Seminars and Colloquia Schedule

Simultaneous Random and Optimized Sources and Detectors for Efficient Optimization in Inverse Problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 19, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eric de Sturler – Department of Mathematics, Virginia Tech –
In nonlinear inverse problems, we often optimize an objective function involving many sources, where each source requires the solution of a PDE. This leads to the solution of a very large number of large linear systems for each nonlinear function evaluation, and potentially additional systems (for detectors) to evaluate or approximate a Jacobian. We propose a combination of simultaneous random sources and detectors and optimized (for the problem) sources and detectors to drastically reduce the number of systems to be solved. We apply our approach to problems in diffuse optical tomography.This is joint work with Misha Kilmer and Selin Sariaydin.

The infinite topology of the hyperelliptic locus

Series
Geometry Topology Seminar
Time
Monday, October 19, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin Kordek – Texas A&M
The hyperelliptic Torelli group of a genus g reference surface S_g is the subgroup of the mapping class group whose elements both commute with a fixed hyperelliptic involution of S_g and act trivially on the integral homology of S_g . This group is an important object in geometric topology and group theory, and also in algebraic geometry, where it appears as the fundamental group of the moduli space of genus g hyperelliptic curves with a homology framing. In this talk, we summarize what is known about the (infinite) topology of these moduli spaces, describe a few open problems, and report on some recent partial progress.

Perturbation of linear forms of singular vectors under Gaussian noise

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, October 20, 2015 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Dong Xia – Georgia Inst. of Technology
Let A be a mxn matrix with singular value decomposition A=UDV', where the columns of U are left singular vectors and columns of V are right singular vectors of A. Suppose X is a mxn noise matrix whose entries are i.i.d. Gaussian random variables and consider A'=A+X. Let u_k be the k-th left singular vector of A and u'_k be its counterpart of A'. We develop sharp upper bounds for concentration of linear forms for the right singular vectors of A'.The talk is based on a joint work with Vladimir Koltchinskii.

Regularity theory for surfaces in geometric optics and other Generated Jacobian Equations

Series
PDE Seminar
Time
Tuesday, October 20, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nestor Guillen – University of Massachusetts at Amherst
The study of reflector surfaces in geometric optics necessitates the analysis of nonlinear equations of Monge-Ampere type. For many important examples (including the near field reflector problem), the equation no longer falls within the scope of optimal transport, but within the class of "Generated Jacobian equations" (GJEs). This class of equations was recently introduced by Trudinger, motivated by problems in geometric optics, however they appear in many others areas (e.g. variations of the Minkowski problem in convex geometry). Under natural assumptions, we prove Holder regularity for the gradient of weak solutions. The results are new in particular for the near-field point source reflector problem, but are applicable for a broad class of GJEs: those satisfying an analogue of the A3-weak condition introduced by Ma, Trudinger and Wang in optimal transport. Joint work with Jun Kitagawa.

Scaling limits of Polynomials are fairly universal

Series
Research Horizons Seminar
Time
Wednesday, October 21, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Doron Lubinsky – School of Mathematics, Georgia Institute of Technology –

Food and Drinks will be provided before the seminar.

In elementary calculus, we learn that (1+z/n)^n has limit exp(z) as n approaches infinity. This type of scaling limit arises in many contexts - from approximation theory to universality limits in random matrices. We discuss some examples.

Small-Time Asymptotic Methods for Levy-Based Jump-Diffusion Models

Series
Mathematical Finance/Financial Engineering Seminar
Time
Wednesday, October 21, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruoting Gong – Illinois Institute of Technology
In recent years, small-time asymptotic methods have attracted much attention in mathematical finance. Such asymptotics are especially crucial for jump-diffusion models due to the lack of closed- form formulas and efficient valuation procedures. These methods have been widely developed and applied to diverse areas such as short-time approximations of option prices and implied volatilities, and non-parametric estimations based on high-frequency data. In this talk, I will discuss some results on the small-time asymptotic behavior of some Levy functionals with applications in finance.

Random Matrices, the GUE and the distribution of eigenvalues

Series
Other Talks
Time
Wednesday, October 21, 2015 - 17:00 for 1.5 hours (actually 80 minutes)
Location
Skies 006
Speaker
Inoel Popescu – Georgia Tech
This is the fourth meeting in a series of a reading seminars. In this lecture we will analyze the distribution of the eigenvalues of GUE ensembles. We will use Hermite polynomials to get very concrete computations. This way we will recover the semicircular law and we will also discuss a little bit the top eigenvalue.

