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Seminars, Colloquia, and Conferences

Colloquium

The colloquium meets on Fridays at 4:00pm in Bromfield-Pearson 101, unless otherwise indicated.

Spring 2014

Date Speaker Title
January 17 Petra Schwer (née Hitzelberger)
Mathematisches Institut, Universität Münster
Non-positive curvature, braids and buildings
January 31 Genevieve Walsh
Tufts University
3-manifolds, surgery, and a graph
February 7 Fulton Gonzalez
Tufts University
The world of mean value operators
February 28 George McNinch
Tufts University
The Frobenius map (or: Why study things mod p?)
March 7 Todd Quinto
Tufts University
Microlocal Analysis in Limited Data Tomography
March 14 Arvind Krishna Saibaba
Tufts University
Computational Challenges in Hyperspectral Diffuse Optical Tomography
March 28 Tim Mitchell
Courant Institute, New York University
(Tufts Alumnus)
Fast approximation of the H_∞ norm via hybrid expansion-contraction using spectral value sets
April 4 Jürgen Frikel
Technische Universität München
Reconstructions in limited data tomography
April 11 Yusuf Mustopa
Northeastern University
Clifford Algebras, Old and New
April 18 Frank Filbir
Helmholtz Center München
Parameter Estimation in Exponential Sums
April 25 Michael Overton
Courant Institute, New York University
Investigation of Crouzeix's Conjecture via Optimization

Abstracts

January 17, 2014
Non-positive curvature, braids and buildings
Petra Schwer (née Hitzelberger), Mathematisches Institut, Universität Münster

We will present the notion of a CAT(0) and more generally CAT(k)-space which are spaces with certain upper curvature bounds. While these curvature properties are in general hard to verify one can deduce many algebraic properties of a group from an action on such a space. We show that braid groups with at most 6 strands are CAT(0), i.e. act geometrically on a CAT(0) space, using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings.

January 31, 2014
3-manifolds, surgery, and a graph
Genevieve Walsh, Tufts University

We will describe a construction which allows one to go from one 3-manifold to another, called Dehn surgery. Then we will use this construction to build an infinite graph whose vertices correspond to 3-manifolds. Along the way, properties of 3-manifolds and properties of infinite metric graphs will be discussed. New results stated are joint with Neil Hoffman.

February 7, 2014
The world of mean value operators
Fulton Gonzalez, Tufts University

The spherical mean value of a continuous function f on a Riemannian manifold M is the average of f over a sphere of any radius r. We'll explore the relation between the spherical mean value operator and harmonic functions and, more generally, eigenfunctions of invariant differential operators. Then, we'll look at the spherical mean value operator as an integral transform and examine questions about its kernel and range. Finally, we look at how we can use the mean value operator to solve certain differential equations. We'll show that a common theme that occurs throughout is the action of (mostly) compact groups on manifolds.

February 28, 2014
The Frobenius map (or: Why study things mod p?)
George McNinch, Tufts University

The talk hopes to gives some motivation(s) for why one should study algebraic and algebro-geometric objects defined "over fields of characteristic p > 0", for a prime number p. The talk will be almost exclusively expository: it will (at least loosely) explain a number of examples, constructions, and perspectives. Here are some examples I hope to discuss or at least mention: the Frobenius morphism and the Galois group of a rational polynomial, finite simple groups, normality of Schubert varieties, the Hasse principle, and possibly the Weil conjectures.

March 7, 2014
Microlocal Analysis in Limited Data Tomography
Todd Quinto, Tufts University

In this talk, I will give an overview of how microlocal analysis is used to understand limited data problems in tomography, in particular, the limited angle problem. All of the challenges of limited data tomography are illustrated by this classical problem.

First, the physical problem will be described and reconstructions presented and evaluated. Then, we introduce microlocal analysis, a deep mathematical framework to characterize singularities. We give a paradigm to analyze limitations in all limited angle problems and use this to analyze the limited angle reconstructions. Finally, we describe our artifact reduction procedure and put the work in the context of singular pseudodifferential operators.

March 14, 2014
Computational Challenges in Hyperspectral Diffuse Optical Tomography
Arvind Krishna Saibaba, Tufts University

Diffusive Optical Tomography (DOT) is an emerging technology for breast tumour detection and brain imaging in which the region of interest is illuminated with near infrared light at a specific wavelength and the data are comprised of observations of the resulting scattered diffuse fields. The characterization of tumours as well as reconstruction of the large scale structure of the breast can be mathematically described as an ill-posed, non-linear inverse problem. New technology allows for the collection of hyperspectral data. Although we anticipate that the availability of more information using multiple wavelengths increases the accuracy of the reconstruction, the use of hyperspectral data poses a significant computational burden in the context of image recovery. In this talk, I will discuss various computational challenges and our approach to tackle the large-scale inverse problem.

