Past

In recent years, limit order books have been adopted as the pricing mechanism in more than half of the world's financial markets. Thanks to recent technological advances, traders around the globe also now have real-time access to limit order book trading platforms and can develop trading strategies that make use of this "ultimate microscopic level of description". In this talk I will briefly describe the limit order book trade-matching mechanism, and explain how the extra flexibility it provides has vastly impacted the problem of how a market participant should optimally behave in a given set of circumstances. I will then discuss the findings from my recent statistical analysis of real limit order book data for spot trades of 3 highly liquid currency pairs (namely, EUR/USD, GBP/USD, and EUR/GBP) on a large electronic trading platform during May and June 2010, and discuss how a number of my findings highlight weaknesses in current models of limit order books.

In this seminar I will expose some results obtained jointly with P. Marcati, concerning the global existence of weak solutions for the Quantum Hydrodynamics System in the space of energy. We don not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.

I will then discuss some possible future extensions of the theory.

This is the first of two talks about Derived Algebraic Geometry. Due to the vastity of the theory, the talks are conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as it might reveal interesting new insights also on classical objects, rather than a detailed and precise exposition. We will start with an introduction on the very basic idea of this theory, and we will expose some motivations for introducing it. After a brief review on the existing literature and a speculation about homotopy theories and higher categorical structures, we will review the theory of dg-categories, model categories, S-categories and Segal categories. This is the technical part of the seminar and it will give us the tools to understand the basic setting of Topos theory and Homotopical Algebraic Geometry, whose applications will be exploited in the next talk.

I will present a history of the problem, relate it to other conjectures, and, with time permitting, indicate recent developments. The focus will primarily be group-theoretic and intended for the non-specialist.

In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H=C(Z/p) of complex valued functions on Z/p={0,...,p-1}, the integers modulo a prime number p>>0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form

R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t),

where W(t) in H is a white noise, and τ,ω in ℤ/p, encode the distance from, and velocity of, the object.

Problem (digital radar problem) Extract τ,ω from R and S.

I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p^2⋅log(p) operations. I will then explain how to use techniques from group representation theory to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations. I will demonstrate additional applications to mobile communication, and global positioning system (GPS).

This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley).

Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, I will explain relations to homological mirror symmetry and the Gross-Siebert construction.

Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, I will explain relations to homological mirror symmetry and the Gross-Siebert construction.

The WZW functional for a map from a surface to a Lie group has a role in the theory of harmonic maps, and it also arises as the determinant of a d-bar operator on the surface, as the action functional for a 2-dimensional quantum field theory, as the partition function of 3-dimensional Chern-Simons theory on a manifold with boundary, and as the norm-squared of a state-vector. It is intimately related to the quantization of the symplectic manifold of flat bundles on the surface, a fascinating test-case for different approaches to geometric quantization. It is also interesting as an example of interpolation between commutative and noncommutative geometry. I shall try to give an overview of the area, focussing on the aspects which are still not well understood.

Chataur and Menichi showed that the homology of the free loop space of the classifying space of a compact Lie group admits a rich algebraic structure: It is part of a homological field theory, and so admits operations parametrised by the homology of mapping class groups.  I will present a new construction of this field theory that improves on the original in several ways: It enlarges the family of admissible Lie groups.  It extends the field theory to an open-closed one.  And most importantly, it allows for the construction of co-units in the theory.  This is joint work with Anssi Lahtinen.

I will present a two-parameter family of Wilson loop operators in N = 4 supersymmetric Yang-Mills theory which interpolates smoothly between the 1/2 BPS line or circle and a pair of antiparallel lines. These observables capture a natural generalization of the quark-antiquark potential. These loops are calculated on the gauge theory side to second order in perturbation theory and in a semiclassical expansion in string theory to one-loop order. The resulting determinants are given in integral form and can be evaluated numerically for general values of the parameters or analytically in a systematic expansion around the 1/2 BPS configuration. I will comment about the feasibility of deriving all-loop results for these Wilson loops.

