yield more general results that way (e.g. the Bott element implementing the famous periodicity theorem). Equivalence is then merely a matter of happily translating back and forth between both pictures.

Ravi Shroff: An Introduction to Rigidity in CR Geometry

We will introduce the idea of a CR manifold, modelled after a real hypersurface embedded in complex space. Two interesting questions in CR geometry are: when do there locally exist embeddings from one manifold to another, and how are they related. We will discuss the latter question, and how such rigidity is related to the intrinsic geometry of the source and target manifold.

Applicable Mathematics

Yilei Hu: Evolution of natural language: a stochastic model

We study a generalized version of the signaling process originally introduced and studied by Argiento, Pemantle, Skyrms and Volkov(2009), which models how two interacting agents learn to signal each other and thus create a common language. We show that the process asymptotically leads to the emergence of a graph of connections between signals and states which has the property that no signal-state correspondance could be associated both to a synonym and an informational bottleneck.

Matas Šileikis: Finite set combinatorics yields optimal probability inequalities

Consider a sum S of n independent random variables. Exponential bounds such as ones for the tail probability P(S>x) is already an established subfield of probability theory. The famous Hoeffding inequality (1963) gives the best possible bound (assuming the summands are bounded) that can be obtained by moment generating function method. However, obtaining optimal bounds for P(S>x), i.e., a supremum over the class of random variables in question, is in general a hard problem. Essentially no solutions for general n exist to date. We solve the problem under a natural additional condition that the summands are symmetric. The problem then reduces to the class of linear combinations of Rademacher random variables (fair coin tosses). In addition to our straighforward solution, we deduce it from Katona's intersection theorem on maximal t-intersecting families of subsets of {1,...,n}. This deduction is in spirit close to the way Erdos (1945) solved Littlewood-Offord problem using Sperners lemma on maximal antichain of subsets of {1,...,n}. Ironically, Littlewood-Offord problem can as well be solved circumventing combinatorics.

Sylvestre Burgos: Computing Greeks using Monte Carlo simulations

Monte Carlo simulations are popular methods in computational finance. They provide powerful and efficient techniques for the pricing of financial derivatives. My research focuses on the computation of the so-called Greeks, i.e. the sensitivities of derivatives to market parameters such as interest rates or market volatility. A proper estimation of those values is crucial for hedging financial portfolios against risk.

A naive implementation of classic Monte Carlo techniques [Gla04] for the computation of Greeks is often very costly and may not even be possible in the case of derivatives with non-differentiable payoffs. We will explore new algorithms to get around those limitations.

Differentiability-related problems can be addressed in three different ways:

smoothing the payoff using conditional expectations before maturity [Gla04].
using a hybrid combination of pathwise sensitivity and the likelihood ratio method, the Vibrato Monte Carlo [Gil09].

We also investigate the use of the recent multilevel Monte Carlo [Gil08] method to reduce the computational complexity of Greeks’ computations. We will discuss the strengths and weaknesses of the aforementioned alternatives in a multilevel Monte Carlo setting.

References

[Gil08] M.B. Giles. Multilevel Monte Carlo path simulation. Operations Research, 56(3):607–617, 2008.
[Gil09] M.B. Giles. Vibrato Monte Carlo sensitivities. In P. L’Ecuyer and A. Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008, pages 369–382. Springer, 2009.
[Gla04] P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, New York, 2004.

Lifeng Gao: Sampling Importance Re-sampling for Bayesian Dynamic Models

The dynamic linear models (DLMs), one of the most commonly used time series models in applications, have been investigated from different viewpoints and methodologies. However, the current stufies are almost concentrated on some special restrictions, such as normal priors or linear transfer functions, which limit the DLMs to some narrow situations. An approach is outlined here which exerts the sampling importance re-sampling (SIR) to yield all samples we need. Using these samples one can make the posterior inferences, the observation predictions, and model selection through simulations, etc. Furthermore, our methods may be easily generalized to be used in nonlinear dynamic models (NDMs), which can be applied in more wide situations.

