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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.

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.

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: 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: 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.

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: 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.

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: 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.

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: 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.

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.

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: 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.

Series: Combinatorics Seminar

An old question in additive number theory is determining the length of the longest progression in a sumset A+B = {a + b : a in A, b in B}, given that A and B are "large" subsets of {1,2,...,n}. I will survey some of the results on this problem, including a discussion of the methods, and also will discuss some open questions and conjectures.