Past

I'll explain (following Kontsevich and Soibelman) how cluster transformations intertwine non-commutative DT invariants for CY3 algebras related by a tilt.

We have recently found a way to describe large-scale neural

activity in terms of non-equilibrium statistical mechanics.

This allows us to calculate perturbatively the effects of

fluctuations and correlations on neural activity. Major results

of this formulation include a role for critical branching, and

the demonstration that there exist non-equilibrium phase

transitions in neocortical activity which are in the same

universality class as directed percolation. This result leads

to explanations for the origin of many of the scaling laws

found in LFP, EEG, fMRI, and in ISI distributions, and

provides a possible explanation for the origin of various brain

waves. It also leads to ways of calculating how correlations

can affect neocortical activity, and therefore provides a new

tool for investigating the connections between neural

dynamics, cognition and behavior

In this talk, Professor Lions will first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving partial differential equations and he will briefly explain the nature, history and role of such equations. Examples showing the importance of the mathematical analysis (i.e. ‘understanding’) of those models will be presented, concluding with a few trends and perspectives.

Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself become a renowned mathematician, making numerous important contributions to the theory of non-linear partial differential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPerna giving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Other awards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently he holds the position of Chair of Partial Differential Equations and their Applications at the prestigious Collège de France in Paris.

This lecture is given as part of the 7th ISAAC Congress • www.isaac2009.org

Clore Lecture Theatre, Huxley Building, Imperial College London,
South Kensington Campus, London SW7 2AZ

RSVP: Attendance is free, but with registration in advance
Michael Ruzhansky • @email

I will describe some recent joint work with Davide Gaiotto and Greg Moore, in which we explain the origin of the wall-crossing formula of Kontsevich and Soibelman, in the context of N=2 supersymmetric field theories in four dimensions. The wall-crossing formula gives a recipe for constructing the smooth hyperkahler metric on the moduli space of the field theory reduced on a circle to 3 dimensions. In certain examples this moduli space is actually a moduli space of ramified Higgs bundles, so we obtain a new description of the hyperkahler structure on that space.

When two inclusions (in a composite) get closer and their conductivities degenerate

to zero or infinity, the gradient of the solution to the

conductivity equation blows up in general. We show

that the solution to the conductivity equation can be decomposed

into two parts in an explicit form: one of them has a bounded

gradient and the gradient of the other part blows up. Using the

decomposition, we derive the best possible estimates for the blow-up

of the gradient. The decomposition theorem and estimates have an

important implication in computation of electrical field in

the presence of closely located inclusions.

This short course runs from Monday 29th June to Friday 3rd July. For details of the course and how to register, please visit http://www2.maths.ox.ac.uk/oxmos/meetings/moms/.

Preview of problems to be solved at the study Group in Limerick taking place in the following week.

I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.

Martin Bridson will give a "repeat" performance of his Abel Lecture which he delivered a few weeks ago in Oslo as part of the scientific programme in honour of Abel Prize laureate Mikhail Gromov.

Abstract:

Gromov has illuminated great swathes of mathematics with the bright light of geometry. By means of example, I hope to convey the sense of wonder that his work engenders and something of the profound influence he has had on the way my generation thinks about mathematics.

I shall focus particularly on Geometric Group Theory. Gromov's ideas turned the study of discrete groups on its head, infusing it with an array of revolutionary ideas and unveiling deep connections to many other branches of mathematics.

We generalize rings, Banach algebras and C*-algebras to ringoids, Banach algebroids and C*-algebroids. We construct algebraic and topological K-theory of these objects. As an application we can formulate Farrell-Jones Conjecture in algebraic K-theory, Bost- and Baum-Connes-Conjecture in topological K-theory

I will describe joint work with Mohammed Abouzaid, in which we complete the proof of homological mirror symmetry for the standard four-torus and consider various applications. A key tool is the recently-developed holomorphic quilt theory of Mau-Wehrheim-Woodward.

We resume former discussions of the question, whether the spin-spin repulsion and the gravitational attraction of two aligned black holes can balance each other. To answer the question we formulate a boundary value problem for two separate (Killing-) horizons and apply the inverse (scattering) method to solve it. Making use of results of Manko, Ruiz and Sanabria-Gómez and a novel black hole criterion, we prove the non-existence of the equilibrium situation in question.