Analysis and Partial Differential Equations

Daniele Garrisi: A splitting property for two-variable functions

In the literature, there are many results concerned with the behaviour of an arbitrary minimizing sequence of a continuously differentiable function on H^1 (R^n). The kind of sequence we hope to deal with is either converging or can be obtained from a converging sequence by means of translations of the argument. We show how it is possible to extend such conclusions to two-variable H^1 (R^n;R^2) functions. Such properties are crucial to prove that certain pairs of standing waves, solutions of a non-linear wave system, are orbitally stable.

Zunwei Fu: Commutators of weighted Hardy operators

In this talk, we establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO$(\mathbb{R}^{n})$) on $L^{p}(\mathbb{R}^n)$, where $1 < p < \infty$.

Francisco Javier Suárez-Grau: Homogenization of the Navier-Stokes system posed in periodic rugose domains

The goal of this work is to study the asymptotic behavior of partial differential systems in rugous domains. More precisely we are interested in the case of Navier-Stokes system.

A well-accepted hypothesis in the fluid dynamics is that if the boundary $\Gamma_\varepsilon$ of the physical domain is impermeable then the viscous fluid adheres completely to it. On the contrary, we assume that the fluid satisfies the slip condition given by Navier's law $$u_\varepsilon\cdot n_\varepsilon=0,\quad (Du_\varepsilon\cdot n_\varepsilon)_{tan}+\gamma u_\varepsilon=0\quad \hbox{ on } \Gamma_\varepsilon,$$ where $u_\varepsilon$ is the velocity, $n_\varepsilon$ is the normal vector to $\Gamma_\varepsilon$, $\gamma$ is the friction coefficient and the subscript $_{tan}$ denotes the tangential component.

Assuming $\Gamma_\varepsilon$ periodic with period $\varepsilon$ and amplitude $\delta_\varepsilon$ we show that if $\delta_\varepsilon>>\varepsilon^{3/2}$ then Navier's condition implies adherence condition in the limit. This justifies the use of the adherence condition in fluid dynamics. It doesn't happen the same in the cases $\delta_\varepsilon\sim\varepsilon^{3/2}$ and $\delta_\varepsilon < < \varepsilon^{3/2}$.

Moreover, we extend these results for PDE systems (not necessarily in fluid mechanics) with rugose boundaries, not necessarily periodic.

Felipe Rivero: Evolution processes: Results and applications for pullback attractors

In the framework of autonomous dynamical systems, the concept of global attractors plays an important role in the study of their asymptotic behaviour. However in the non-autonomous context it is necessary to extend this theory for evolution processes and pullback attractors. Our aim in this talk is to show some results, introducing new concepts and bringing up some important differences between the asymptotic properties of semigroups and evolution processes. We give non-trivial examples possessing pullback attractors with concrete structures.

Filip Rindler: Characterization of generalized gradient Young measures in W^{1,1} and BV

Generalized Young measures (or DiPerna/Majda measures) are an extension of classical Young measures that are able to quantitatively account for both oscillation and concentration phenomena in L1-bounded sequences of functions or in a sequence of Radon measures (uniformly bounded in the total variation norm). After explaining the basic framework in a functional-analytic spirit, the main result of this work is presented: A characterization of the class of generalized Young measures that can be generated by W^{1,1}-gradients or BV-derivatives. This characterization is in terms of Jensen-type inequalities for quasiconvex functions and can be considered a natural generalization of the Kinderlehrer-Pedregal Theorem for classical Young measures. This is joint work with Jan Kristensen.

Emil Wiedermann: Measure-Valued Solutions of the Euler Equations

The incompressible Euler equations model the flow of an ideal incompressible fluid. They are a system of nonlinear partial differential equations and have been, like the related Navier-Stokes equations, of great interest to mathematicians, physicists and engineers throughout several centuries. Nevertheless our rigorous mathematical knowledge about the Euler equations is still very restricted. For example, it is not known (not even for Navier-Stokes) whether for smooth initial data there exists a smooth solution for all times (this is, in fact, one of the Millennium Problems). Several attempts have therefore been made to weaken the notion of solution so that at least the existence of a very weak form of solution could be proved. One such concept, which I will focus on in my talk, was developed in the 80's by DiPerna and Majda and uses so-called Young measures. In my talk I will present some recent results, and work in progress, on these measure-valued solutions.

María Anguiano: Pullback attractors for reaction-diffusion equations

Several aspects of reaction-diffusion equations have been analyzed over the last years, particularly, their asymptotic behaviour.

