Past

In this talk we will discuss geometric quantization. First of all we will discuss what it is, but shall also see that it has relations to many other parts of mathematics. Especially shall we see how the Hitchin connection in geometric quantization can give us representations of a certain group associated to a surface, the mapping class group. If time permits we will discuss some recent results about these groups and their representations, results that are essentially obtained from geometrically quantizing a moduli space of flat connections on a surface."

In this seminar, we discuss three questions related to the finite difference computation of early exercise options, one of which has a useful answer, one an interesting one, and one is open.

We begin by showing that a simple iteration of the exercise strategy of a finite difference solution is efficient for practical applications and its convergence can be described very precisely. It is somewhat surprising that the method is largely unknown.

We move on to discuss properties of a so-called penalty method. Here we show by means of numerical experiments and matched asymptotic expansions that the approximation of the value function has a very intricate local structure, which is lost in functional analytic error estimates, which are also derived.

Finally, we describe a gap in the analysis of the grid convergence of finite difference approximations compared to empirical evidence.

This is joint work with Jan Witte and Sam Howison.

Pseudo-differential operators (PDO's) are primarily defined in the familiar setting of the Euclidean space. For four decades, they have been standard tools in the study of PDE's and it is natural to attempt defining PDO's in other settings. In this talk, after discussing the concept of PDO's on the Euclidean space and on the torus, I will present some recent results and outline future work regarding PDO's on Lie groups as well as some of the applications to PDE's

The lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL all share a nice kind of geometry. We'll see how the Cayley graph of a lamplighter group is a Diestel-Leader graph, that is a horocyclic product of two trees. The geometry of the solvable Baumslag-Solitar groups has been studied by Farb and Mosher and they showed that these groups are quasi-isometric to spaces which are essentially the horocyclic product of a tree and the hyperbolic plane. Finally, lattices in the Lie groups SOL can be seen to act on the horocyclic product of two hyperbolic planes. We use these spaces to measure the length of short conjugators in each type of group.

Many swimming microorganisms inhabit heterogeneous environments consisting of solid particles immersed in viscous fluid. Such environments require the organisms attempting to move through them to negotiate both hydrodynamic forces and geometric constraints. Here, we study this kind of locomotion by first observing the kinematics of the small nematode and model organism Caenorhabditis elegans in fluid-filled, micro-pillar arrays. We then compare its dynamics with those given by numerical simulations of a purely mechanical worm model that accounts only for the hydrodynamic and contact interactions with the obstacles. We demonstrate that these interactions allow simple undulators to achieve speeds as much as an order of magnitude greater than their free-swimming values. More generally, what appears as behavior and sensing can sometimes be explained through simple mechanics.

This is a report on a joint work (in progress) with Pantev, Vaquie and Vezzosi. After some

reminders on derived algebraic geometry, I will present the notion of shifted symplectic structures, as well as several basic examples. I will state existence results: mapping spaces towards a symplectic targets, classifying spaces of reductive groups, Lagrangian intersections, and use them to construct many examples of (derived) moduli spaces endowed with shifted symplectic forms. In a second part, I will explain what "Quantization" means in the shifted context. The general theory will be illustrated by the particular examples of moduli of sheaves on oriented manifolds, in dimension 2, 3 and higher.

The phase transition deals with sudden global changes and is observed in many fundamental random discrete structures arising from statistical physics, mathematics and theoretical computer science, for example, Potts models, random graphs and random $k$-SAT. The phase transition in random graphs refers to the phenomenon that there is a critical edge density, to which adding a small amount results in a drastic change of the size and structure of the largest component. In the Erdős--R\'enyi random graph process, which begins with an empty graph on $n$ vertices and edges are added randomly one at a time to a graph, a phase transition takes place when the number of edges reaches $n/2$ and a giant component emerges. Since this seminal work of Erdős and R\'enyi, various random graph processes have been introduced and studied. In this talk we will discuss new approaches to study the size and structure of components near the critical point of random graph processes: key techniques are the classical ordinary differential equations method, a quasi-linear partial differential equation that tracks key statistics of the process, and singularity analysis.

