Seminars and Colloquia by Series

spectral equivalence classes based on isospectral reductions

Series
Dynamical Systems Working Seminar
Time
Friday, February 22, 2019 - 03:05 for 1 hour (actually 50 minutes)
Location
Skiles 246
Speaker
Longmei Shu – GT Math
Isospectral reductions on graphs remove certain nodes and change the weights of remaining edges. They preserve the eigenvalues of the adjacency matrix of the graph, their algebraic multiplicities and geometric multiplicities. They also preserve the eigenvectors. We call the graphs that can be isospectrally reduced to one same graph spectrally equivalent. I will give examples to show that two graphs can be spectrally equivalent or not based on the feature one picks for the equivalence class.

Stationary coalescing walks on the lattice

Series
Stochastics Seminar
Time
Thursday, February 21, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arjun Krishnan – University of Rochester –
Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of families of semi-infinite geodesics in first- and last-passage percolation that come from Busemann functions. For d>=2, we describe the behavior of bi-infinite trajectories, and show that they carry an invariant measure. We also construct several examples displaying unexpected behavior. One of these examples lets us answer a question of C. Hoffman's from 2016. (joint work with Jon Chaika)

On Bounding the Number of Automorphisms of a Tournament

Series
Graph Theory Working Seminar
Time
Wednesday, February 20, 2019 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Wigal – Georgia Tech
Let $g(n) = \max_{|T| = n}|\text{Aut}(T)|$ where $T$ is a tournament. Goldberg and Moon conjectured that $g(n) \le \sqrt{3}^{n-1}$ for all $n \ge 1$ with equality holding if and only if $n$ is a power of 3. Dixon proved the conjecture using the Feit-Thompson theorem. Alspach later gave a purely combinatorial proof. We discuss Alspach's proof and and some of its applications.

Minimal gaussian partitions with applications

Series
High Dimensional Seminar
Time
Wednesday, February 20, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Steven Heilman – USC

A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air. When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps. The double-bubble minimizes total surface area among all sets enclosing two fixed volumes. This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s. The analogous case of three or more Euclidean sets is considered difficult if not impossible. However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems. We also use the calculus of variations. Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking. http://arxiv.org/abs/1901.03934

The symmetric Gaussian isoperimetric inequality

Series
Analysis Seminar
Time
Wednesday, February 20, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Steven Heilman – USC
It is well known that a Euclidean set of fixed Euclidean volume with least Euclidean surface area is a ball. For applications to theoretical computer science and social choice, an analogue of this statement for the Gaussian density is most relevant. In such a setting, a Euclidean set with fixed Gaussian volume and least Gaussian surface area is a half space, i.e. the set of points lying on one side of a hyperplane. This statement is called the Gaussian Isoperimetric Inequality. In the Gaussian Isoperimetric Inequality, if we restrict to sets that are symmetric (A= -A), then the half space is eliminated from consideration. It was conjectured by Barthe in 2001 that round cylinders (or their complements) have smallest Gaussian surface area among symmetric sets of fixed Gaussian volume. We discuss our result that says this conjecture is true if an integral of the curvature of the boundary of the set is not close to 1. https://arxiv.org/abs/1705.06643 http://arxiv.org/abs/1901.03934

AWM Lunch Talk Series - Anna Kirkpatrick: Markov Chain Monte Carlo and RNA Secondary Structure

Series
Other Talks
Time
Wednesday, February 20, 2019 - 12:00 for 30 minutes
Location
005
Speaker
Anna Kirkpatrick – Georgia Tech –
Understanding the structure of RNA is a problem of significant interest to biochemists. Thermodynamic energy functions are often key to this pursuit, but it is well-established that these energy functions do not perform well when applied to longer RNA sequences. This work specifically investigates the branching properties of RNA secondary structures, viewed as plane trees. By employing Markov chain Monte Carlo techniques, we sample from the probability distributions determined by these thermodynamic energy functions. We also investigate some of the challenges in employing Markov chain Monte Carlo, in particular the existence of local energy minima in transition graphs. This talk will give background, share preliminary results, and discuss future avenues of investigation.

Topological Data Analysis, Automating Mapper for Novel Data

Series
Research Horizons Seminar
Time
Wednesday, February 20, 2019 - 00:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jennifer Kloke – Ayasdi
The Mapper algorithm constructs compressed representations of the underlying structure of data but involves a large number of parameters. To make the Mapper algorithm accessible to domain experts, automation of the parameter selection becomes critical. This talk will be accessible to graduate students.

On Bounding the Number of Automorphisms of a Tournament

Series
Graph Theory Working Seminar
Time
Tuesday, February 19, 2019 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Wigal – Georgia Tech
Let $g(n) = \max_{|T| = n}|\text{Aut}(T)|$ where $T$ is a tournament. Goldberg and Moon conjectured that $g(n) \le \sqrt{3}^{n-1}$ for all $n \ge 1$ with equality holding if and only if $n$ is a power of 3. Dixon proved the conjecture using the Feit-Thompson theorem. Alspach later gave a purely combinatorial proof. We discuss Alspach's proof and and some of its applications.

Heegaard Floer and the homology cobordism group

Series
Geometry Topology Seminar
Time
Monday, February 18, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jen Hom – Georgia Tech
We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. Our proof relies on the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

Low-rank matrix completion for the Euclidean distance geometry problem and beyond

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 18, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rongjie Lai – Rensselaer Polytechnic Institute –
Abstract: The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. Instead of directly reconstruct the incomplete distance matrix, we consider a low-rank matrix completion method to reconstruct the associated Gram matrix with respect to a suitable basis. Computationally, simple and fast algorithms are designed to solve the proposed problem. Theoretically, the well known restricted isometry property (RIP) can not be satisfied in the scenario. Instead, a dual basis approach is considered to theoretically analyze the reconstruction problem. Furthermore, by introducing a new condition on the basis called the correlation condition, our theoretical analysis can be also extended to a more general setting to handle low-rank matrix completion problems under any given non-orthogonal basis. This new condition is polynomial time checkable and holds for many cases of deterministic basis where RIP might not hold or is NP-hard to verify. If time permits, I will also discuss a combination of low-rank matrix completion with geometric PDEs on point clouds to understanding manifold-structured data represented as incomplete inter-point distance data. This talk is based on:1. A. Tasissa, R. Lai, “Low-rank Matrix Completion in a General Non-orthogonal Basis”, arXiv:1812.05786 2018. 2. A. Tasissa, R. Lai, “Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion”, accepted, IEEE. Transaction on Information Theory, 2018. 3. R. Lai, J. Li, “Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance”, SIAM Journal on Scientific Computing, 39(5), pp. 2231-2256, 2017.

Pages