Longest Subsequences Problems and Maximal Eigenvalues of Gaussian Random Matrices

Series
Stochastics Seminar
Time
Thursday, October 22, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian Houdre – School of Mathematics, Georgia Tech
This is survey talk where, both for random words and random permutations, I will present a panoramic view of the subject ranging from classical results to recent breakthroughs. Throughout, equivalencies with some directed last passage percolation models with dependent weights will be pointed out.

Global Smooth Solutions in R^3 to Short Wave-Long Wave Interactions in Magnetohydrodynamics

Series
PDE Seminar
Time
Thursday, October 22, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hermano Frid – Institute de Matematica Pura e Aplicada (IMPA)
We consider a Benney-type system modeling short wave-long wave interactions in compressible viscous fluids under the influence of a magnetic field. Accordingly, this large system now consists of the compressible MHD equations coupled with a nonlinear Schodinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R^3 when the initial data are small smooth perturbations of an equilibrium state. An important point here is that, instead of the simpler case having zero as the equilibrium state for the magnetic field, we consider an arbitrary non-zero equilibrium state B for the magnetic field. This is motivated by applications, e.g., Earth's magnetic field, and the lack of invariance of the MHD system with respect to either translations or rotations of the magnetic field. The usual time decay investigation through spectral analysis in this non-zero equilibrium case meets serious difficulties, for the eigenvalues in the frequency space are no longer spherically symmetric. Instead, we employ a recently developed technique of energy estimates involving evolution in negative Besov spaces, and combine it with the particular interplay here between Eulerian and Lagrangian coordinates. This is a joint work with Junxiong Jia and Ronghua Pan.

Label optimal regret bounds for online local learning

Series
ACO Student Seminar
Time
Friday, October 23, 2015 - 13:05 for 30 minutes
Location
Skiles 005
Speaker
Kevin Lai – Georgia Tech
We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items at each step, chosen from a set of n items. The learner then predicts a label for each item, from a label set of size L and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as collaborative filtering, online gambling, and online max cut among others. (Christiano, 2014a) designed an efficient online learning algorithm for this problem achieving a regret of O((nL^3T)^(1/2)), where T is the number of rounds. Information theoretically, one can achieve a regret of O((n log LT)^(1/2)). One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of (Christiano, 2014a), in fact achieves a regret of O((nLT)^(1/2)). Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well. Joint work with Pranjal Awasthi, Moses Charikar, and Andrej Risteski at Princeton.

The Complexity of Counting Poset and Permutation Patterns

Series
ACO Student Seminar
Time
Friday, October 23, 2015 - 13:30 for 30 minutes
Location
Skiles 005
Speaker
Anna Kirkpatrick – Georgia Tech
We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems – particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for unconstrained dimension it is GI-complete; counting not necessarily induced, injective occurrences among dimension 2 posets is #P-hard; counting injective or not necessarily injective occurrences of an arbitrary pattern in a dimension 1 text is #P-hard, although it is in FP if the pattern poset is constrained to have bounded intrinsic width; and counting injective occurrences of a dimension 1 pattern in an arbitrary text is #P-hard, while it is in FP for bounded dimension texts. This framework easily leads to a number of open questions, chief among which are (1) is it #P-hard to count the number of occurrences of a dimension 2 pattern in a dimension 1 text, and (2) is it #P-hard to count the number of texts which avoid a given pattern?

On the inverse of some sign matrices and on the Moments sliding vector field on the intersection of several manifolds: nodally attractive case

Series
SIAM Student Seminar
Time
Friday, October 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fabio Difonzo – Georgia Institute of Technology
In this paper, we consider selection of a sliding vector fieldof Filippov type on a discontinuity manifold $\Sigma$ of co-dimension 3(intersection of three co-dimension 1 manifolds). We propose an extension of the “moments vector field”to this case, and - under the assumption that $\Sigma$ is nodally attractive -we prove that our extension delivers a uniquely definedFilippov vector field. As it turns out, the justification of our proposed extension requiresestablishing invertibility of certain sign matrices. Finally,we also propose the extension of the moments vector field todiscontinuity manifolds of co-dimension 4 and higher.

Georgia Algebraic Geometry Symposium

Series
Other Talks
Time
Friday, October 23, 2015 - 16:30 for 1 hour (actually 50 minutes)
Location
Room 208 Emory Math and Science Center
Speaker
Valery Alexeev – University of Georgia
Friday October 23 through Sunday October 25 Emory will host the Georgia Algebraic Geometry symposium featuring the following invited speakers: Valery Alexeev (University of Georgia); Brian Conrad (Stanford University); Brian Lehman (Boston College); Max Lieblich (University of Washington); Alexander Merkurjev (UCLA); Alena Pirutka (Ecole Polytechnique); Aaron Pixton (Harvard University); Tony Varilly-Alvarado (Rice University); Olivier Wittenberg (CNRS - Ecole Normale Superieure).