March 28, 2014
Fast approximation of the H_∞ norm via hybrid expansion-contraction using spectral value sets
Tim Mitchell, Courant Institute, New York University (Tufts Alumnus)

The H_∞ norm is a well known and important measure arising in many applications that quantifies the least amount of perturbation a linear dynamical system with inputs and outputs can incur such that it may no longer be asymptotically stable. Unfortunately, merely calculating it is an expensive proposition of finding a global optimum to a nonconvex and nonsmooth optimization problem, with the standard algorithm requiring cubic cost per iteration to compute it.

In 2013, Guglielmi, Gürbüzbalaban, and Overton presented the first fast algorithm to approximate the H_∞ norm using a spectral value set based approach that does not require reducing the dimension of the control system, only that it be sparse and have relatively few inputs and outputs. However, despite its favorable scaling properties and apparent ability to usually provide good approximations to the H_∞ norm, the method can sometimes critically break down in practice.

We present an improved spectral value set based algorithm using a novel hybrid expansion-contraction scheme that, under no additional assumptions, guarantees convergence to a stationary point of the optimization problem but importantly does not break down. Furthermore, we present new optimizations to accelerate both methods and handle cases where the number of inputs and/or outputs may be large.

Joint work with Michael L. Overton

April 4, 2014
Reconstructions in limited data tomography
Jürgen Frikel, Technische Universität München

The reconstruction of tomographic slices from x-ray CT data is an interesting and challenging problem in medical imaging. In particular, there are many applications, such as digital breast tomosynthesis, dental tomography, electron microscopy etc., where the data is available at a limited angular range only. In this case the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process.

In this talk, we will give an overview of limited angle tomography and introduce a reconstruction framework which is based on curvelet expansions. Moreover, we will show that curvelet expansions allow to integrate microlocal information about visible singularities into the reconstruction and, simultaneously, to achieve preservation of edges in the reconstruction. In particular, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will present the practical relevance of these results.

April 11, 2014
Clifford Algebras, Old and New
Yusuf Mustopa, Northeastern University

Clifford algebras have long been important in representation theory, Riemannian geometry and physics in addition to the theory of quadratic forms. During the past few decades they been fruitfully generalized in a number of directions, with consequences for fields as diverse as convex optimization and complex algebraic geometry. In this talk, which is largely based on joint projects with Emre Coskun, Rajesh S. Kulkarni, and Ian Shipman, I will survey some of these generalized Clifford algebras and what is currently known about their structures.

April 18, 2014
Parameter Estimation in Exponential Sums
Frank Filbir, Technische Universität and Helmholtz Center München

Let I ≥ 1 be an integer, \(ω_0 = 0 < ω_1 < ... < ω_I \leq π\), and for \(j = 0,...,I\), \(a_j ∈ ℂ\), \(a_ -j = \overline{a_j}\), \(ω_− j = −ω_j\), and \(a_j≠ 0\) if \(j≠ 0\). We consider the following problem:

Given finitely many noisy samples of an exponential sum of the form
\(˜x(k)= \sum_{j=−I}^I  a_j exp(−iω_jk)+ ε(k),\)   \(k = −2N,... ,2N,\)

where ε(k) are random variables with mean zero, each in the range [−ε,ε] for some ε > 0, determine approximately the frequencies ωj. We combine the features of several recent works to use the available information to construct the moments \(˜y_N (k)\) of a positive measure on the unit circle. In the absence of noise, the support of this measue is exactly \({exp(−iω_j) : aj ≠ 0}\). This support can be recovered as the zeros of the monic orthogonal polynomial of an appropriate degree on the unit circle with respect to this measure. In the presence of noise, this orthogonal polynomial structure allows us to provide error bounds in terms of ε and N. It is not our intention to propose a new algorithm. Instead, we prove that a preprocessing of the raw moments \(˜x(k)\) to obtain \(˜y_N (k)\) enable us to obtain rigorous performance guarantees for existing algorithms. We demonstrate also that the proposed preprocessing enhances the performance of existing algorithms.

April 25, 2014
Investigation of Crouzeix's Conjecture via Optimizatio
Michael Overton (Courant Institute, New York University)

Crouzeix's conjecture is a fascinating open problem in matrix theory. We present a new approach to its investigation using optimization. Let p be a polynomial ofany degree and let A be a square matrix of any order.