Constantinides (1986) finds that transaction cost has only a second order effect on liquidity premia. In this paper, we show that simply incorporating the well-established time-varying return dynamics across trading and nontrading periods generates a first order effect that is much greater than that found by the existing literature and comparable to empirical evidence. Surprisingly, the higher liquidity premium is Not from higher trading frequency, but mainly from the substantially suboptimal (relative to the no transaction case) trading strategy chosen to control transaction costs. In addition, we show that adopting strategies prescribed by standard models that assume a continuously open market and constant return dynamics can result in significant utility loss. Furthermore, our model predicts that trading volume is greater at market close and market open than the rest of trading times.

I will talk about two pieces of work on finite covers:

(i) Work of Hrushovski which, for a stable theory, links splitting of certain finite covers with higher amalgamation properties;

(ii) Joint work of myself and Elisabetta Pastori which uses group cohomology to investigate some non-split finite covers of the set of k-sets from a disintegrated set.

Platelet ice may be an important component of Antarctic land-fast sea

ice. Typically, it is found at depth in first-year landfast sea ice

cover, near ice shelves. To explain why platelet ice is not commonly

observed at shallower depths, we consider a new mechanism. Our

hypothesis is that platelet ice eventually appears due to the sudden

deposition of frazil ice against the fast ice-ocean interface,

providing randomly oriented nucleation sites for crystal growth.

Brine rejected in plumes from land-fast ice generates stirring

sufficient to prevent frazil ice from attaching to the interface,

forcing it to remain in suspension until ice growth rate and brine

rejection slow to the point that frazil can stick. We calculate a

brine plume velocity, and match this to frazil rise velocity.

We consider both laminar and turbulent environments. We find that

brine plume velocities are generally powerful enough to prevent most

frazil from sticking in the case of laminar flow, and that in the

turbulent case there may be a critical ice thickness at which most

frazil suddenly settles.

Suppose a power series $f(x):= \sum_{n=0}^{\infty} a_{n} x^{n}$ has radius of convergence equal to $1$ and that $lim_{x\rightarrow 1}f(x) = s$. Does it therefore follow that $\sum_{n=0}^{\infty} a_{n} = s$? Tauber's Theorem answers in the affirmative, \textit{if} one imposes a certain growth condition (a \textit{Tauberian Condition}) on the coefficients $a_{n}$. Without such a condition it is clear that this cannot be true in general - take, for example, $f(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n}.$

Current community models in the geosciences employ a variety of numerical methods from finite-difference, finite-volume, finite- or spectral elements, to pseudospectral methods. All have specialized strengths but also serious weaknesses. The first three methods are generally considered low-order and can involve high algorithmic complexity (as in triangular elements or unstructured meshes). Global spectral methods do not practically allow for local mesh refinement and often involve cumbersome algebra. Radial basis functions have the advantage of being spectrally accurate for irregular node layouts in multi-dimensions with extreme algorithmic simplicity, and naturally permit local node refinement on arbitrary domains. We will show test examples ranging from vortex roll-ups, modeling idealized cyclogenesis, to the unsteady nonlinear flows posed by the shallow water equations to 3-D mantle convection in the earth’s interior. The results will be evaluated based on numerical accuracy, stability and computational performance.

The AJ conjecture relates two different knot invariants, namely the coloured Jones polynomial and the A-polynomial. The approach we will use will be that of 2+1 dimensional Topological Quantum Field Theory. Indeed, the coloured Jones polynomial is constructed in Reshetikhin and Turaev's formulation of a TQFT using quantum groups. The A-polynomial is defined by a subvariety of the moduli space of flat SL(2,C) connections of a torus.  Geometric quantization on this moduli space also gives a TQFT, and the correspondence between these provides a framework where the knot invariants can be compared. In the talk I will sketch the above constructions and show how we can do explicit calculations for simple knots. This is work in progress joint with J. E. Andersen.

This is a special talk outside the normal Computational Mathematics and Application seminar series. Please note it takes place on a Wednesday.

In some options markets (eg. commodities), options are listed with only a single maturity for each underlying.

In others, (eg. equities, currencies),

options are listed with multiple maturities.

In this paper, we assume that the risk-neutral process for the underlying futures price is a pure jump Markov martingale and that European option prices are given at a continuum of strikes and at one or more maturities. We show how to construct a time-homogeneous process which meets a single smile and a piecewise time-homogeneous process, which can meet multiple smiles.