This is a report on a joint work (in progress) with Pantev, Vaquie and Vezzosi. After some

reminders on derived algebraic geometry, I will present the notion of shifted symplectic structures, as well as several basic examples. I will state existence results: mapping spaces towards a symplectic targets, classifying spaces of reductive groups, Lagrangian intersections, and use them to construct many examples of (derived) moduli spaces endowed with shifted symplectic forms. In a second part, I will explain what "Quantization" means in the shifted context. The general theory will be illustrated by the particular examples

of moduli of sheaves on oriented manifolds, in dimension 2, 3 and higher.

Quantile forecasting of wind power using variability indices
Abstract: Wind power forecasting techniques have received substantial attention recently due to the increasing penetration of wind energy in national power systems.  While the initial focus has been on point forecasts, the need to quantify forecast uncertainty and communicate the risk of extreme ramp events has led to an interest in producing probabilistic forecasts. Using four years of wind power data from three wind farms in Denmark, we develop quantile regression models to generate short-term probabilistic forecasts from 15 minutes up to six hours ahead. More specifically, we investigate the potential of using various variability indices as explanatory variables in order to include the influence of changing weather regimes. These indices are extracted from the same  wind power series and optimized specifically for each quantile. The forecasting performance of this approach is compared with that of some benchmark models. Our results demonstrate that variability indices can increase the overall skill of the forecasts and that the level of improvement depends on the specific quantile.

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time ``splash'' singularity, wherein the evolving 2-D hypersurface intersects itself at a point. Our approach is based on the Lagrangian description of the free-boundary problem, combined with novel approximation scheme. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems. This is joint work with Daniel Coutand.

We consider the prime k-tuples conjecture, which predicts that a system of linear forms are simultaneously prime infinitely often, provided that there are no obvious obstructions. We discuss some motivations for this and some progress towards proving weakened forms of the conjecture.

I shall describe the construction of the four-brane giant graviton on $\mathrm{AdS}_4\times \mathbb{CP}^3$ (extended and moving in the complex projective space), which is dual to a subdeterminant operator in the ABJM model. This dynamically stable, BPS configuration factorizes at maximum size into two topologically stable four-branes (each wrapped on a different $\mathbb{CP}^2 \subset \mathbb{CP}^3$ cycle) dual to ABJM dibaryons. Our study of the spectrum of small fluctuations around this four-brane giant provides good evidence for a dependence in the spectrum on the size, $\alpha_0$, which is a direct result of the changing shape of the giant’s worldvolume as it grows in size. I shall finally comment upon the implications for operators in the non-BPS, holomorphic sector of the ABJM model.

We develop the idea of using Monte Carlo sampling of random portfolios to solve portfolio investment problems. We explore the need for more general optimization tools, and consider the means by which constrained random portfolios may be generated. DeVroye’s approach to sampling the interior of a simplex (a collection of non-negative random variables adding to unity) is already available for interior solutions of simple fully-invested long-only systems, and we extend this to treat, lower bound constraints, bounded short positions and to sample non-interior points by the method of Face-Edge-Vertex-biased sampling. A practical scheme for long-only and bounded short problems is developed and tested. Non-convex and disconnected regions can be treated by applying rejection for other constraints. The advantage of Monte Carlo methods is that they may be extended to risk functions that are more complicated functions of the return distribution, without explicit gradients, and that the underlying return distribution may be modeled parametrically or empirically based on general distributions. The optimization of expected utility, Omega, Sortino ratios may be handled in a similar manner to quadratic risk, VaR and CVaR, irrespective of whether a reduction to LP or QP form is available. Robustification is also possible, and a Monte Carlo approach allows the possibility of relaxing the general maxi-min approach to one of varying degrees of conservatism. Grid computing technology is an excellent platform for the development of such computations due to the intrinsically parallel nature of the computation. Good comparisons with established results in Mean-Variance and CVaR optimization are obtained, and we give some applications to Omega and expected Utility optimization. Extensions to deploy Sobol and Niederreiter quasi-random methods for random weights are also proposed. The method proposed is a two-stage process. First we have an initial global search which produces a good feasible solution for any number of assets with any risk function and return distribution. This solution is already close to optimal in lower dimensions based on an investigation of several test problems. Further precision, and solutions in 10-100 dimensions, are obtained by invoking a second stage in which the solution is iterated based on Monte-Carlo simulation based on a series of contracting hypercubes.