The study of reaction-diffusion equations in a bounded domain in the autonomous case, or in the non-autonomous one but with strong uniformity properties on the time-dependent terms, can be found in the literature.

Our aim is to analyse a more general problem. Namely, let us consider the following problem for a non-autonomous reaction-diffusion equation with zero Dirichlet boundary condition in $\Omega$, $$ \left\{ \begin{array} [c]{l} \displaystyle\frac{\partial u}{\partial t}-\bigtriangleup u=f(u)+h(t)\text{ \ in \ }\Omega\times(\tau,+\infty)\text{,}\\ u=0\text{ \ on \ }\partial\Omega\times(\tau,+\infty)\text{,}\\ u(x,\tau)=u_{\tau}(x),\text{ \ }x\in\Omega\text{,}% \end{array} \right. $$ where $\Omega\subset\mathbb{R}^{N}$ is a nonempty open set, not necessarily bounded, and suppose that $\Omega$ satisfies the Poincaré inequality. The non-autonomous forcing term $h$ takes values in the space $H^{-1}$, and the assumptions on $f$ ensure uniqueness of solutions.

We will use the pullback theory for non-autonomous dynamical systems, since this allows for more generality in the non-autonomous terms, to prove the existence of a pullback attractor.

Donal Connolly: Pseudo Differential Operators on Compact Lie Group

Pseudo differential operators are an extension of linear partial differential operators and have roots deeply entwined in the solution of differential equations. The basic idea in pseudo differential operator theory is to represent an operator as a multiplication operator in phase space.

In this talk we will begin by studying classical results in the Euclidean and Toroidal setting. We will then survey analogues of these results in the setting of compact Lie groups.

Combinatorics and Analytic Number Theory

Konrad Dabrowski: Fixed Parameter Tractability and The Independent Set Problem

Many algorithmic graph problems are hard to solve in general. However, for many of these problems, all is not lost. Sometimes we know some extra information about the structure of the graphs we want to solve the problem for. We introduce a parameter, which encodes some of this information. Often, we can come up with algorithms where the "non-polynomial behaviour" is somehow restricted by this parameter.

We will give some examples of graph problems where this approach works, in particular looking at techniques to find large induced independent sets in a graph (N.B This is the same as looking for large cliques in the complement of the graph). The talk will focus on graph-theoretic proofs and techniques and will be very light on technical details of algorithms.

Chris Dowden: Random planar graphs with n vertices and m edges

Let P_{n,m} denote a graph taken uniformly at random from the set of all labelled planar graphs with n vertices and m(n) edges. We shall use elementary counting arguments to investigate the probability that P_{n,m} has components/subgraphs isomorphic to H, for various fixed H, as n tends to infinity. We will provide a fairly complete picture of exactly when the probability is bounded away from 0 and/or 1, showing that there is different behaviour depending on both the graph H and the ratio m/n.

Richard Mycroft: A proof of Sumner's universal tournament conjecture for large n

A tournament is an orientation of a complete graph. Sumner conjectured in 1971 that any tournament $G$ on $2n-2$ vertices contains any directed tree $T$ on $n$ vertices. Taking $G$ to be a regular tournament on $2n-3$ vertices and $T$ to be an outstar shows that this conjecture, if true, is best possible. Many partial results have been obtained towards this conjecture.

In this talk I shall outline how a randomised embedding algorithm can be used to prove an approximate version of Sumner's conjecture, by first proving a stronger result for the case when $T$ has bounded maximum degree. Furthermore, I will demonstrate that by considering the extremal cases of this proof we may deduce that Sumner's conjecture holds for all sufficiently large $n$.

This is joint work with Daniela K\"uhn and Deryk Osthus.

Alexey Pokrovskiy: Growth of graph powers

For a graph G its rth power is constructed by placing an edge between two vertices if they are within distance r of each other. How many new edges are added to a graph by taking its rth power? It turns out that under certain conditions, the rth power is either complete or "many" new edges are added. The motivation for studying this comes from the Cauchy-Davenport theorem which implies a similar result for Cayley graphs of cyclic groups.

