## Seminars and Colloquia Schedule

### New mechanisms of instability in Hamiltonian systems

Series
CDSNS Colloquium
Time
Monday, October 21, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Tere M. SearaUniv. Politec. de Catalunya

In this talk we present some recent results which allow to prove
instability in near integrable Hamiltonian systems. We will show how
these mechanisms are suitable to apply to concrete systems but also are
useful to obtain Arnold diffusion in a large set  of Hamiltonian systems.

### Groups as geometric objects

Series
Geometry Topology Seminar Pre-talk
Time
Monday, October 21, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

Gromov revolutionized the study of finitely generated groups by showing that an intrinsic metric on a group is intimately connected with the algebra of the group. This point of view has produced deep applications not only in group theory, but also topology, geometry, logic, and dynamical systems. We will start at the beginning of this story with the definitions of these metrics on groups and how notions from classical geometry can be generalized to this context.  The focus will be on how the "hyperbolic groups" exhibit geometric and dynamical feature reminiscent of the hyperbolic plane and its isometries.

### Tropical convex hulls of convex sets

Series
Student Algebraic Geometry Seminar
Time
Monday, October 21, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Cvetelina HillGeorgia Tech

This talk is based on work in progress with Sara Lamboglia and Faye Simon. We study the tropical convex hull of convex sets and of tropical curves. Basic definitions of tropical convexity and tropical curves will be presented, followed by an overview of our results on the interaction between tropical and classical convexity. Lastly, we study a tropical analogue of an inequality bounding the degree of a projective variety in classical algebraic geometry; we show a tropical proof of this result for a special class of tropical curves.

### The geometry of subgroup combination theorems

Series
Geometry Topology Seminar
Time
Monday, October 21, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker

While producing subgroups of a group by specifying generators is easy, understanding  the structure of such a subgroup is notoriously difficult problem.  In the case of hyperbolic groups, Gitik utilized a local-to-global property for geodesics to produce an elegant condition that ensures a subgroup generated by two elements (or more generally generated by two subgroups) will split as an amalgamated free product over the intersection of the generators. We show that the mapping class group of a surface and many other important groups have a similar local-to-global property from which an analogy of Gitik's result can be obtained.   In the case of the mapping class group, this produces a combination theorem for the dynamically and topologically important convex cocompact subgroups.  Joint work with Davide Spriano and Hung C. Tran.

### Surfaces: BIG and small

Series
Time
Monday, October 21, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Dr. Marissa LovingGeorgia Tech

As a geometric group theorist, my favorite type of manifold is a surface and my favorite way to study surfaces is by considering the mapping class group, which is the collection of symmetries of a surface. In this talk, we will think bigger than your average low-dimensional topologist and consider surfaces of infinite type and their associated “big” mapping class groups.

### Mixing and Explosions for the Generalized Recurrent Set

Series
CDSNS Colloquium
Time
Monday, October 21, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Jim WisemanAgnes Scott

We consider the strong chain recurrent set and the generalized recurrent set for continuous maps of compact metric spaces.  Recent work by Fathi and Pageault has shown a connection between the two sets, and has led to new results on them.  We discuss a structure theorem for transitive/mixing maps, and classify maps that permit explosions in the size of the recurrent sets.

### Proof of Kac's conjecture for the hard sphere gas

Series
Math Physics Seminar
Time
Monday, October 21, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LossGeorgia Tech
This talk will be about the master equation approach to kinetic theory pioneered by Mark Kac. Specifically, the physically realistic case of three dimensional hard spheres will be considered.  This process describes an ensemble of  hard spheres undergoing binary energy and momentum preserving collisions.  One measure for the speed of approach to equilibrium is the gap which was conjectured by Kac to be bounded below by a positive constant independent of the number of particles. In this talk a proof of this conjecture  will be presented. This is joint work with Eric Carlen and Maria Carvalho.

### The Mori Dream Space property for blow-ups of projective spaces at points and lines

Series
Algebra Seminar
Time
Tuesday, October 22, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zhuang HeNortheastern University

Mori Dream Spaces are generalizations of toric varieties and, as the name suggests, Mori's minimal model program can be run for every divisor. It is known that for n5, the blow-up of Pn at r very general points is a Mori Dream Space iff rn+3. In this talk we proceed to blow up points as well as lines, by considering the blow-up X of P3 at 6 points in very general position and all the 15 lines through the 6 points. We find that the unique anticanonical section of X is a Jacobian K3 Kummer surface S of Picard number 17. We prove that there exists an infinite-order pseudo-automorphism of X, whose restriction to S is one of the 192 infinite-order automorphisms constructed by Keum.  A consequence is that there are infinitely many extremal effective divisors on X; in particular, X is not a Mori Dream Space. We show an application to the blow-up of Pn (n3) at (n+3) points and certain lines.  We relate this pseudo-automorphism to the structure of the birational automorphism group of P3. This is a joint work with Lei Yang.