Three short talks by the authors of essays on topics related to c3 Algebraic topology: Whitehead's theorem, Cohomology of fibre bundles, Division algebras

• Cameron Hall - Dislocations and discrete-to-continuum asymptotics: the summary
• Kostas Zygalakis - Multi scale methods: theory numerics and applications
• Lian Duan - Barcode Detection and Deconvolution in Well Testing

Nonnormality is a well studied subject in the context of partial differential operators. Yet, only little is known for boundary integral operators. The only well studied case is the unit ball, where the standard single layer, double layer and conjugate double layer potential operators in acoustic scattering diagonalise in a unitary basis. In this talk we present recent results for the analysis of spectral decompositions and nonnormality of boundary integral operators on more general domains. One particular application is the analysis of stability constants for boundary element discretisations. We demonstrate how these are effected by nonnormality and give several numerical examples, illustrating these issues on various domains.

The aim of this talk is to explain how to construct solutions to a

relativistic transport equation via a time discrete scheme based on an

optimal transportation problem.

First of all, I will present a joint work with J. Bertrand, where we prove the existence of an optimal map

for the Monge-Kantorovich problem associated to relativistic cost functions.

Then, I will explain a joint work with Robert McCann, where

we study the limiting process between the discrete and the continuous

equation.

We will consider a simplified model for on-chip power distribution networks of array bonded integrated circuits. In this model the voltage is the solution of a Poisson equation in an infinite planar domain whose boundary is an array of circular or square pads of size $\epsilon$. We deal with the singular limit as $\epsilon\to 0$ and we are interested in deriving an explicit formula for the maximum voltage drop in the domain in terms of a power series in $\epsilon$. A procedure based on the method of matched asymptotic expansions will be presented to compute all the successive terms in the approximation, which can be interpreted as using multipole solutions of equations involving spatial derivatives of $\delta$-functions.

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

Many iterative algorithms for large sparse matrix problems are based on orthogonality (or $A$-orthogonality, bi-orthogonality, etc.), but these properties can be lost very rapidly using vector orthogonalization (subtracting multiples of earlier supposedly orthogonal vectors from the latest vector to produce the next orthogonal vector). Yet many of these algorithms are some of the best we have for very large sparse problems, such as Conjugate Gradients, Lanczos' method for the eigenproblem, Golub and Kahan bidiagonalization, and MGS-GMRES.

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Here we describe an ideal form of orthogonal matrix that arises from any sequence of supposedly orthogonal vectors. We illustrate some of its fascinating properties, including a beautiful measure of orthogonality of the original set of vectors. We will indicate how the ideal orthogonal matrix leads to expressions for new concepts of stability of such iterative algorithm. These are expansions of the concept of backward stability for matrix transformation algorithms that was so effectively developed and applied by J. H. Wilkinson (FRS). The resulting new expressions can be used to understand the subtle and effective performance of some (and hopefully eventually all) of these iterative algorithms.

Please note that this is taking place in the afternoon - partly to avoid a clash with the OCCAM group meeting in the morning.

• Ian Griffiths - Control and optimization in filtration and tissue engineering
• Vladimir Zubkov - Comparison of the Navier-Stokes and the lubrication models for the tear film dynamics
• Victor Burlakov - Applying the ideas of 1-st order phase transformations to various nano-systems

There is much current concern over the future evolution of climate under conditions of increased atmospheric carbon. Much of the focus is on a bottom-up approach in which weather/climate models of severe complexity are solved and extrapolated beyond their presently validated parameter ranges. An alternative view takes a top-down approach, in which the past Earth itself is used as a laboratory; in this view, ice-core records show a strong association of carbon with atmospheric temperature throughout the Pleistocene ice ages. This suggests that carbon variations drove the ice ages. In this talk I build the simplest model which can accommodate this observation, and I show that it is reasonably able to explain the observations. The model can then be extrapolated to offer commentary on the cooling of the planet since the Eocene, and the likely evolution of climate under the current industrial production of atmospheric carbon.

In this article we propose a novel approach to reduce the computational

complexity of the dual method for pricing American options.

We consider a sequence of martingales that converges to a given

target martingale and decompose the original dual representation into a sum of

representations that correspond to different levels of approximation to the

target martingale. By next replacing in each representation true conditional expectations with their

Monte Carlo estimates, we arrive at what one may call a multilevel dual Monte

Carlo algorithm. The analysis of this algorithm reveals that the computational

complexity of getting the corresponding target upper bound, due to the target martingale,

can be significantly reduced. In particular, it turns out that using our new

approach, we may construct a multilevel version of the well-known nested Monte

Carlo algorithm of Andersen and Broadie (2004) that is, regarding complexity, virtually

equivalent to a non-nested algorithm. The performance of this multilevel

algorithm is illustrated by a numerical example. (joint work with Denis Belomestny)

The standard mathematical treatment of risk combines numerical measures of uncertainty (usually probabilistic) and loss (money and other natural estimators of utility). There are significant practical and theoretical problems with this interpretation. A particular concern is that the estimation of quantitative parameters is frequently problematic, particularly when dealing with one-off events such as political, economic or environmental disasters. Practical decision-making under risk, therefore, frequently requires extensions to the standard treatment.

