Seminars and Colloquia by Series

Seifert conjecture in the even convex case

Series
CDSNS Colloquium
Time
Monday, March 30, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Slikes 005
Speaker
Chungen Liu – Nankai University, China
The iteration theory for Lagrangian Maslov index is a very useful tool in studying the multiplicity of brake orbits of Hamiltonian systems. In this talk, we show how to use this theory to prove that there exist at least $n$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface in $\R^{2n}$ satisfying the reversible condition. As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer $n$.

Furstenberg sets and Furstenberg schemes over finite fields

Series
Algebra Seminar
Time
Friday, March 27, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jordan Ellenberg – University of Wisconsin, Madison

Please Note: Useful background:The paper I’m discussing: http://arxiv.org/abs/1502.03736Terry Tao’s blog post on Dvir’s theorem: https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel...My earlier paper with Terry and Richard Oberlin about Kakeya restriction over finite fields: http://arxiv.org/abs/0903.1879

The study of extremal configurations of points and subspaces sits at the boundary between combinatorics, harmonic analysis, and number theory; since Dvir’s 2008 resolution of the Kakeya conjecture over finite fields, it has been clear that algebraic geometry is also part of the story.We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant non-reduced subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer. It will, however, try to convince everyone in the room that it can be useful to be an algebraic geometer.This is joint work with Daniel Erman.

Introduction to regularity theory of second order Hamilton-Jacobi-Bellman equations

Series
PDE Working Seminar
Time
Friday, March 27, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Andrzej Swiech – Georgia Tech
I will give a series of elementary lectures presenting basic regularity theory of second order HJB equations. I will introduce the notion of viscosity solution and I will discuss basic techniques, including probabilistic techniques and representation formulas. Regularity results will be discussed in three cases: degenerate elliptic/parabolic, weakly nondegenerate, and uniformly elliptic/parabolic.

Science Matters lecture series - How Not to Be Wrong

Series
Other Talks
Time
Thursday, March 26, 2015 - 19:00 for 1 hour (actually 50 minutes)
Location
Clary Theater, Bill Moore Student Success Center
Speaker
Jordan Ellenberg – University of Wisconsin, Department of Mathematics

Please Note: A reception will follow the talk and giving time for visitors to chat with Ellenberg and each other.

The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how wrong this view is: Math touches everything we do, allowing us to see the hidden structures beneath the messy and chaotic surface of our daily lives. It’s a science of not being wrong, worked out through centuries of hard work and argument.

The Euclidean Distance Degree

Series
Algebra Seminar
Time
Wednesday, March 25, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bernd Sturmfels – UC Berkeley
The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. The Euclidean distance degree is the number of critical points for this optimization problem. We focus on projective varieties seen in engineering applications, and we discuss tools for exact computation. Our running example is the Eckart-Young Theorem which relates the nearest point map for low rank matrices with the singular value decomposition. This is joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani, Rekha Thomas.

Quantum representations of braids

Series
Geometry Topology Student Seminar
Time
Wednesday, March 25, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan Paprocki – Georgia Tech
Solutions to the Yang-Baxter equation are one source of representations of the braid group. Solutions are difficult to find in general, but one systematic method to find some of them is via the theory of quantum groups. In this talk, we will introduce the Yang-Baxter equation, braided bialgebras, and the quantum group U_q(sl_2). Then we will see how to obtain the Burau and Lawrence-Krammer representations of the braid group as summands of natural representations of U_q(sl_2).

Functional Completions and Complex Vector Lattices

Series
Analysis Seminar
Time
Wednesday, March 25, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris Schwanke – University of Mississippi –
In this talk, we demonstrate how to use convexity to identify specific operations on Archimedean vector lattices that are defined abstractly through functional calculus with more concretely defined operations. Using functional calculus, we then introduce functional completions of Archimedean vector lattices with respect to continuous, real-valued functions on R^n that are positively homogeneous. Given an Archimedean vector lattice E and a continuous, positively homogeneous function h on R^n, the functional completion of E with respect to h is the smallest Archimedean vector lattice in which one is able to use functional calculus with respect to h. It will also be shown that vector lattice homomorphisms and positive linear maps can often be extended to such completions. Combining all of the aforementioned concepts, we characterize Archimedean complex vector lattices in terms of functional completions. As an application, we construct the Fremlin tensor product for Archimedean complex vector lattices.

Global well-posedness for some cubic dispersive equations

Series
PDE Seminar
Time
Tuesday, March 24, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin Dodson – Johns Hopkins University
In this talk we examine the cubic nonlinear wave and Schrodinger equations. In three dimensions, each of these equations is H^{1/2} critical. It has been showed that such equations are well-posed and scattering when the H^{1/2} norm is bounded, however, there is no known quantity that controls the H^{1/2} norm. In this talk we use the I-method to prove global well posedness for data in H^{s}, s > 1/2.

The view from the other side of the table

Series
Professional Development Seminar
Time
Tuesday, March 24, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Adam Fox – Western New England University
Faculty jobs at smaller teaching schools are highly sought after by those who value their balance of education, research, and service. Hear what it takes to succeed in this market from a former GT postdoc, who is a new assistant professor and recent veteran of many JMM interviews --- from the employer side of the table!

A Non-convex Approach for Signal and Image Processing

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, March 24, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Yifei Lou – UT Dallas
A fundamental problem in compressed sensing (CS) is to reconstruct a sparsesignal under a few linear measurements far less than the physical dimensionof the signal. Currently, CS favors incoherent systems, in which any twomeasurements are as little correlated as possible. In reality, however, manyproblems are coherent, in which case conventional methods, such as L1minimization, do not work well. In this talk, I will present a novelnon-convex approach, which is to minimize the difference of L1 and L2 norms(L1-L2) in order to promote sparsity. Efficient minimization algorithms areconstructed and analyzed based on the difference of convex functionmethodology. The resulting DC algorithms (DCA) can be viewed as convergentand stable iterations on top of L1 minimization, hence improving L1 consistently. Through experiments, we discover that both L1 and L1-L2 obtain betterrecovery results from more coherent matrices, which appears unknown intheoretical analysis of exact sparse recovery. In addition, numericalstudies motivate us to consider a weighted difference model L1-aL2 (a>1) todeal with ill-conditioned matrices when L1-L2 fails to obtain a goodsolution. An extension of this model to image processing will be alsodiscussed, which turns out to be a weighted difference of anisotropic andisotropic total variation (TV), based on the well-known TV model and naturalimage statistics. Numerical experiments on image denoising, imagedeblurring, and magnetic resonance imaging (MRI) reconstruction demonstratethat our method improves on the classical TV model consistently, and is onpar with representative start-of-the-art methods.

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