Distances Between Approximately Unitary Equivalence Classes of Self-adjoints in C*-algebras

Committee: A. Toms(Ch), L. Brown, M. Dadarlat, B. McReynolds

Bose Einstein Condensation, The Nonlinear Schrodinger Equation, and a Central Limit Theorem

Abstract: Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body dynamics, a step towards large deviations for Bose-Einstein condensation.

The Ovsyannikov Theorem and Applications to Nonlocal Evolution Equations

Abstract: In the first part of this talk we shall presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces. In addition to existence and uniqueness of solution, we shall provide an estimate about the analytic lifespan of the solution. In the second part we shall present applications to the Cauchyproblem for Camassa-Holm type equations and systems with initial data in spaces of analytic function on both the circle and the line. Finally, we shall discuss the continuity of the data-to-solution map in spaces of analytic functions.

Refreshments served in the Math Library Lounge at 4:00 P.M.

Detecting association between copy number variation and multiple myeloma: a functional data analytic approach

The relationship between the copy number profiles and clinical biomarkers are often known to be very complicated, especially in a heterogeneous disease such as cancer. Such a problem is further complicated by the serial correlation and measurement error present in the copy number measurements. In this talk, we consider a semiparametric functional regression model that relates a continuous outcome of interest (e.g., measurements on beta2-microblobulin) to copy number profile observed with measurement errors, adjusting for other demographic covariates. In this framework, we develop a statistical procedure to test for association between the observed copy number profiles and the outcome of interest, while accounting for possibly complex nonlinear underlying relationships and interactions. We investigate the finite sample performance of our procedure via a simulation study, and illustrate our by analyzing a Multiple Myeloma data set collected by the Multiple Myeloma Research Consortium (MMRC).

Co-authors: Adrian Coles, Ganiraju Manyam, Veera Baladandayuthapani

Associated Reading:

Baladandayuthapani, V., Ji, Y., Talluri, R., Nieto-Barajas, L. E., and Morris, J. S. (2010), "Bayesian Random Segmentation Models to Identify Shared Copy Number Aberrations for Array CGH Data," Journal of the American Statistical Association, 105, 1358-1375.

Click here for a full schedule of BIOINFORMATICS SEMINARS, past and present.

Boundary Components of Mumford-Tate domains

Abstract: Mumford-Tate groups arise as the natural symmetry groups of Hodge structures and their variations. I describe recent work with Matt Kerr on computing the Mumford-Tate group of the Kato-Usui boundary components of a degeneration of Hodge structure.

Drunken Birds, Brownian Motion, and Other Random Fun

Abstract: Flip a coin. If it lands heads up, take a step forward. Otherwise, take astep back. Wait a second, then repeat this process infinitely many times and you have a process called a Simple Symmetric Random Walk. The Simple symmetric random walk is the prototype of a Martingale or "fair game". Brownian Motion is the analogous process in continuous space and time, (i.e. it is real-valued rather than integer-valued and our "random walker" doesn't wait a second between steps).

I will give a brief introduction to random walks, Brownian motion, andmartingales, and a couple quick examples of how these tools can be used to prove familiar theorems in complex analysis.

Breuil's Conjecture on Strongly Divisible Lattices in the r = p-1 Unipotent Case

Committee: T. Liu(Ch), E. Goins, D. Goldberg, F. Shahidi

Quasidiagonality and KK-theory of Continuous Fields of C*-algebras

Committee: M. Dadarlat(Ch), S. Bell, A. Toms, M. Torres

Universal Operations in Feynman Categories and Relations to Other Theories

Abstract: First we will briefly recall the definition of a Feynman category. We will then give the relations to colored operads and patterns which were defined by Getzler. Then we will discuss newer examples and the relation of Feynman categories to Lavwere theories and crossed simplicial groups. Finally, we discuss how universal operations natural arise from Feynman categories by taking colimits in a cocompletion.This includes the pre-Lie operations for operads, the Lie admissible operations for di-operads, the Kontsevich Soibelman minimal operad for operads with A_∝ multiplication.