An intuitive approach to reasoning under uncertainty has recently become established in computer science and cognitive science in which general theories (formalised in a non-classical first-order logic) are applied to descriptions of specific situations in order to construct arguments for and/or against claims about possible events. Collections of arguments can be aggregated to characterize the type or degree of risk, using the logical grounds of the arguments to explain, and assess the credibility of, the supporting evidence for competing claims. Discussions about whether a complex piece of equipment or software could fail, the possible consequences of such failure and their mitigation, for example, can be  based on the balance and relative credibility of all the arguments. This approach has been shown to offer versatile risk management tools in a number of domains, including clinical medicine and toxicology (e.g. www.infermed.com; www.lhasa.com). Argumentation frameworks are also being used to support open discussion and debates about important issues (e.g. see debate on environmental risks at www.debategraph.org).

Despite the practical success of argument-based methods for risk assessment and other kinds of decision making they typically ignore measurement of uncertainty even if some quantitative data are available, or combine logical inference with quantitative uncertainty calculations in ad hoc ways. After a brief introduction to the argumentation approach I will demonstrate medical risk management applications of both kinds and invite suggestions for solutions which are mathematically more satisfactory. 

Definitions (Hubbard:  http://en.wikipedia.org/wiki/Risk)

Uncertainty: The lack of complete certainty, that is, the existence of more than one possibility. The "true" outcome/state/result/value is not known.

Measurement of uncertainty: A set of probabilities assigned to a set of possibilities. Example:"There is a 60% chance this market will double in five years"

Risk: A state of uncertainty where some of the possibilities involve a loss, catastrophe, or other undesirable outcome.

Measurement of risk: A set of possibilities each with quantified probabilities and quantified losses. Example: "There is a 40% chance the proposed oil well will be dry with a loss of $12 million in exploratory drilling costs".

The conceptual background to the argumentation approach to reasoning under uncertainty is reviewed in the attached paper “Arguing about the Evidence: a logical approach”.

Tsunami asymptotics: For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as diffraction-decorated caustics in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth can focus tsunamis onto cusped caustics, and this 'singularity on a singularity' constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude.

(Joint work with P. Corvaja and D.

Masser.)

The topic of the talk arises from the

Manin-Mumford conjecture and its extensions, where we shall

focus on the case of (complex connected) commutative

algebraic groups $G$ of dimension $2$. The `Manin-Mumford'

context in these cases predicts finiteness for the set of

torsion points in an algebraic curve inside $G$, unless the

curve is of `special' type, i.e. a translate of an algebraic

subgroup of $G$.

In the talk we shall consider not merely the set of torsion

points, but its topological closure in $G$ (which turns out

to be also the maximal compact subgroup). In the case of

abelian varieties this closure is the whole space, but this is

not so for other $G$; actually, we shall prove that in certain

cases (where a natural dimensional condition is fulfilled) the

intersection of this larger set with a non-special curve

must still be a finite set.

We shall conclude by stating in brief some extensions of

this problem to higher dimensions.

I'll present the work of Gaitsgory arXiv:1108.1741. In it he uses Beilinson-Drinfeld factorization techniques in order to uniformize the moduli stack of G-bundles on a curve. The main difference with the gauge theoretic technique is that the the affine Grassmannian is far from being contractible but the fibers of the map to Bun(G) are contractible.

• Sufficient conditions for bifurcation from points that are not isolated eigenvalues of the linearisation.

• Odd potential operators.

• Defining min-max critical values using sets of finite genus.

• Formulating some necessary conditions for bifurcation.

The fundamental task in climate variability research is to eke

out structure from climate signals. Ideally we want a causal

connection between a physical process and the structure of the

signal. Sometimes we have to settle for a correlation between

these. The challenge is that the data is often poorly

constrained and/or sparse. Even though many data gathering

campaigns are taking place or are being planned, the very high

dimensional state space of the system makes the prospects of

climate variability analysis from data alone impractical.