Crouzeix's conjecture is the inequality

||p(a)|| ≤ ||p||W(A)

Here the left-hand side is the 2-norm of the matrix p(A), while the norm on the righthandside is the maximum of jp(z)j over z 2W(A), the field of values (or numericalrange) of A. It is known that the conjecture holds if 2 is replaced by 11.08 (Crouzeix2007).

Joint work with Anne Greenbaum, Adrian S. Lewis and Lloyd N. Trefethen


Fall 2013

Date Speaker Title
September 27 Alex Barnett
Dartmouth College
"Efficient and robust integral equation methods for acoustic scattering from periodic media in 2D and 3D"
October 18 Koichi Kaizuka
University of Tsukuba, Japan
"The Strichartz conjecture on symmetric spaces"
October 25
*new date
Christoph Börgers
Department of Mathematics, Tufts University
"What is computational neuroscience, and is it good for anything?"
November 1 Kye Taylor
Department of Mathematics, Tufts University
"Image analysis with graphs"
November 8 Bruce Boghosian
Department of Mathematics, Tufts University
"Asset exchange models, the origin of Pareto's Law, and the origin of oligarchy"
November 15 Tim Atherton
Department of Physics, Tufts University
"Why topology suddenly matters in physics"
December 6 Umberto Villa
Lawrence Livermore National Laboratory
"Numerical Upscaling and Algebraic Multigrid for Mixed Finite Element Discretizations"

Abstracts

September 27, 2013
Efficient and robust integral equation methods for acoustic scattering from periodic media in 2D and 3D
Alex Barnett, Department of Mathematics (Dartmouth College)

A growing number of our technologies (telecommunications, radar, solar energy, etc) rely on the manipulation of linear waves at the wavelength scale, and advances in numerical modeling continue to be key to such progess. We study the common grating diffraction problem where time-harmonic scalar waves scatter from a periodic medium. We develop new solvers for 2D and 3D cases, for either isolated obstacles or a connected "bumpy surface". After reviewing the integral equation method for solving PDEs, I will explain the two innovations that allow our solvers to be highorder and optimal O(N) complexity: 1) new surface quadrature schemes compatible with the fast multipole method (extending our recent QBX scheme to 3D), and 2) robust ways to "periodize" the integral equation so that the unknowns live only on a single period of the geometry. (This includes joint work with Leslie Greengard, Zydrunas Gimbutas, Adrianna Gillman, Andreas Klockner, and Mike O'Neil.)
 

October 18, 2013
The Strichartz conjecture on symmetric spaces
Koichi Kaizuka (University of Tsukuba, Japan)

A famous classical result characterizes harmonic functions on the unit disk by the Poisson integral formula. This can be generalized to the Poisson transform, which is the integral transform by the complex powers of the Poisson kernel. In 1970, Helgason proved that any eigenfunction of the Laplacian on the Poincar´e disk is given by the image of the Poisson transform of a boundary value on the unit circle. It is well-known that via the group action of M¨obius transformations (which preserve the unit disk), the Poincar´e disk can be regarded as the real hyperbolic surface, i.e a symmetric space of noncompact type. Using Lie group techniques, this can be generalized to a similar characterization problem for symmetric spaces of noncompact type. Helgason (Advances in Math. (1970)) conjectured a characterization of joint eigenfunctions in terms of the Poisson transform in the general case, and this conjecture was proved by six Japanese mathematicians (Ann. of Math. (1978)). After that, from the point of view of spectral theory, Strichartz (J. Funct. Anal. (1989)) formulated a conjecture concerning a different image characterization of the Poisson transform of the L2-space on the boundary. Except for a special case, the Strichartz conjecture had remain unsolved up to the present moment. In this talk, we employ techniques in scattering theory to present a positive answer to the Strichartz conjecture.
 

October 25, 2013
What is computational neuroscience, and is it good for anything?
Christoph Börgers, Department of Mathematics (Tufts University)

A large part of computational neuroscience (not all of it) is the study of differential equations models of nerve cells and brain networks. Nobody can know at this point whether this sort of modeling will play an important role in understanding the brain. In this talk, I will give examples from my own work.
 

November 1, 2013
Image analysis with graphs
Kye Taylor, Department of Mathematics (Tufts University)

Extracting the salient features from a dataset that consists of many datapoints in highdimensional space is a problem of wide contemporary interest. Many techniques for extracting such information rely on the assumption that the high-dimensional observations are created by a system that depends on a relatively small number of variables. These techniques are collectively referred to as nonlinear dimensionality reduction techniques, and most of the associated theoretical results assume the data lies on a smooth, compact submanifold of R^n.