Matas Sileikis: Finite set combinatorics yields optimal probability inequalities

Lutz Warnke: On the Evolution of K_k-free graphs

Kolaitis, Prömel and Rothschild proved in 1986 that a random K_k-free graph on n vertices is almost surely (k-1)-colorable, extending a previous result by Erdös, Kleitman and Rothschild. In this talk we consider the question whether a similar result holds for random K_k-free graphs on n vertices with a prescribed number m=m(n) of edges. We show that the evolution of random K_k-free graphs exhibits two phase transitions with respect to being (k-1)-colorable as m increases from 0 to n^2: first it is almost surely (k-1)-colorable, then it is not, and then it is once again. In particular, we prove that there is a sharp threshold at t_k(n) = c_k n^{2-2/(k+1)} (log n)^{2/(k^2-k-2)}. Our results hold provided the so-called KLR-Conjecture is true for K_k. The latter has been verified for K_3, K_4, and K_5.

Joint work with Angelika Steger.

Ping Zhao: Colouring theory of mixed hypergraphs

This paper is a introduction on the coloring theory of mixed hypergraphs.

A mixed hypergraph ${\cal H}=(X, {\cal C}, {\cal D})$ is a triple system on $X$, where $X$ is a finite set, ${\cal C}$ and ${\cal D}$ are both families of subsets of $X$, ${\cal C}$ (resp. ${\cal D}$) is called the $C$-edge set (resp. $D$-edge set) of ${\cal H}$, every $C\in {\cal C}$ (resp. $D\in {\cal D}$) is called a $C$-edge (resp. $D$-edge). The difference between $C$-edges and $D$-edges is in the requirements for a coloring. In a strict coloring, each $C$-edge has at least two vertices with a Common color, each $D$-edge has at least two vertices with Different colors, in this way, "C" stands for "Common" color and "D" stands for "Different" colors. In a mixed hypergraph it makes sense to ask for both the minimum and the maximum numbers of colors required for a strict coloring. The maximum (minimum) integer $t$ for which there exists a strict $t$-coloring of a mixed hypergraph ${\cal H}$ is called the \it upper (lower) chromatic number \rm of ${\cal H}$ and denoted by $\overline \chi ({\cal H})$ ($\chi ({\cal H})$). Therefore the classical coloring theory of hypergraphs becomes a special case, i.e., the case of mixed hypergraphs with only $D$-edges and lower chromatic numbers.

V. Voloshin initiated this new theory in 1992, and the first paper about this theory was published in "Discrete Mathematics" in 1995. There are many problems waiting for us to investigate. At present, the study on this theory are mainly on the following directions: colorability; uniquely colorable mixed hypergraphs; $C$-perfectness of mixed hypergrphs; colorings of block designs; colorings of $C$-hypergraphs; integer programming and fractional colorings of mixed hypergraphs; colorings of planar mixed hypergraphs, etc. In this paper, we mainly introduce colorings of $C$-hypergraphs, colorings of Steiner systems, or more generally, colorings of block designs.

Applied Mathematics

Alireza Yazdani: The Mathematics of Water Distribution Networks

Water distribution networks (WDNs) are usually modelled by graphs in which the pipes are represented by graph edges and the hydraulic junctions and pipe-intersections by graph nodes. Improving reliability and robustness in a cost-effective fashion is one of the important challenges to the water industry researchers and engineers during design, maintenance and rehabilitation of the networks. Reliability (the probability of non-failure) in water distribution systems is expressed as the likelihood that the system will meet its objectives for water demand, for a specifed period of time. Failures in this context are two types: (i) Physical component failures (e.g. pipe breakages) and (ii) Low hydraulic fow and pressure. Quantitative analysis of the cost-effective reliability in WDNs with respect to these failures, takes place by extensive simulation processes that involve minimization of the cost as a function of the pipe sizes, subject to constraints of hydraulic fow and pressure. The graph representation, on the other hand, gives rise to the study of these systems in the context of planar graphs with complex network characteristics consisting of multiple interconnected elements. From a topological point of view, the likelihood and impact of the failures in the networks, is dependent on the layout and strength of the connections (robustness) as well as availability of the loops which are formed by the alternative supply pipes with suffcient capacity (redundancy). In other words, by studying the structure of cycles in the network and measuring cohesion and other connectivity aspects, it is possible to compare different network layouts and comment on their vulnerability against failures. In particular, the spectrum of graph adjacency matrix in addition to the building blocks and statistical properties of the networks reveal invaluable information on robustness and the degree of redundancy in the network. Furthermore, critical locations such as bridges and bottlenecks are identifed and the impacts of the single or multiple failures in water distribution networks are analysed.

