## Seminars and Colloquia by Series

Friday, October 13, 2017 - 13:05 , Location: Skiles 005 , , CS, Georgia Tech , , Organizer: He Guo
We show variants of spectral sparsification routines can preserve thetotal spanning tree counts of graphs, which by Kirchhoff's matrix-treetheorem, is equivalent to determinant of a graph Laplacian minor, orequivalently, of any SDDM matrix. Our analyses utilizes thiscombinatorial connection to bridge between statistical leverage scores/ effective resistances and the analysis of random graphs by [Janson,Combinatorics, Probability and Computing `94]. This leads to a routinethat in quadratic time, sparsifies a graph down to about $n^{1.5}$edges in ways that preserve both the determinant and the distributionof spanning trees (provided the sparsified graph is viewed as a randomobject). Extending this algorithm to work with Schur complements andapproximate Choleksy factorizations leads to algorithms for countingand sampling spanning trees which are nearly optimal for dense graphs.We give an algorithm that computes a $(1\pm \delta)$ approximation tothe determinant of any SDDM matrix with constant probability in about$n^2\delta^{−2}$ time. This is the first routine for graphs thatoutperforms general-purpose routines for computing determinants ofarbitrary matrices. We also give an algorithm that generates in about$n^2\delta^{−2}$ time a spanning tree of a weighted undirected graphfrom a distribution with total variation distance of $\delta$ fromthe w-uniform distribution.This is joint work with John Peebles, Richard Peng and Anup B. Rao.
Friday, October 13, 2017 - 10:00 , Location: Skiles 114 , Libby Taylor , GA Tech , Organizer: Timothy Duff
We will give an overview of divisor theory on curves and give definitions of the Picard group and the Jacobian of a compact Riemann surface.  We will use these notions to prove Plucker’s formula for the genus of a smooth projective curve.  In addition, we will discuss the various ways of defining the Jacobian of a curve and why these definitions are equivalent.  We will also give an extension of these notions to schemes, in which we define the Picard group of a scheme in terms of the group of invertible sheaves and in terms of sheaf cohomology.
Wednesday, October 11, 2017 - 13:55 , Location: Skiles 005 , Akram Aldroubi , Vanderbilt University , Organizer: Shahaf Nitzan
Dynamical sampling is the problem of recovering an unknown function from a set of space-time samples. This problem has many connections to problems in frame theory, operator theory and functional analysis.  In this talk, we will state the problem and discuss its relations to various areas of functional analysis and operator theory, and  we will give a brief review of previous results and present several new ones.
Wednesday, October 11, 2017 - 13:55 , Location: Skiles 006 , Justin Lanier , Georgia Tech , Organizer: Jennifer Hom
We will discuss the mapping class groupoid, how it is generated by handle slides, and how factoring in the mapping class groupoid can be used to compute Heegaard Floer homology. This talk is based on work by Lipshitz, Ozsvath, and Thurston.
Series: Other Talks
Wednesday, October 11, 2017 - 11:30 , Location: Skiles 005 , , Georgia Tech , , Organizer: Samantha Petti

Lunch will be provided. The talk will be the first 25 minutes of the hour and then will be followed by discussion.&nbsp;

In a recent article to appear in the American Mathematical Mothly next year, we use the Lambert series generating function for Euler’s totient function to introduce a new identity for the number of 1’s in the partitions of n. New expansions for Euler’s partition function p(n) are derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions. We will define partitions and several variants of Euler's partition function in the talk to state our new results.
Monday, October 9, 2017 - 13:55 , Location: Skiles 006 , None , None , Organizer: Jennifer Hom
Friday, October 6, 2017 - 15:00 , Location: Skiles 005 , Matas Sileikis , Charles University Prague , Organizer: Lutz Warnke
Given a (fixed) graph H, let X be the number of copies of H in the random binomial graph G(n,p). In this talk we recall the results on the asymptotic behaviour of X, as the number n of vertices grows and pis allowed to depend on. In particular we will focus on the problem of estimating probability that X is significantly larger than its expectation, which earned the name of the 'infamous upper tail'.
Friday, October 6, 2017 - 15:00 , Location: Skiles 154 , Prof. Rafael de la Llave , School of Mathematics, Georgia Tech , Organizer: Jiaqi Yang
We will present an introduction to the results of S. Aubry and J. Mather who used variational methods to prove the existence of quasi-periodic orbits in twist mappings and in some models appearing in solid state Physics.
Friday, October 6, 2017 - 15:00 , Location: Skiles 154 , Sergio Mayorga , Georgia Tech , Organizer: Jiaqi Yang
We will look at a system of hamiltonian equations on the torus, with an initial condition in momentum and a terminal condition in position, that arises in mean field game theory. Existence of and uniqueness of solutions will be shown, and a few remarks will be made in regard to its connection to the minimization problem of a cost functional. This is the second part of lasrt week's talk.
Friday, October 6, 2017 - 13:05 , Location: Skiles 005 , Josh Daymude , Arizona State University/GaTech theory lab , , Organizer: He Guo
In a self-organizing particle system, an abstraction of programmable matter, simple computational elements called particles with limited memory and communication self-organize to solve system-wide problems of movement, coordination, and configuration. In this paper, we consider stochastic, distributed, local, asynchronous algorithms for 'shortcut bridging', in which particles self-assemble bridges over gaps that simultaneously balance minimizing the length and cost of the bridge. Army ants of the genus Eticon have been observed exhibiting a similar behavior in their foraging trails, dynamically adjusting their bridges to satisfy an efficiency tradeoff using local interactions. Using techniques from Markov chain analysis, we rigorously analyze our algorithm, show it achieves a near-optimal balance between the competing factors of path length and bridge cost, and prove that it exhibits a dependence on the angle of the gap being 'shortcut' similar to that of the ant bridges. We also present simulation results that qualitatively compare our algorithm with the army ant bridging behavior. Our work presents a plausible explanation of how convergence to globally optimal configurations can be achieved via local interactions by simple organisms (e.g., ants) with some limited computational power and access to random bits. The proposed algorithm demonstrates the robustness of the stochastic approach to algorithms for programmable matter, as it is a surprisingly simple extension of a stochastic algorithm for compression. This is joint work between myself/my professor Andrea Richa at ASU and Sarah Cannon and Prof. Dana Randall here at GaTech.