The Week That Changed Everything

The week of September 8, 2008 was one for the record books on Wall St. By the time that week ended, two of the most respected and well-known names in the investment banking industry were gone. But how did we get to that point? And how did banks and hedge funds survive arguably the most volatile time in US financial history? I'll give you a firsthand perspective of my experience at Merrill Lynch prior to its collapse and then at a hedge fund while it was all occurring. I'll share some stories of frustration, relief, fear, and satisfaction, all common emotions during such a trying time.

Fractional Eigenvalues

Abstract: I will discuss a non-local eigenvalue problem that arises as the Euler-Lagrange equation of Rayleigh quotients in the fractional Sobolev spaces. This can be seen as a non-local or fractional version of the eigenvalue problem for the p-Laplacian. In particular, I will talk about the limiting case when p goes to infinity.

Cohomology of Operad Algebras and Deligne's Conjecture

Committee: R. Kaufmann(Ch), McClure, Toms, McReynolds

Business Analytics and the Opportunity for Applied Statisticians

The field of Business Analytics has grown quickly and has become a relatively new career choice for applied statisticians. A recent McKinsey Global Institute report projects the demand for ‘deep analytical’ positions will exceed the supply by 50-60% by 2018. Gartner Research suggests that data analytics is expected to create 4.4 million jobs worldwide by 2015 but that the available supply of skilled workers will only be able to fill a third of those projected openings. Clearly this creates an exciting professional opportunity for applied statisticians.

This talk will explore the background and foundations of the new and developing field of Business Analytics and suggest why this has become such a dynamic and progressive area of growth. We will review some of the key skills and capabilities required to be successful in this new field and provide an example to demonstrate these points. Finally we will consider some the implications of Business Analytics for Applied Statistics and applied statisticians.

There will be a reception before the seminar in HAAS 111, 9:30 am-10:15 am.

Compressed Sensing without Incoherence

Abstract: Compressed sensing is a theory and collection of techniques for the recovery of sparse signals and images from seemingly highly incomplete sets of measurements. To achieve this, it exploits a property known as incoherence between the measurement device and the sparsity system (the basis or frame in which the object to be recovered is assumed to be sparse). Unfortunately, many practical problems in which one wishes to apply compressed sensing techniques are not incoherent. This is the case in Magnetic Resonance Imaging (MRI), for example.

In this talk I will commence with an overview of compressed sensing, and explain the limitations of the current theory. Next I will introduce a new theory of compressed sensing, where incoherence is replaced by so-called asymptotic incoherence. As I will explain, this new concept is far more realistic in applications, and leads to a mathematical theory that explains why compressed sensing can be successful in practice, even for coherent problems such as MRI. In the final part of the talk, I will discuss some important conclusions for the applicability of compressed sensing that arise out of this new theory.

This is joint work with Anders C. Hansen, Clarice Poon and Bogdan Roman.

Why Should I Care About Lie Groups?

Abstract: Sometimes differential equations have an obvious symmetry which leads to a natural guess for its solution. The Norwegian mathematician Marius Sophus Lie (1842 -- 1899) spent most of his career attempting to generalize ideas of fellow Norwegian Niels Henrik Abel (1802 -- 1829) from discrete groups of symmetries of algebraic objects to continuous groups of symmetries of topological objects. In the process, Lie created a new branch of mathematics which united differential geometry and abstract algebra. In this talk, we give a brief introduction to the pulchritude of Lie's ideas. From the geometric nature of manifolds to the analytic nature of differential equations, we discuss the natural group action of the space of vector fields of a manifold on itself. We conclude the talk with a discussion of the computation of Lie group of the real line.

Approximation of Functions of Many Variables on Sparse Grids

Abstract: We formulate approximation of functions in many variables on sparse grids as the evaluation of a dimenssion reducible sum. This framework allows us to develop fast and acurate algorithms for constructing approximations by the Fouries basis, orthogonal polynomial basis and B-spline quasi-interpolation.

Deligne-Mumford-Knudsen Compactification of M₀,n

Abstract: M0,n (n>=3) is the moduli space of genus zero curve with n marked points over complex numbers. It can be thought of as the configuration space of n points on a sphere up to automorphisms that respect marked points. We will describe Deligne-Mumford-Knudsen compactification of this space, and explain about the topological and algebraic operad structure that it exhibits. This leads to the notion of (formal) Frobenius manifold. Quantum cohomology is a manifestation of this structure.