Axel Kroener: Semi-Smooth Newton Methods for Optimal Boundary Control of Wave Equations

In this talk numerical methods for solving PDE constrained optimization problems are considered.

Optimal Neumann and Dirichlet boundary control problems governed by wave equations with control constraints are analyzed. For treating inequality constraints semi–smooth Newton methods are discussed and their convergence properties are investigated. For numerical realization a space-time fnite element discretization is introduced. Numerical examples illustrate the results.

Florian Rupp: On Stochastic Differential Equation Models for Epidemics

These days the spread of epidemics, viz h1n1 influenca, is broadly discussed. In this talk we will summarize key results on the behavior of solutions of epidemic models given by stochastic differential equations (SDEs). We will discuss specific ways to derive these models from discrete considerations and well-known ordinary differential equation (ODE) models. The main part will be devoted to the analysis of the differences between corresponding ODE and SDE models for diseases. Thereby, we will study the issues of persistence, coexistence of multiple infections or total elimination of an epidemic. For instance, we will show that environmental noise can – in contrast to the results of the corresponding ODE model – lead to a stabilization of the infection and thus prohibit (deterministic) blow-up in finite time.

Andreas Klaiber: Internal Waves in Stratified Fluids

We are interested in studying so-called internal waves which travel in the interior of lakes or oceans, as opposed to the surface waves being visible to the naked eye.

To this end, we consider an inviscid incompressible density-stratified fluid within an infinitely long channel of finite depth. The motion, which we assume to be two-dimensional, is governed by the Euler equations and it is well known that, under pertinent assumptions, this system of coupled partial differential equations can be approximated by two scalar equations, one of which is given by the famous Korteweg-de Vries (KdV) model.

The KdV equation has been investigated comprehensively by mathematicians in the 20th century; we review some of the important properties and results related to the existence and behaviour of KdV waves. With the help of these results, many observations and measurements of oceanographers and limnologists have been explained to high precision, and for that reason one naturally expects the original Euler model to contain these waves, too. Indeed, to some extent this has already proven to be true, as publications from the 90s show.

We present work of K. Kirchgässner, K. Lankers, and others who recasted the travelling wave problem as a dynamical system in a function space and showed the existence of periodic and solitary waves for the Euler equations.

Mathematical Biology

Nafiu Hussaini: Role of public health education campaign on HIV transmission dynamics

In this talk I will present a nonlinear deterministic model for assessing the impact of public health education campaign on curtailing the spread of the HIV pandemic in a population. Rigorous qualitative analysis of the model, which includes the phenomenon of backward bifurcation and its epidemiological implication, will be discussed. Furthermore, the model will be used to assess the potential impact of some targeted public health education campaigns using data from numerous countries.

This is joint work with A. Gumel (Manitoba, Canada) and M. Winter (Brunel University, UK).

Jasmina Panovska-Griffiths: Mathematics for Biology or Biology for Mathematics: application to gene transcription networks

Morphogens are secreted signalling molecules that form long-range concentration gradients over feld of cells and elicit different responses depending on the duration and signalling doses received by the target cells. Specifcally, in the vertebrate central nervous system and limbs, both the pattern of cellular differentiation and the neuronal fate are controlled by the duration and the concentration of the morphogen Sonic hedgehog (Shh). In particular graded Shh signalling controls the identity of the neural cell type by regulating the expression of a series of transcription factors, e.g. the homeobox proteins Pax6 and Nkx2.2 and the basic helix-loop-helix protein Olig2.

