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Series: Algebra Seminar

According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

Friday, September 14, 2018 - 13:55 ,
Location: Skiles 006 ,
Peter Lambert-Cole ,
Georgia Insitute of Technology ,
Organizer: Peter Lambert-Cole

The Oka-Grauert principle is one of the first examples of an
h-principle. It states that for a Stein domain X and a complex Lie
group G, the topological and holomorphic classifications of principal
G-bundles over X agree. In particular, a complex vector bundle over X
has a holomorphic trivialization if and only if it has a continuous
trivialization. In these talks, we will discuss the complex geometry of
Stein domains, including various characterizations of Stein domains,
the classical Theorems A and B, and the Oka-Grauert principle.

Series: ACO Student Seminar

We study the dynamic graph connectivity problem in the massively
parallel computation model. We give a data structure for maintaining a
dynamic undirected graph that handles batches of updates and
connectivity queries in constant rounds, as long as the queries fit on a
single machine. This assumption corresponds to the gradual buildup of
databases over time from sources such as log files and user
interactions. Our techniques combine a distributed data structure for
Euler Tour (ET) trees, a structural theorem about rapidly contracting
graphs via sampling n^{\epsilon} random neighbors, as well as
modifications to sketching based dynamic connectivity data structures.
Joint work with David Durfee, Janardhan Kulkarni, Richard Peng and Xiaorui Sun.

Series: Stochastics Seminar

Let (A_n) be a sequence of random matrices, such that for every n, A_n
is n by n with i.i.d. entries, and each entry is of the form b*x, where b
is a Bernoulli random variable with probability of success p_n, and x
is an independent random variable of unit variance. We show that, as
long as n*p_n converges to infinity, the appropriately rescaled spectral
distribution of A_n converges to the uniform measure on the unit disc
of complex plane. Based on joint work with Mark Rudelson.

Thursday, September 13, 2018 - 13:30 ,
Location: Skiles 006 ,
Trevor Gunn ,
Georgia Tech ,
Organizer: Trevor Gunn

Tropical geometry is a blend of algebraic geometry and polyhedral combinatorics that arises when one looks at algebraic varieties over a valued field. I will give a 50 minute introduction to the subject to highlight some of the key themes.

Series: Other Talks

This is an interdisciplinary event using puzzles, story-telling, and original music and dance to interpret Euler's analysis of the problem of the Seven Bridges of Königsberg, and the birth of graph theory. Beginning at 11:00, students from GT's Club Math will be on the plaza between the Howie and Mason Buildings along Atlantic Dr., with information and hands-on puzzles related to Euler and to graphs. At 12:00 the performance will begin, as the GT Symphony Orchestra and a team of dancers interpret the story of the Seven Bridges. For more information see the news article at http://hg.gatech.edu/node/610095.

Series: Graph Theory Working Seminar

Gallai
conjectured in 1968 that the edges of a connected graph on n vertices
can be decomposed into at most (n+1)/2 edge-disjoint paths. This
conjecture
is still open, even for planar graphs. In this talk we will discuss some
related results and special cases where it is known to hold.

Series: Math Physics Seminar

We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $E\to 0$ and $t\to t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere.Our results are based on splitting the system's evolution into a ``slow'' process and an independent ``noise''. We show that the noise, suitably rescaled, converges to a Brownian motion. Then we employ the Ito-Lyons continuity theorem to identify the limit of the slow process.

Wednesday, September 12, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyunki Min ,
Georgia Tech ,
Organizer: Hyun Ki Min

In 1957, Smale proved a striking result: we can turn a sphere inside out without any singularity. Gromov in his thesis, proved a generalized version of this theorem, which had been the starting point of the h-principle. In this talk, we will prove Gromov's theorem and see applications of it.

Series: Analysis Seminar

Koldobsky showed that for an arbitrary measure on R^n, the measure of the largest section of a symmetric convex body can be estimated from below by 1/sqrt{n}, in with the appropriate scaling. He conjectured that a much better result must hold, however it was recemtly shown by Koldobsky and Klartag that 1/sqrt{n} is best possible, up to a logarithmic error. In this talk we will discuss how to remove the said logarithmic error and obtain the sharp estimate from below for Koldobsky's slicing problem. The method shall be based on a "random rounding" method of discretizing the unit sphere. Further, this method may be effectively applied to estimating the smallest singular value of random matrices under minimal assumptions; a brief outline shall be mentioned (but most of it shall be saved for another talk). This is a joint work with Bo'az Klartag.