Seminars and Colloquia by Series

Eigenvalues and the shapes of very small things

Series
Research Horizons Seminar
Time
Wednesday, September 22, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Evans Harell – School of Mathematics - Georgia Institute of Technology

Please Note: Hosts: Yao Li and Ricardo Restrepo

When an object is small enough that quantum mechanics matters, many of its physical properties, such as energy levels, are determined by the eigenvalues of some linear operators. For quantum wires, waveguides, and graphs, geometry and topology show up in the operators and affect the set of eigenvalues, known as the spectrum. It turns out that the spectrum can't be just any sequence of numbers, both because of some general theorems about the eigenvalues and because of inequalities involving the shape. I'll discuss some of the extreme cases that test the theorems and inequalities and connect them to the shapes of the structures and to algebraic properties of the operators.To understand this lecture it would be helpful to know a little about PDEs and eigenvalues, but no knowledge of quantum mechanics will be needed.

Non-archimedean amoebas and tropical geometry

Series
Tropical Geometry Seminar
Time
Wednesday, September 22, 2010 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Matt Baker – Georgia Tech
I will discuss the correspondence between non-archimedean amoebas and tropical varieties, which is a generalization of the theory of Newton polygons to polynomials in several variables.

An unusual duality principle for fully nonlinear equations

Series
PDE Seminar
Time
Tuesday, September 21, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Professor Scott Armstrong – University of Chicago
We discuss how to solve a Hamilton-Jacobi-Bellman equation ``at resonance." Our characterization is in terms of invariant measures and is analogous to the Fredholm alternative in the linear case.   

The left frontal lobe´s role in language

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 20, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 002
Speaker
Christopher Rorden – Center for Advanced Brain Imaging (Gatech/GSU)
This talk showcases how we can use emerging methods to understand brainfunction. Many of the techniques described could be optimized usingtechniques being developed by researchers in the GT Mathematicsdepartment. A primary tenet of neuroscience is that the left frontal lobeis crucial for speech production and the posterior regions of the lefthemisphere play a critical role in language comprehension and wordretrieval. However, recent work shows suggests the left frontal lobe mayalso aid in tasks classically associated with posterior regions, such asvisual speech perception. We provide new evidence for this notion based onthe use brain imaging (structural and functional MRI) and brainstimulation techniques (TMS and tDCS) in both healthy individuals andpeople with chronic stroke. Our work takes these theoretical findings andtests them in a clinical setting. Specifically, our recent work suggeststhat stimulation of the frontal cortex may complement speech therapy inchronic stroke. Our recent brain stimulation work using transcranialdirect current stimulation supports this hypothesis, illustrating smallbut statistically significant benefits in anomia following brainstimulation.

Diamond-free Families

Series
Combinatorics Seminar
Time
Friday, September 17, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Jerry Griggs, Carolina Distinguished Professor and Chair – Mathematics, University of South Carolina
Given a finite poset $P$, we consider the largest size ${\rm La}(n,P)$ of a family of subsets of $[n]:=\{1,\ldots,n\}$ that contains no  subposet $HP.  Sperner's Theorem (1928) gives that ${\rm La}(n,P_2)= {n\choose{\lfloor n/2\rfloor}}$,  where $P_2$ is the two-element chain.    This problem has been studied intensively in recent years, and it is conjectured that $\pi(P):=  \lim_{n\rightarrow\infty} {\rm La}(n,P)/{n\choose{\lfloor n/2\rfloor}}$  exists for general posets $P$, and, moreover, it is an integer. For $k\ge2$ let $D_k$ denote the $k$-diamond poset $\{A< B_1,\ldots,B_k < C\}$. We study the average number of times a random full chain meets a $P$-free family, called the Lubell function, and use it for $P=D_k$ to determine  $\pi(D_k)$ for infinitely many values $k$.  A stubborn open problem is to show that $\pi(D_2)=2$; here we prove $\pi(D_2)<2.273$ (if it exists).    This is joint work with Wei-Tian Li and Linyuan Lu of University of South Carolina.

Curve complexes and mapping class groups II

Series
Geometry Topology Working Seminar
Time
Friday, September 17, 2010 - 14:00 for 2 hours
Location
Skiles 171
Speaker
Dan Margalit – Georgia Tech
We will prove that the mapping class group is finitely presented, using its action on the arc complex. We will also use the curve complex to show that the abstract commensurator of the mapping class group is the extended mapping class group. If time allows, we will introduce the complex of minimizing cycles for a surface, and use it to compute the cohomological dimension of the Torelli subgroup of the mapping class group. This is a followup to the previous talk, but will be logically independent.

Von Neumann Entropy Penalization and Estimation of Low Rank Matrices

Series
Stochastics Seminar
Time
Thursday, September 16, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 002
Speaker
Vladimir Koltchinskii – School of Mathematics, Georgia Tech
We study a problem of estimation of a large Hermitian nonnegatively definite matrix S of unit trace based on n independent measurements Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n, where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean zero random variables independent of {X_j}. Problems of this nature are of interest in quantum state tomography, where S is an unknown density matrix of a quantum system. The estimator is based on penalized least squares method with complexity penalty defined in terms of von Neumann entropy. We derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state S by low-rank matrices. We will discuss these results as well as some of the tools used in their proofs (such as generic chaining bounds for empirical processes and noncommutative Bernstein type inequalities).

Geometry of lattices associated to a given graph

Series
Graph Theory Seminar
Time
Thursday, September 16, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Omid Amini – CNRS-École Normale Supérieure
We present some geometric properties of the Laplacian lattice and the lattice of integer flows of a given graph and discuss some applications and open problems.

Non-homogeneous Harmonic Analysis and randomized Beylkin--Coifman--Rokhlin algorithm (BCR): an application for the solutions of A2 conjecture.

Series
Analysis Seminar
Time
Wednesday, September 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexander Volberg – Michigan State
A2 conjecture asked to have a linear estimate for simplest weighted singular operators in terms of the measure of goodness of the weight in question.We will show how the paradigm of non-homogeneous Harmonic Analysis (and especially its brainchild, the randomized BCR) was used to eventually solve this conjecture.

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