Series: School of Mathematics Colloquium
The piecewise linear objects appearing in tropical geometry are shadows, or skeletons, of nonarchimedean analytic spaces, in the sense of Berkovich, and often capture enough essential information about those spaces to resolve interesting questions about classical algebraic varieties. I will give an overview of tropical geometry as it relates to the study of algebraic curves, touching on applications to moduli spaces.
Wednesday, March 1, 2017 - 14:05 , Location: Skiles 006 , Hyun Ki Min , Georgia Tech , Organizer: Justin Lanier
There is no general h-principle for Legendrian embeddings in contact manifolds. In dimension 3, however, Legendrian knots in the complement of an overtwisted disc, which are called loose, satisfy an h-principle. We will discuss the high dimensional analog of loose knots.
Series: Analysis Seminar
For a compact subset $A$ of $R^n$ , let $A(k)$ be the Minkowski sum of $k$ copies of $A$, scaled by $1/k$. It is well known that $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to infinity. A few years ago, Bobkov, Madiman and Wang conjectured that the volume of $A(k)$ is non-decreasing in $k$, or in other words, that when the volume deficit between the convex hull of $A$ and $A(k)$ goes to $0$, it actually does so monotonically. While this conjecture holds true in dimension $1$, we show that it fails in dimension $12$ or greater. Then we consider whether one can have monotonicity of convergence of $A(k)$ when its non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of $A(k)$ converges monotonically to $0$ as $k$ increases; even the convergence does not seem to have been known before. We also obtain some results for the Hausdorff distance to the convex hull, along the way clarifying various properties of these notions of non-convexity that may be of independent interest.Joint work with Mokshay Madiman, Matthieu Fradelizi and Arnaud Marsiglietti.
Tuesday, February 28, 2017 - 11:05 , Location: Skiles 006 , Tomasz Łuczak , Adam Mickiewicz University , firstname.lastname@example.org , Organizer: Lutz Warnke
The talk is meant to be a gentle introduction to a part of combinatorial topology which studies randomly generated objects. It is a rapidly developing field which combines elements of topology, geometry, and probability with plethora of interesting ideas, results and problems which have their roots in combinatorics and linear algebra.
Series: Job Candidate Talk
We consider the problem of estimating pairwise comparison probabilities in a tournament setting after observing every pair of teams play with each other once. We assume the true pairwise probability matrix satisfies a stochastic transitivity condition which is popular in the Social Sciences.This stochastic transitivity condition generalizes the ubiquitous Bradley- Terry model used in the ranking literature. We propose a computationally efficient estimator for this problem, borrowing ideas from recent work on Shape Constrained Regression. We show that the worst case rate of our estimator matches the best known rate for computationally tractable estimators. Additionally we show our estimator enjoys faster rates of convergence for several sub parameter spaces of interest thereby showing automatic adaptivity. We also study the missing data setting where only a fraction of all possible games are observed at random.
Monday, February 27, 2017 - 14:00 , Location: Skiles 005 , Gunay Dogan , National Institute of Standards and Technology , Organizer: Sung Ha Kang
For many problems in science and engineering, one needs to quantitatively compare shapes of objects in images, e.g., anatomical structures in medical images, detected objects in images of natural scenes. One might have large databases of such shapes, and may want to cluster, classify or compare such elements. To be able to perform such analyses, one needs the notion of shape distance quantifying dissimilarity of such entities. In this work, we focus on the elastic shape distance of Srivastava et al. [PAMI, 2011] for closed planar curves. This provides a flexible and intuitive geodesic distance measure between curve shapes in an appropriate shape space, invariant to translation, scaling, rotation and reparametrization. Computing this distance, however, is computationally expensive. The original algorithm proposed by Srivastava et al. using dynamic programming runs in cubic time with respect to the number of nodes per curve. In this work, we propose a new fast hybrid iterative algorithm to compute the elastic shape distance between shapes of closed planar curves. The asymptotic time complexity of our iterative algorithm is O(N log(N)) per iteration. However, in our experiments, we have observed almost a linear trend in the total running times depending on the type of curve data.
Series: Other Talks
Mozghan Entekhabi (Wichita State University) Radial Limits of Bounded Nonparametric Prescribed Mean Curvature Surfaces ; Miyuki Koiso (Kyushu University) Stability and bifurcation for surfaces with constant mean curvature ; Vladimir Oliker (Emory University) Freeform lenses, Jacobian equations, and supporting quadric method(SQM) ; Sungho Park (Hankuk University of Foreign Studies) Circle-foliated minimal and CMC surfaces in S^3 ; Yuanzhen Shao (Purdue University) Degenerate and singular elliptic operators on manifolds with singularities ; Ray Treinen (Texas State University) Surprising non-uniqueness for the 2D floating ball ; See http://www.math.uab.edu/sgs/ for abstracts and further details.
Saturday, February 25, 2017 - 09:00 , Location: University of Georgia, Paul D. Coverdell Center for Biomedical & Health Sciences, Athens, GA 30602 , Haomin Zhou , GT Math , Organizer: Sung Ha Kang
The Georgia Scientific Computing Symposium (GSCS) is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community. The format of the day-long symposium is a set of invited presentations, poster sessions and a poster blitz, and plenty of time to network with other attendees. More information at http://euler.math.uga.edu/cms/GSCS-2017
Series: Combinatorics Seminar
In enumerative combinatorics, it is quite common to have in hand a number of known initial terms of a combinatorial sequence whose behavior you'd like to study. In this talk we'll describe two techniques that can be used to shed some light on the nature of a sequence using only some known initial terms. While these methods are, on the face of it, experimental, they often lead to rigorous proofs. As we talk about these two techniques -- automated conjecturing of generating functions, and the method of differential approximation -- we'll exhibit their usefulness through a variety of combinatorial topics, including matchings, permutation classes, and inversion sequences.
Friday, February 24, 2017 - 15:05 , Location: Skiles 254 , Simon Berman , School of Physics , Organizer: Jiaqi Yang
In a high harmonic generation (HHG) experiment, an intense laser pulse is sent through an atomic gas, and some of that light is converted to very high harmonics through the interaction with the gas. The spectrum of the emitted light has a particular, nearly universal shape. In this seminar, I will describe my efforts to derive a classical reduced Hamiltonian model to capture this phenomenon. Beginning with a parent Hamiltonian that yields the equations of motion for a large collection of atoms interacting self-consistently with the full electromagnetic field (Lorentz force law + Maxwell's equations), I will follow a sequence of reductions that lead to a reduced Hamiltonian which is computationally tractable yet should still retain the essential physics. I will conclude by pointing out some of the still-unresolved issues with the model, and if there's time I will discuss the results of some preliminary numerical simulations.