Please note that this is a joint seminar with the William Dunn School of Pathology and will take place in the EPA Seminar Room, which is located inside the Sir William Dunn School of Pathology and must be entered from the main entrance on South Parks Road. link: http://g.co/maps/8cbbx

This is the second of two talks about Derived Algebraic Geometry. We will go through the various geometries one can develop from the Homotopical Algebraic Geometry setting. We will review stack theory in the sense of Laumon and Moret-Bailly and higher stack theory by Simpson from a new and more general point of view, and this will culminate in Derived Algebraic Geometry. We will try to point out how some classical objects are actually secretly already in the realm of Derived Algebraic Geometry, and, once we acknowledge this new point of view, this makes us able to reinterpret, reformulate and generalize some classical aspects. Finally, we will describe more exotic geometries. In the last part of this talk, we will focus on two main examples, one addressed more to algebraic geometers and representation theorists and the second one to symplectic geometers.

We will demonstrate the following. Category theory, usually conceived as some very abstract form of metamathematics, is present everywhere around us. Explicitly, we show how it provides a kindergarten version of quantum theory, an how it will help Google to understand sentences rather than words.

Some references are:

-[light] BC (2010) "Quantum picturalism". Contemporary Physics 51, 59-83. arXiv:0908.1787 
-[a bit heavier] BC and Ross Duncan (2011) "Interacting quantum observables: categorical algebra and diagrammatics". New Journal of Physics 13, 043016. arXiv:0906.4725
-[light] New Scientist (8 December 2010) "Quantum links let computers understand language". www.cs.ox.ac.uk/people/bob.coecke/NewScientist.pdf
-[a bit heavier] BC, Mehrnoosh Sadrzadeh and Stephen Clark (2011) "Mathematical foundations for a compositional distributional model of meaning". Linguistic Analysis - Lambek Festschrift. arXiv:1003.439

Recall that a difference field is a field with a distinguished automorphism. ACFA is the theory of existentially closed difference fields. I will discuss results on groups definable in models of ACFA, in particular when they are one-based and what are the consequences of one-basedness.

It is an inherent premise in Boltzmann's formulation of linear viscoelasticity, that for shear deformations at constant pressure and constant temperature, every material has a unique continuous relaxation spectrum. This spectrum defines the memory kernel of the material. Only a few models for representing the continuous spectrum have been proposed, and these are entirely empirical in nature.

Extensive laboratory time is spent worldwide in collecting dynamic data from which the relaxation spectra of different materials may be inferred. In general the process involves the solution of one or more exponentially ill-posed inverse problems.

In this talk I shall present rigorous models for the continuous relaxation spectrum. These arise naturally from the theory of continuous wavelet transforms. In solving the inverse problem I shall discuss the role of sparsity as one means of regularization, but there is also a secondary regularization parameter which is linked, as always, to resolution. The topic of model-induced super-resolution is discussed, and I shall give numerical results for both synthetic and real experimental data.

The talk is based on joint work with Neil Goulding (Cardiff University).

I will discuss joint work with Matthew Emerton on geometric
approaches to the Breuil-Mézard conjecture, generalising a geometric
approach of Breuil and Mézard. I will discuss a proof of the geometric
version of the original conjecture, as well as work in progress on a
geometric version of the conjecture which does not make use of a fixed
residual representation.

This is a sequel to Lecture I (given in the algebra seminar, Tuesday). It will be slightly more specialized. The finite Weil representation is the algebra object that governs the symmetries of the Hilbert space H =C(Z/p): The main objective of this talk is to introduce the geometric Weil representation which is an algebra-geometric (l-adic perverse

Weil sheaf) counterpart of the finite Weil representation. Then, I will explain how the geometric Weil representation is used to prove the main technical results stated in Lecture I. In the course, I will explain the Grothendieck geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects. This is a joint work with R. Hadani (Austin).

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