Parabolic Equations in Sub-Riemannian Spaces: Boundary Behavior of Non-negative Solutions, Curvature-dimension Inequalities, Li-Yau Inequalities and Volume Growth

Committee: F. Baudoin(Ch), N. Garofalo(Co-Chair), D. Danielli-Garofalo, N-K. Yip

Local topology of loop spaces in sub-Riemannian manifolds

Abstract: Given a step 2 Subriemannian manifold, we consider the space Omega(x,y) of admissible paths with bounded energy, joining a fixed initial point x to a point y. By Morse theory the topology of this space is closely related to the number of geodesics joining the twopoints, geodesics being extremals of the energy functional. We study in particular the behaviour of the topology of Omega(x,y) as we movex closer to y; it appears that its topological complexity grows unbounded, having as a side effect that the number of bounded geodesics grows unbounded as well (by Morse inequalities).This is a joint work with A. A. Agrachev and A. Lerario

TBA

Refreshments served in the Math Library Lounge at 4:00 P.M.

Partial Desingularization

Abstract: Resolution of singularities consists in constructing a non-singular model of an algebraic variety. This is done by applying a proper birational map that is a local isomorphism at the smoothpoints. Often too much information is lost if all singularities are modified. In these cases, a desingularization preserving some minimal singularities is necessary. This suggests the question of whether, given a class of singularity types S, it is possible to remove with abirational map all singularities not in S while still having a local isomorphism over the singularities of type S. We will talk about several instances of this problem and techniques that can be used tosolve them.

Domains of Holomorphy in Several Complex Variables

Abstract: Moving from one complex variable to several introduces a new depth to the theory of holomorphic functions which far exceeds the simplejuggling of appropriate multi-indices. Entirely new phenomena present themselves from the outset. As an introduction to the subject, I will outline the concept of a “domain of holomorphy,” beginning with Hartogs' Phenomenon, a result which places a surprising restriction on the possible singularities of analytic functions.

Strong Unique Continuation for Hyperbolic Operators

Committee: A. Sa Barreto(Ch), S. Bell, L. Lempert, A. Petrosyan

TBA

When You Can't Have It All: Knowing What to Give Up and When

Abstract: Closed, complete, or compact spaces are often good mathematical spaces to work in because existence theorems follow more easily. Spaces where limits exist are good. Unfortunately, often the "natural" spaces we are given don't have these properties. However, if we are willing to give up some structure, then certain other properties such as existence may follow more easily. A simple example is moving from the rationals to the real number line by giving up the requirement that a number be the ratio of two integers. In this talk we'll discuss how this concept of giving up structure has become a powerful tool in many areas of mathematics. We'll touch on "weak solutions" and "distributions" in differential equations as well as "currents" and "varifolds" in geometric measure theory.

TBA

Thin Free Boundary Problems

Committee: A. Petrosyan(Ch), D. Danielli-Garofalo, N. Garofalo, N-K. Yip

Efficient Estimation of Failure Probability

Committee: D. Xiu(Co-Ch), J. Shen(Co-Ch), G. Buzzard, S. Dong

TBA

Refreshments served in the Math Library Lounge at 4:00 P.M.

Analysis of the Inverse Boundary Value Prolem and Iterative Methods in Banach Spaces

Committee: M. De Hoop(Ch), P. Li, A. Sa Barreto, P. Stefanov

TBA

Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials

Abstract: The practical application of integral equation methods requires the evaluation of boundary integrals with singular, weakly singular or nearly singular kernels in complicated domains. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by the construction of corrections to high order non-singular rules for specific kernels, by singularity subtraction/cancellation, or by kernel regularization and asymptotic analysis. We have developed a systematic, high order approach that works for any singularity (including hypersingular kernels), based only the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method.

Refreshments will be served in the Math Library Lounge at 3:00 P.M.

Signal/Image Registration via Polynomial System Solution Method and the Pascal Triangle of a Discrete Image

Committee: M. Boutin(Ch), B. Adcock, S. Basu, U. Walther

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Refreshments will be served in the Math Library Lounge at 3:00 P.M.