In this paper we formulate a mathematical model to explore the mechanisms by which Shh concentration-dependent induction of Pax6, Olig2 and Nkx2.2, their cross-repression and the noise in the network due to extrinsic signal and/or intrinsic cellular fuctuations can explain the progenitor patterning in the vertebrate neural tube. Numerical simulations and stability analysis suggest that the cross-repressive interactions between the three transcription factors and the asymmetries in the strength of repressive connections in the circuit are able to produce a ‘French Flag’ morphogen-like response. Furthermore, the feedback generates switch-like changes in the expression of the transcription factors that respond to both changes in the duration and level of signaling. Strikingly the temporal response is independent of the Shh concentration and instead is controlled by the cross-repression between Olig2 and Nkx2.2. We use the model to predict consequences for Nkx2.2 expression of removing either Pax6 or Olig2 from the system or changing the strength of repression between the transcription factors. We also demonstrate that the system exhibits hysteresis to Shh concentration, the extent of which is dependent on the strength of cross-repressive interactions between Olig2 and Pax6. Furthermore, we identify the key mechanisms that can buffer the noise in the gene expression as well as describe the boundary precision between progenitors types.

Paul Kirk: Modelling biology... despite the noise

The modern biological and life sciences have given rise to a host of novel challenges for applied mathematicians and statisticians. We are routinely presented with datasets comprising measurements on hundreds or thousands of different interacting variables, and seek to propose useful models and make meaningful inferences about the complex biological processes that generated them. The high throughput experimental technologies used to collect these data have been both a blessing and a curse: without them, we would stand little chance of being able to elucidate the full complexity of biological systems, yet the datasets they generate are frequently beset by problems of high noise, systematic error and missing measurements. In order to have any confidence about the models that we fit to these data or the inferences that we draw from them, the development of generic mathematical methods for quantifying (and, as much as possible, reducing) the effects of these problems is vital. We here consider this problem in the context of estimating the parameters of ordinary differential equation (ODE) models of cell signalling pathways, and assessing our confidence in molecular biomarkers identified from mass spectral proteomic data. The techniques that we develop make use of ideas from Bayesian nonparametrics, as well as frequentist bootstrapping approaches. We apply our methods to a number of real experimental datasets and demonstrate that taking into account the effects of data limitations can have important consequences upon the conclusions that we draw.

William Kelly: Statistical analysis of protein interaction networks

Protein interaction networks describe the reported protein interactions found in an organism, and their organisation is of interest throughout biology. The amount of interaction data has expanded dramatically since the advent of high-throughput experimental technologies. However, interaction data are in general believed to be noisy and incorporating other biological knowledge will provide more reliable information. We focus on data from baker's yeast (S. cerevisiae) and present an overview of its important biological and network characteristics.

Phylogenetic trees of S. cerevisiae proteins are compared in order to assess possible evolutionary linkage of protein-protein interactions. The comparisons are made across a variety of different network null models to assess their importance.

A (capture-recapture) model is also presented which can help to find the interactome size and false discovery rate for reported protein interaction data.

Algebraic Geometry

Hamid Ahmadinezhad: Cox rings and del Pezzo fibrations

Traditionally there are two ways of studying the geometry of an algebraic variety. One by looking at the embeddings of the variety and the other by studying the embedded varieties into it. Minimal model program of Mori gives a birational classification of threefold varieties by studying the rational curves inside the variety and classifies them into varieties of general type, Calabi--Yau varieties and Mori fibre spaces. The uniqueness (rigidity) of a Mori fibre space in some classes remains open. On the other hand, Cox's construction of toric varieties allows one to study toric varieties in the language of GIT. In this talk we look at some families of Mori fibre spaces embedded into such toric varieties and study their birational geometry by comparing with the one of the ambient space. The main ideas will be illustrated by examples of threefold fibrations embedded into rank two Cox varieties. The birational geometry of the ambient space in these cases induces a particularly nice birational link from the initial variety to a different minimal model.

Will Donovan: Flops and derived categories

Flops are a special sort of codimension 2 operation which can be performed on a variety, yielding a new 'flopped' variety. I will explain a straightforward example known as the standard flop using a local model, and we will see how this flop can naturally induce a functor on the associated derived categories of coherent sheaves. When we focus on the standard line bundles on the model, we find a 'window' of line bundles within which the functor acts in a very simple manner, and we can use this to show that it is in fact an equivalence of derived categories. If there is time, I will explain how I am trying to generalize these ideas to more complicated flops, and the equivalences associated with them.

