Seminars and Colloquia by Series

Sparse sum-of-squares certificates on finite abelian groups

Series
Algebra Seminar
Time
Monday, March 30, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hamza Fawzi – MIT
We consider functions on finite abelian groups that are nonnegative and also sparse in the Fourier basis. We investigate conditions under which such functions admit sparse sum-of-certificates certificates of nonnegativity, i.e., certificates where the functions in the sum of squares decomposition have a small common sparsity pattern. Our conditions are purely combinatorial in nature, and are based on finding particularly nice chordal covers of a certain Cayley graph. These techniques allow us to show that any nonnegative quadratic function in binary variables is a sum of squares of functions of degree at mostceil(n/2), resolving a conjecture of Laurent. After discussing the connection with semidefinite programming lifts of polytopes, we also see how our techniques provide an example of separation between sizes ofsemidefinite programming lifts and linear programming lifts. This is joint work with James Saunderson and Pablo Parrilo.

A method of computation of 2D Fourier transforms and diffraction integrals with applications in vision science

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 30, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Andrei Martinez-Finkelshtein – University of Almería
The importance of the 2D Fourier transform in mathematical imaging and vision is difficult to overestimate. For instance, the impulse response of an optical system can be defined in terms of diffraction integrals, that are in turn Fourier transforms of a function on a disk. There are several popular competing approaches used to calculate diffraction integrals, such as the extended Nijboer-Zernike (ENZ) theory. In this talk, an alternative efficient method of computation of two dimensional Fourier-type integrals based on approximation of the integrand by Gaussian radial basis functions is discussed. Its outcome is a rapidly converging series expansion for the integrals, allowing for their accurate calculation. The proposed method yields a reliable and fast scheme for simultaneous evaluation of such kind of integrals for several values of the defocus parameter, as required in the characterization of the through-focus optics.

Independence of Whitehead Doubles of Torus Knots in the Smooth Concordance Group

Series
Geometry Topology Seminar
Time
Monday, March 30, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Juanita Pinzon-Caicedo – University of Georgia
In the 1980’s Furuta and Fintushel-Stern applied the theory of instantons and Chern-Simons invariants to develop a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In turn, using the fact that the 2-fold cover of S^3 branched over the Whitehead double of a positive torus knot is negatively cobordant to a Seifert fibred homology sphere, Hedden-Kirk establish conditions under which an infinite family of Whitehead doubles of positive torus knots are independent in the smooth concordance group. In the talk, I will review some of the definitions and constructions involved in the proof by Hedden and Kirk and I will introduce some topological constructions that greatly simplify their argument. Time permiting I will mention some ways in which the result could be generalized to include a larger set of knots.

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.

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