Past

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