Matthew Fletcher: The resolution of the adjoint orbits of real semisimple Lie algebras

The set of adjoint orbits of a real semisimple Lie algebra form a foliation of the algebra. This foliation will be singular. For a compact Lie algebra we show that a single Nash blow-up resolves the foliation. The Nash blow-up is dominated by the Grothendieck resolution. We will then briefly explore the noncompact case.

David Holmes: Heights and Arakelov intersection theory.

Given a curve on a regular surface, the Gysin map sends cycles on the> surface to cycles on the curve, in particular we can pull back the curve to cycle on itself. The local height of an integral point on an elliptic curve over a prime of good reduction is zero. We relate these statements via Arakelov intersection theory.

Julian Holstein: Schematic Homotopy Type and the Fundamental Group

This talk will give a brief introduction to a model Bertrand Toen suggested for associating to a topological space a schematic homotopy type, an idea going back to Grothendieck. The crucial building blocks for these homotopy types are Eilenberg-Mac Lane spaces and in order to understand the homotopy type of these we will conisder completions of the fundamental group.

Markus Severitt: Natural Bundles of Frobenius Twists

Let $k$ be a field of prime characteristic $p$ and $X$ a smooth variety of dimension $n$. Then the relative Frobenius morphism is a fppf-fiber bundle with fibers $k[x_1,\ldots,x_n]/(x_1^p,\ldots,x_n^p)$. So we can consider the automorphism group $G_n$ of this fiber. Now we can associate to each $G_n$-representation a vector bundle over the Frobenius twist of $X$. We call the vector bundles obtained in this way \emph{natural bundles}. Two examples of natural bundles are given by the pushforward and pullback of the cotangent bundle of $X$ over its Frobenius twist by the relative Frobenius and by the arithmetic Frobenius, respectively. The aim of the talk is to show that these two examples essentially provide all natural bundles. Finally, we will indicate how this generalizes to $r$-th Frobenius morphisms for $r\geq2$.

Marcus Zwibrowius: Stiefel-Whitney classes in algebraic geometry

In topology, Stiefel-Whitney classes are the real counterparts of Chern classes: universal cohomological invariants attached to real vector bundles. In algebraic geometry, the notion of a real vector bundle has to be replaced by that of a vector bundle equipped with a quadratic form. The theory of quadratic forms is interesting even over a point (a field), and Milnor constructed Stiefel-Whitney invariants in this context. Over more general varieties one can construct Stiefel-Whitney classes that are the exact analogues of their topological incarnations using étale cohomology.

Mathematical Physics

Neville Boon: Mathematical Methods for Modeling Myosin-V Processivity and the Framework for the Comparison of Stepping Models

A molecular motor is a nano-scale protein which converts chemical energy, typically obtained from the hydrolysis of adenosine triphosphate (ATP), into mechanical work. Myosin-V is a double headed processive molecular motor which transports a variety of cargos within cells. It achieves this by walking head-over-head along an actin filament, passing through a sequence of biochemical reactions and mechanical motions, taking several successive steps before detaching.

The exact nature of the stepping mechanism of myosin-V remains unresolved due to the noise to which nano-scale measurements are subject. However, some experimental quantities can be observed. Average velocities and run lengths of the molecules have been measured against varying cellular conditions and single molecule experiments have shown that in one stepping cycle, myosin-V can take at least one large step and perhaps also a smaller step. Our work focuses on theoretical methods to extract more information - such as the stepping mechanism - from these experimental results.

A particular stepping mechanism can be encoded into a discrete stochastic model describing the processes a molecule undergoes in taking one step. Several possible different versions have previously been proposed, each encoding a different sequence of biochemical and mechanical processes. Run lengths, velocities and dwell times can be calculated for different chemical concentrations and compared against experiment.

Mathematical methods which use experimental evidence to determine the parameters of such models are discussed. Probable energetic relationships are postulated through agreement across different models. Methods for assessing the degree to which each model can reproduce the experimental evidence are presented. A quantitative analysis of competing stepping cycles and their agreement with experimental results is also discussed.

Sebastian Schmittner: Supersymmetry Methods in Condensed Matter Physics: How Fermionic Integrals can help treating Disordered Bosons

I will explain how Supersymmetry Methods, in particular Superbosonization, can be used to calculate the Greens function of a toy model for a generic disordered Bosonic system. If time permits, I will also outline how the results can be generalized to obtain a Field Theory of disordered Bosons, which is the aim of my current Diploma work. No previous knowledge of Supersymmetry will be assumed.

