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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.

Series: High Dimensional Seminar

The concentration of Lipschitz functions around their expectation is a classical topic and continues to be very active. In these talks, we will discuss some recent progress in detail, including: A tight log-Sobolev inequality for isotropic logconcave densities A unified and improved large deviation inequality for convex bodies An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. (Warning: the talk will involve elementary calculus on the board, sometimes at an excruciatingly slow pace). Joint work with Yin Tat Lee.

Series: Research Horizons Seminar

Random and irregular growth is all around us. We see it in the form of
cancer growth, bacterial infection, fluid flow through porous rock, and
propagating flame fronts. In this talk, I will introduce several
different models for random growth and
the different shapes that can arise from them. Then I will talk in more
detail about one model, first-passage percolation, and some of the main
questions that researchers study about it.

Series: Geometry Topology Seminar

We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(X; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

Monday, September 10, 2018 - 13:55 ,
Location: Skiles 005 ,
Sergei Avdonin ,
University of Alaska Fairbanks ,
s.avdonin@alaska.edu ,
Organizer: Wenjing Liao

Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs

is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.

In this talk we describe some new controllability and identifability results for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more

regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems

for such systems more diffcult.

We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of thestrings, lengths of the edges, attached masses, and the topology of the graph. The proofs are based on the boundary control and leaf peeling methods developed in our previous papers. The boundary control method is a powerful

method in inverse theory which uses deep connections between controllability and identifability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.

Friday, September 7, 2018 - 15:05 ,
Location: Skiles 156 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk we will discuss the paper of McGehee titled "The stable manifold theorem via an isolating block," in which a proof of the theorem is made using only elementary topology of Euclidean spaces and elementary linear algebra.