Seminars and Colloquia by Series

Nodal count of eigenfunctions as index of instability

Series
Math Physics Seminar
Time
Monday, February 27, 2012 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gregory Berkolaiko – Texas A&M Univ. –
Zeros of vibrational modes have been fascinating physicists for several centuries. Mathematical study of zeros of eigenfunctions goes back at least to Sturm, who showed that, in dimension d=1, the n-th eigenfunction has n-1 zeros. Courant showed that in higher dimensions only half of this is true, namely zero curves of the n-th eigenfunction of the Laplace operator on a compact domain partition the domain into at most n parts (which are called "nodal domains"). It recently transpired that the difference between this "natural" number n of nodal domains and the actual values can be interpreted as an index of instability of a certain energy functional with respect to suitably chosen perturbations. We will discuss two examples of this phenomenon: (1) stability of the nodal partitions of a domain in R^d with respect to a perturbation of the partition boundaries and (2) stability of a graph eigenvalue with respect to a perturbation by magnetic field. In both cases, the "nodal defect" of the eigenfunction coincides with the Morse index of the energy functional at the corresponding critical point. Based on preprints arXiv:1107.3489 (joint with P.Kuchment and U.Smilansky) and arXiv:1110.5373

Transitivity of nilpotent extensions of hyperbolic systems

Series
CDSNS Colloquium
Time
Monday, February 27, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew Torok – Univ. of Houston
Consider a hyperbolic basic set of a smooth diffeomorphism. We are interested in the transitivity of Holder skew-extensions with fiber a non-compact connected Lie group. In the case of compact fibers, the transitive extensions contain an open and dense set. For the non-compact case, we conjectured that this is still true within the set of extensions that avoid the obvious obstructions to transitivity. Within this class of cocycles, we proved generic transitivity for extensions with fiber the special Euclidean group SE(2n+1) (the case SE(2n) was known earlier), general Euclidean-type groups, and some nilpotent groups. We will discuss the "correct" result for extensions by the Heisenberg group: if the induced extension into its abelinization is transitive, then so is the original extension. Based on earlier results, this implies the conjecture for Heisenberg groups. The results for nilpotent groups involve questions about Diophantine approximations. This is joint work with Ian Melbourne and Viorel Nitica.

Discrete Mathematical Biology Working Seminar

Series
Other Talks
Time
Monday, February 27, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Shel Swenson – Georgia Tech
The paper "Abstract shapes of RNA" (2004) by Giegerich, Voss, and Rehmsmeier will be discussed.

Atlanta Lecture Series in Combinatorics and Graph Theory

Series
Other Talks
Time
Saturday, February 25, 2012 - 13:00 for 1 hour (actually 50 minutes)
Location
Math and Science Center, Emory University
Speaker
Jacob Fox – Massachusetts Institute of Technology
The 5th in a series of 9 mini-conferences features Jacob Fox as the prominent researcher who will give 2 fifty-minute lectures and 4 other outstanding researchers each giving one fifty-minute lecture. There will also be several 25-minute lectures by young researchers and graduate students. The lectures will begin at 1 PM on Saturday, February 25 and conclude at at noon on Sunday, February 26.To register, please send an email to rg@mathcs.emory.edu and for complete details, see the website. Registration is free.

4th Georgia Scientific Computing Symposium

Series
Other Talks
Time
Saturday, February 25, 2012 - 08:30 for 1 hour (actually 50 minutes)
Location
Room S175, Coverdell Center, University of Georgia
Speaker
Georgia Scientific Computing Symposium – University of Georgia
The purpose of the GSC Symposium is to provide an opportunity for professors, postdocs, and graduate students in the Atlanta area to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. Certainly, the symposium is open to whole mathematics and computer sciences communities. Three previous meetings were held at Emory University (2009), Georgia Institute of Technology (2010), and Emory University (2011). The 2012 GSC Symposium will be held at the University of Georgia campus and is organized by Dept. of Mathematics, University of Georgia. The following researchers have agreed to give invited plenary lectures: David Bader, College of Computing, Georgia Institute of Technology; Michele Benzi, Dept. of Mathematics and Computer Science, Emory University; Sung Ha Kang, School of Mathematics, Georgia Institute of Technology; Tianming Liu, Dept. of Computer Sciences, University of Georgia; Andrew Sornborger, Dept. of Mathematics, University of Georgia. There will be two poster sessions. Anyone attending this symposium may present a poster. We especially encourage graduate students and postdocs to use this opportunity displaying their research results. Please register at the  website.

A Discrepancy based Approach to Integer Programming

Series
ACO Student Seminar
Time
Friday, February 24, 2012 - 13:00 for 1 hour (actually 50 minutes)
Location
Klaus 1116W
Speaker
Karthekeyan Chandrasekaran – CoC, Georgia Tech
I will show a new approach based on the discrepancy of the constraint matrix to verify integer feasibility of polytopes. I will then use this method to show a threshold phenomenon for integer feasibility of random polytopes. The random polytope model that we consider is P(n,m,x0,R) - these are polytopes in n-dimensional space specified by m "random" tangential hyperplanes to a ball of radius R centered around the point x0. We show that there exist constants c_1 < c_2 such that with high probability, the random polytope P(n,m,x0=(0.5,...,0.5),R) is integer infeasible if R is less than c_1sqrt(log(2m/n)) and the random polytope P(n,m,x0,R) is integer feasible for every center x0 if the radius R is at least c_2sqrt(log(2m/n)). Thus, a transition from infeasibility to feasibility happens within a constant factor increase in the radius. Moreover, if the polytope contains a ball of radius Omega(log (2m/n)), then we can find an integer solution with high probability (over the input) in randomized polynomial time. This is joint work with Santosh Vempala.

A stochastic approach to parabolicity and area growth of minimal ends

Series
Stochastics Seminar
Time
Thursday, February 23, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
006
Speaker
Robert W. Neel – Lehigh University –
We wish to understand ends of minimal surfaces contained in certain subsets of R^3. In particular, after explaining how the parabolicity and area growth of such minimal ends have been previously studied using universal superharmonic functions, we describe an alternative approach, yielding stronger results, based on studying Brownian motion on the surface. It turns out that the basic results also apply to a larger class of martingales than Brownian motion on a minimal surface, which both sheds light on the underlying geometry and potentially allows applications to other problems.

Risk neutral and risk averse approaches to multistage stochastic programming

Series
School of Mathematics Colloquium
Time
Thursday, February 23, 2012 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander Shapiro – ISyE, Georgia Tech
In many practical situations one has to make decisions sequentially based on data available at the time of the decision and facing uncertainty of the future. This leads to optimization problems which can be formulated in a framework of multistage stochastic programming. In this talk we consider risk neutral and risk averse approaches to multistage stochastic programming. We discuss conceptual and computational issues involved in formulation and solving such problems. As an example we give numerical results based on the Stochastic Dual Dynamic Programming method applied to planning of the Brazilian interconnected power system.

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