Progress in the analysis is possible by the use of models and

data. Data assimilation is one such strategy. In this talk we

will describe the methodology, illustrate some of its

challenges, and highlight some of the ways our group has

proposed to improving the methodology.

I will talk about $W^{2,1}$ regularity for strictly convex Aleksandrov solutions to the Monge Amp\`ere equation

\[

\det D^2 u =f

\]

where $f$ satisfies $\log f\in L^{\infty} $. Under the previous assumptions in the 90's Caffarelli was able to prove that $u \in C^{1,\alpha}$ and that $u\in W^{2,p}$ if $|f-1|\leq \varepsilon(p)$. His results however left open the question of Sobolev regularity of $u$ in the general case in which $f$ is just bounded away from $0$ and infinity. In a joint work with Alessio Figalli we finally show that actually $|D^2u| \log^k |D^2 u| \in L^1$ for every positive $k$.

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If time will permit I will also discuss some question related to the $W^{2,1}$ stability of solutions of Monge-Amp\`ere equation and optimal transport maps and some applications of the regularity to the study of the semi-geostrophic system, a simple model of large scale atmosphere/ocean flows (joint works with Luigi Ambrosio, Maria Colombo and Alessio Figalli).

After recalling some definitions and facts about spectra from the previous two "respectra" talks, I will explain what Thom spectra are, and give many examples. The cohomology theories associated to various different Thom spectra include complex cobordism, stable homotopy groups, ordinary mod-2 homology.......

I will then talk about Thom's theorem: the ring of homotopy groups of a Thom spectrum is isomorphic to the corresponding cobordism ring. This allows one to use homotopy-theoretic methods (calculating the homotopy groups of a spectrum) to answer a geometric question (determining cobordism groups of manifolds with some specified structure). If time permits, I'll also describe the structure of some cobordism rings obtained in this way.

Building on the previous talk, we continue the exploration of techniques required to understand Wise's results. We present an overview of classical small cancellation theory running in parallel with the newer one for cubical complexes.

I will give an overview of some of the most interesting algebraic-lattice theoretical results on bilattices. I will focus in particular on the product construction that is used to represent a subclass of bilattices, the so-called 'interlaced bilattices', mentioning some alternative strategies to prove such a result. If time allows, I will discuss other algebras of logic related to bilattices (e.g., Nelson lattices) and their product representation.

Turbidity currents are fast-moving streams of sediment in the ocean 
which have the power to erode the sea floor and damage man-made
infrastructure anchored to the bed. They can travel for hundreds of
kilometres from the continental shelf to the deep ocean, but they are
unpredictable and can occur randomly without much warning making them
hard to observe and measure. Our main aim is to determine the distance
downstream at which the current will become extinct. We consider the
fluid model of Parker et al. [1986] and derive a simple shallow-water
description of the current which we examine numerically and analytically
to identify supercritical and subcritical flow regimes. We then focus on
the solution of the complete model and provide a new description of the
turbulent kinetic energy. This extension of the model involves switching
from a turbulent to laminar flow regime and provides an improved
description of the extinction process. 

In recent decades, quantum field theory (QFT) has become the framework for

several basic and outstandingly successful physical theories. Indeed, it has

become the lingua franca of entire branches of physics and even mathematics.

The universal scope of QFT opens fascinating opportunities for philosophy.

Accordingly, although the philosophy of physics has been dominated by the

analysis of quantum mechanics, relativity and thermo-statistical physics,

several philosophers have recently undertaken conceptual analyses of QFT.

One common feature of these analyses is the emphasis on rigorous approaches,

such as algebraic and constructive QFT; as against the more heuristic and

physical formulations of QFT in terms of functional (also knows as: path)

integrals.

However, I will follow the example of some recent mathematicians such as

Atiyah, Connes and Kontsevich, who have adopted a remarkable pragmatism and

opportunism with regard to heuristic QFT, not corseted by rigor (as Connes

remarks). I will conceptually discuss the advances that have marked

heuristic QFT, by analysing some of the key ideas that accompanied its

development.  I will also discuss the interactions between these concepts in

the various relevant fields, such as particle physics, statistical

mechanics, gravity and geometry.