We also show that our construction leads to partial differential difference equations (PDDE's), which permit both explicit calibration and fast numerical valuation

I will describe in detail the first construction of infinite, finitely generated torsion groups due to Golod in early 60s --

these groups are special cases of the so-called Golod-Shafarevich groups. If time allows, I will discuss some related constructions and open problems.

Informally speaking, a finitely generated group G is said to be {\it Golod-Shafarevich} (with respect to a prime p) if it has a presentation with a ``small'' set of relators, where relators are counted with different weights depending on how deep they lie in the Zassenhaus p-filtration. Golod-Shafarevich groups are known to behave like (non-abelian) free groups in many ways: for instance, every Golod-Shafarevich group G has an infinite torsion quotient, and the pro-p completion of G contains a non-abelian free pro-p group. In this talk I will extend the list of known ``largeness'' properties of Golod-Shafarevich groups by showing that they always have an infinite quotient with Kazhdan's property (T). An important consequence of this result is a positive answer to a well-known question on non-amenability of Golod-Shafarevich groups.

Let $F$ be a uniformly distributed random $k$-SAT formula with $n$ variables and $m$ clauses. We present a polynomial time algorithm that finds a satisfying assignment of $F$ with high probability for constraint densities $m/n

The orientational order of a nematic liquid crystal in a spatially inhomogeneous flow situation is best described by a Q-tensor field because of the defects that inevitably occur. The evolution is determined by two equations. The flow is governed by a generalised Stokes equation in which the divergence of the stress tensor also depends on Q and its time derivative. The evolution of Q is governed by a convection-diffusion type equation that contains terms nonlinear in Q that stem from a Landau-de Gennes potential.

In this talk, I will show how the most general evolution equations can be derived from a dissipation principle. Based on this, I will identify a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of a Q-dependent body force. This leads to a convenient time-discretised strategy for solving the flow-orientation problem using two alternating steps. First, the flow field of the Stokes flow is computed for a given orientation field. Second, with the given flow field, one time step of the orientation equation is carried out. The new orientation field is then used to compute a new body force which is again used in the Stokes equation and so forth.

For some simple test applications at low Reynolds numbers, it is found that the non-homogeneous orientation of the nematic liquid crystal leads to non-linear flow effects similar to those known from Newtonian flow at high Reynolds numbers.

One of the major open challenges in general relativity is the classification of black hole solutions in higher dimensional theories. I will explain a recent result in this direction in the context of Kaluza-Klein spacetimes admitting a sufficient number N of commuting U(1)-symmetries. It turns out that the black holes in such a theory are characterized by the usual asymptotic charges, together with certain combinatorical data and moduli. The combinatorial data characterize the nature of the U(1)^N-action, and its analysis is closely related to properties of integer lattices and questions in the area of geometric number theory. I will also explain recent results on extremal black holes which show that such objects display remarkable ``symmetry enhancement'' properties

We define a chain complex B_*(C, M) (the "blob complex") associated to an n-category C and an n-manifold M. This is in some sense the derived category version of a TQFT. Various special cases of the blob complex are

familiar: (a) if M = S^1, then the blob complex is homotopy equivalent to the Hochschild complex of the 1-category C; (b) for * = 0, H_0 of the blob complex is the Hilbert space of the TQFT based on C; (c) if C is a commutative polynomial ring (viewed as an n-category), then the blob complex is homotopy equivalent to singular chains on the configuration (Dold-Thom) space of M. The blob complex enjoys various nice formal properties, including a higher dimensional generalization of the Deligne conjecture for Hochschild cohomology.

If time allows I will discuss applications to contact structures on 3-manifolds and Khovanov homology for links in the boundaries of 4-manifolds. This is joint work with Scott Morrison.

We consider the analysis for a class of random differential equations driven by rough noise and with a trajectory that is influenced by its own law. Having described the mathematical setup with great precision, we will illustrate how such equations arise naturally as the limits of a cloud of interacting particles. Finally, we will provide examples to show the ubiquity of such systems across a range of physical and economic phenomena and hint at possible extensions.

Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying efficiency.

In particular they facilitate precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have only been proved for target measures with a product structure, severely limiting their applicability to real applications. The purpose of this talk is to desribe a research program aimed at identifying diffusion limits for a class of naturally occuring problems, found by finite dimensional approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure.

The diffusion limit to a Hilbert space valued SDE (or SPDE) is proved.

Joint work with Natesh Pillai (Warwick) and Jonathan Mattingly (Duke)