In this talk, I will describe a technique that performs nonlinear dimensionality reduction by embedding a dataset of points into a low-dimensional Euclidean space based on discrete approximations to the eigenfunctions of the manifold's Laplace-Beltrami operator. The technique is referred to as the Laplacian eigenmap (which is intimately related to the Diffusion map).

After introducing the Laplacian eigenmap, I will discuss its applications in image analysis. I will describe some of the arguments for and against the existence of smooth and compact image manifolds, then show some embeddings of image data. The intuition gained from our observations will suggest a simple model that can be used to obtain theoretical results, without assuming the existence of any underlying manifold. In particular, I will relate the embedding to a metric on an undirected graph (or network), and examine this metric – referred to as the commute-time distance – in detail.
 

November 8, 2013
Asset exchange models, the origin of Pareto's Law, and the origin of oligarchy
Bruce Boghosian, Department of Mathematics (Tufts University)

A key assumption of neoclassical economic theory is that economic agents trade to further their own best interests without making mistakes. In the real world, agents make mistakes, resulting in losses for some and gains for others. Because the effect of these mistakes can be to anybody's benefit or detriment, one might think that their net effect would average away, and that the principal results of neoclassical theory would be robust in this regard, albeit with some level of superposed noise. It turns out that this intuition is incorrect, and that the effect of a constant rate of mistakes, no matter how small or infrequent they may be, is a gross distortion in the overall distribution of wealth, tending to concentrate it in the hands of a small minority of agents. This effect can be kept in check by some amount of redistribution – for example by taxation and public spending or by price controls. In this work, we show that the combination of these mechanisms is sufficient to explain the general form of Pareto's Law of wealth distribution, a key empirical macroeconomic result that has resisted a microeconomic explanation for over a century. This is interesting, but the consequences of this observation are far broader than that. In essence, contrary to most Western economic policy of the past three decades – but perfectly consistent with its observed consequences – the more "free" the market, the fewer are the mechanisms for wealth redistribution, and the greater is the tendency toward oligarchy. In this light, it is not surprising that the sudden withdrawal of price controls and state subsidies called for by the "shock therapy" imposed on the states of the former Soviet Union in the early 1990s led directly to the oligarchies that currently prevail in many of those states.
 

November 15, 2013
Why topology suddenly matters in physics
Tim Atherton, Department of Physics, Tufts University

The study of materials possessing so-called topological order has been an area of remarkable interest in Physics over the past decade, notably including the award of a Nobel prize to the discoverers of the most famous of such materials, Graphene. It's quite natural to ask: What exactly about these materials is topological? In this talk, I aim to answer that question by presenting a simple toy model of a 1D topological material that has recently been realized experimentally through a photonic analog, and using this to illustrate more complex cases. Along the way, I will make some hopefully entertaining asides revealing unexpected mathematical links between such apparently disparate things as conformal mapping and cloaking devices, Chebyshev polynomials and butterflies.
 

December 6, 2013
Numerical Upscaling and Algebraic Multigrid for Mixed Finite Element Discretizations
Umberto Villa, Lawrence Livermore National Laboratory

The Mixed Finite Element Method shows great potential for the discretization of a large class of partial differential equations (PDEs) that model physical problems of practical relevance in various fields of engineering, including fluid-dynamics, solid mechanics, and electromagnetism. Many applications of these models feature a multi-physics and multi-scale nature that poses a substantial challenge to state-of-theart solvers. Upscaling techniques can reduce computational cost by solving coarse scale models that take into account interactions at different scales.

In this talk, we will introduce a novel numerical upscaling technique that can be applied in the settings of mixed finite element discretizations and unstructured meshes. Our approach is based on a specialized element-based agglomeration technique that allows us to construct hierarchies of coarse spaces that possess stability and approximation properties for wide classes of PDEs. More specifically, the spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of H1-conforming, H(curl)-conforming, H(div)-conforming, and L2 spaces associated with an unstructured fine mesh. The procedure can be recursively applied so that a hierarchy of nested de Rham sequences can be constructed. This hierarchy exhibits approximation properties comparable to those of the original fine-grid spaces and can be employed as a discretization tool in multilevel Monte Carlo processes or as an algebraic multigrid (AMG) preconditioner for iterative solvers.

Numerical results will illustrate the validity of our approach both as a discretization tool (upscaling) and as a solver (AMG). An application to subsurface flow simulation will also be presented.

This is joint work with Dr. P. Vassilevski.

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