### Some basics of Markov chain mixing times

Series
Lorentzian Polynomials Seminar
Time
Tuesday, October 22, 2019 - 14:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker

This is quick tutorial on bounding the mixing time of a finite Markov chain in terms of functional inequalities defining the spectral gap and the entropy constant of a Markov chain. The lecture will include some examples, including bounding the mixing time of the random transposition shuffle and (time permitting) that of the basis-exchange walk on a balanced matroid.

This is intended as a review lecture before Nima Anari’s lectures (during Nov. 4-6) on applications of Lorentzian polynomials, including recent breakthrough analyses of the basis-exchange walk on an arbitrary matroid.

### The seed-to-solution method for the Einstein equations and the asymptotic localization problem

Series
PDE Seminar
Time
Tuesday, October 22, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Philippe G. LeFlochSorbonne University and CNRS

I will present a new method of analysis for Einstein’s
constraint equations, referred to as the Seed-to-Solution Method, which
leads to the existence of asymptotically Euclidean manifolds with
prescribed asymptotic behavior. This method generates a (Riemannian)
Einstein manifold from any seed data set consisting of (1): a Riemannian
metric and a symmetric two-tensor prescribed on a topological manifold
with finitely many asymptotically Euclidean ends, and (2): a density
field and a momentum vector field representing the matter content. By
distinguishing between several classes of seed data referred to as tame
or strongly tame, the method encompasses metrics with the weakest
possible decay (infinite ADM mass) or the strongest possible decay
(Schwarzschild behavior). This analysis is based on a linearization of
the Einstein equations (involving several curvature operators from
Riemannian geometry) around a tame seed data set. It is motivated by
Carlotto and Schoen’s pioneering work on the so-called localization
problem for the Einstein equations. Dealing with manifolds with possibly
very low decay and establishing estimates beyond the critical level of
decay requires significantly new ideas to be presented in this talk. As
an application of our method, we introduce and solve a new problem,
referred to as the asymptotic localization problem, at the critical
level of decay. Collaboration with T. Nguyen. Blog: philippelefloch.org

### Go with the Flow: a parameterized approach to RNA transcript assembly

Series
Mathematical Biology Seminar
Time
Wednesday, October 23, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Blair Sullivan School of Computing, University of Utah

A central pervasive challenge in genomics is that RNA/DNA must be reconstructed from short, often noisy subsequences. In this talk, we describe a new digraph algorithm which enables this "assembly" when analyzing high-throughput transcriptomic sequencing data. Specifically, the Flow Decomposition problem on a directed ayclic graph asks for the smallest set of weighted paths that “cover” a flow (a weight function on the edges where the amount coming into any vertex is equal to the amount leaving). We describe a new linear-time algorithm solving *k*-Flow Decomposition, the variant where exactly *k* paths are used. Further, we discuss how we implemented and engineered a general Flow Decomposition solver based on this algorithm, and describe its performance on RNA-sequence data.  Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic, and we discuss the implications of our results on the original model selection for transcript assembly in this setting.

### Models for DNA-based Tile Self-Assembly

Series
Research Horizons Seminar
Time
Wednesday, October 23, 2019 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel CruzGeorgia Tech
A set of elementary building blocks undergoes self-assembly if local interactions govern how this set forms intricate structures. Self-assembly has been widely observed in nature, ranging from the field of crystallography to the study of viruses and multicellular organisms. In this talk, we give an overview of tile assembly models (TAMs) whose elementary building blocks (i.e. tiles) are polygons which have been defined with rules for local interaction. In particular, we present the basic concepts associated with two of the most well-studied TAMs: the abstract Tile Assembly Model (aTAM) and the Two-Handed Assembly Model (2HAM). We show how TAMs are related to the problem of designing nanoscale structures with DNA. We also present some of the major results within this field of study.

### Uncertainty principles and Schrodinger operators on fractals

Series
Analysis Seminar
Time
Wednesday, October 23, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kasso OkoudjouUniversity of Maryland and M.I.T.

In the first part of this talk, I will give an overview of a theory of harmonic analysis on a class of fractals that includes the Sierpinski gasket. The starting point of the theory is the introduction by J. Kigami of a Laplacian operator on these fractals. After reviewing the construction of this fractal Laplacian, I will survey some of the properties of its spectrum. In the second part of the talk, I will discuss the fractal analogs of the Heisenberg uncertainty principle, and the spectral properties a class of  Schr\"odinger operators.

### Heegaard Floer obstruction to knot surgery

Series
Geometry Topology Student Seminar
Time
Wednesday, October 23, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hongyi ZhouGeorgia Tech

Which manifold can be obtained from surgery on a knot? Many obstructions to this have been studied. We will discuss some of them, and use Heegaard Floer homology to give an infinite family of seifert fibered integer spheres that cannot be obtained by surgery on a knot in S^3. We will also discuss a recipe to compute HF+ of surgery on a knot (Short review on Heegaard Floer homology included).

### Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

Series
High Dimensional Seminar
Time
Wednesday, October 23, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andre WibisonoGeorgia Tech

Sampling is a fundamental algorithmic task. Many modern applications require sampling from complicated probability distributions in high-dimensional spaces. While the setting of logconcave target distribution is well-studied, it is important to understand sampling beyond the logconcavity assumption. We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution on R^n under isoperimetry conditions. We show a convergence guarantee in Kullback-Leibler (KL) divergence assuming the target distribution satisfies log-Sobolev inequality and the log density has bounded Hessian. Notably, we do not assume convexity or bounds on higher derivatives. We also show convergence guarantees in Rényi divergence assuming the limit of ULA satisfies either log-Sobolev or Poincaré inequality. Joint work with Santosh Vempala (arXiv:1903.08568).

### 6-connected graphs are two-three linked

Series
Dissertation Defense
Time
Thursday, October 24, 2019 - 13:40 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Shijie XieSchool of Mathematics, Georgia Tech

Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and Jorgensen's conjecture, Robertson and Seymour asked when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.

Advisor: Dr. Xingxing Yu (School of Mathematics, Georgia Institute of Technology)

Committee: Dr. Robin Thomas (School of Mathematics, Georgia Institute of Technology), Dr. Prasad Tetali (School of Mathematics, Georgia Institute of Technology), Dr. Lutz Warnke (School of Mathematics, Georgia Institute of Technology), Dr. Richard Peng (School of Computer Science, Georgia Institute of Technology)

Reader: Dr. Gexin Yu (Department of Mathematics, College of William and Mary)

### Finite time dynamics of chaotic and random systems

Series
Stochastics Seminar
Time
Thursday, October 24, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leonid BunimovichGeorgia Institute of Technology

Everybody are convinced that everything is known about the simplest random process of coin tossing. I will show that it is not the case. Particularly not everything was known for the simplest chaotic dynamical systems like the tent map (which is equivalent to coin tossing). This new area of finite time predictions emerged from a natural new question in the theory of open dynamical systems.

### High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

Series
ACO Student Seminar
Time
Friday, October 25, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wenlong MouEECS, UC Berkeley

We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of d-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most \varepsilon in Wasserstein distance from the target distribution in O(d^{1/3}/ \varepsilon^{2/3}) steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with α-th order smoothness, we prove that the mixing time scales as O (d^{1/3} / \varepsilon^{2/3} + d^{1/2} / \varepsilon^{1 / (\alpha - 1)} ). Our high-order Langevin diffusion reduces the problem of log-concave sampling to numerical integration along a fixed deterministic path, which makes it possible for further improvements in high-dimensional MCMC problems. Joint work with Yi-An Ma, Martin J, Wainwright, Peter L. Bartlett and Michael I. Jordan.

### The proxy point method for rank-structured matrices

Series
Dissertation Defense
Time
Friday, October 25, 2019 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 311
Speaker
Xin XingSchool of Mathematics, Georgia Tech

Rank-structured matrix representations, e.g., H2 and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their applications is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form.
We focus on the study and application of a class of hybrid analytic-algebraic compression methods, called the proxy point method. We address several critical problems concerning this underutilized method which limit its applicability. A general form of the method is proposed, paving the way for its wider application in the construction of different rank-structured matrices with kernel functions that are more general than those usually used. Further, we extend the applicability of the proxy point method to compress matrices defined by electron repulsion integrals, which accelerates one of the main computational steps in quantum chemistry.

Committee members: Prof. Edmond Chow (Advisor, School of CSE, Georgia Tech), Prof. David Sherrill (School of Chemistry and Biochemistry, Georgia Tech), Prof. Jianlin Xia (Department of Mathematics, Purdue University), Prof. Yuanzhe Xi (Department of Mathematics, Emory University), and Prof. Haomin Zhou (School of Mathematics, Georgia Tech).

### Spin Dynamics: Algorithms and Spin of Planets

Series
SIAM Student Seminar
Time
Friday, October 25, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Renyi ChenGT Math

In this talk, we will focus on the spin dynamics of rigid bodies.
Algorithm part: There are many algorithms designed for N body simulations.
But, to study the climates of a planet, we need to extend the simulation from point mass bodies to rigid bodies.
In the N-rigid-body simulations, we will consider the orientation and angular momentum of the rigid body to understand the spin.
In terms of the algorithm, symplectic integrators are designed by splitting methods.
Physical part: We studied the spin dynamics of an Earth-like planet in circumbinary systems.
Canonical Delaunay variables and Andoyer variables are applied to split the variables to be slow part and fast part.
Applying averaging method, we approximated the spin dynamics.
From the approximated dynamics, we may draw some interesting physical conclusions.