Yanhua Wang: Theoretical Design of Molecular Photonic Materials

The nonmonotonic behavior of the solvatochromic shifts and the first hyperpolarizability of para-nitroaniline (pNA) with respect to the polarity of the solvents have been theoretically confirmed for the first time. The significant contributions of the hydrogen bonding on the electronic structures of pNA are revealed. Vibrational contributions to the linear and nonlinear polarizabilities of methanol, ethanol and propanol have been calculated both at the static limit and in dynamic optical processes. The importance of vibrational contributions to certain nonlinear optical processes have been demonstrated. A series of end-capped triply branched dendritic chromophores have been studied with the result that their second order nonlinear optical properties are found strongly dependent on the mutual orientations of the three chromophores, numbers of caps and the conjugation length of the chromophores. Several possible mechanisms for the origin of the Q-band splitting of aluminum phthalocyanine chloride have been examined. Calculated vibronic one-photon absorption profiles of two molecular systems are found to be in very good agreement with the corresponding experiments, allowing to provide proper assignments for different spectral features. Furthermore, effects of vibronic coupling in the nonradiative decay processes have been considered which helps to understand the aggregation enhanced luminescence of silole molecules. The study of molecular aggregation effects on two-photon absorption cross sections of octupolar molecules has highlighted the need to use a hybrid method that combines density functional response theory and molecular dynamics simulations for the design of molecular materials.

Adrian Hemery: 1-D periodic Schrödinger operators: What does "semifinite-gap" actually mean?

My talk is on the deep and elegant subject of 1-D periodic Schrödinger operators. I will introduce the case of finite-gap potentials using the example of the famous Lame operator and generalise this notion to a new class of "semifinite-gap" potentials. A trigonometric example of this is the Whittaker-Hill operator, whose particularly special properties I will explain and how it is possible to use Darboux transformations to construct many more examples of non-singular trigonometric potentials with the same properties.

George Khachatryan: Representation Varieties

It is possible to define an affine variety parameterizing all of the n-dimensional representations of a finitely generated associative algebra. Representation varieties have been studied for decades, and there are many interesting connections between the representation theory of the algebra and the geometry of its representation varieties. More recently, representation varieties have been used in non-commutative algebraic geometry, with an influential philosophy of Kontsevich and Rosenberg suggesting that the representation varieties form a sort of approximation of the "non-commutative geometry" of an associative algebra.

Foundations

Damien Servais: From Stratification to Ambiguity

We first describe structures of the multi-sorted language of set theory to study stratified formulae. Next, we focus on constructions of models of first order theories based on stratified comprehension. Then, we introduce the notion of ambiguity, which allow us to consider some consistency results of such theories.

Andrew Swan: Realizability, Automorphisms, and AC

Realizability is a way of providing semantics for intuitionist theories using ideas from computability theory. For example, we should be able to use a realizer of the formula A \rightarrow B, to "compute" realizers of B from realizers of A. A particular class of algebraic structures, partial combinatory algebras (pcas), can be used to provide a suitable abstract notion of computability. In this talk I will show a definition of realizability due to Myhill, McCarthy, that uses pcas to define realizability models of IZF (essentially ZF but with only intuitionist logic). This definition is similar to the notion of forcing, famously used by Cohen to show the independence of CH and AC from ZF. I will show that in fact, by using pcas with enough automorphisms, we can find a new proof of the independence of countable choice from IZF that uses realizability rather than forcing.

Stijn Vermeeren: The Entscheidungsproblem

I will talk about the Entscheidungsproblem: does there exist an algorithm that decides whether a given sentence of first order logic is universally valid or not? I will start by discussing the history of the Entscheidungsproblem and its place the mathematics of the early 20th century. This will bring me to the negative solutions by Turing and Church from 1936: there does not exist such an algorithm. It might seem that that's the end of it, but on the contrary, many new related questions came up. For example: if one restricts attention to sentences of first order logic (with equality) containing only one unary function symbol and no relation symbols, is there still no algorithm to decide validity? This particular question was solved in 1969 by Rabin and the proof is very interesting in its own right.