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Friday, November 30, 2018 - 14:00 ,
Location: Skiles 006 ,
Surena Hozoori ,
Georgia Institute of Technology ,
shozoori3@gatech.edu ,
Organizer: Surena Hozoori

In post-geometrization low dimensional topology, we expect to be able to relate any topological theory of 3-manifolds to the Riemannian geometry of those manifolds. On the other hand, originated from reformalization of classical mechanics, the study of contact structures has become a central topic in low dimensional topology, thanks to the works of Eliashberg, Giroux, Etnyre and Taubes, to name a few. Yet we know very little about how Riemannian geometry fits into the theory.In my oral exam, I will talk about "Ricci-Reeb realization problem" which asks which functions can be prescribed as the Ricci curvature of a "Reeb vector field" associated to a contact manifold. Finally motivated by Ricci-Reeb realization problem and using the previous study of contact dynamics by Hofer-Wysocki-Zehnder, I will prove new topological results using compatible geometry of contact manifolds. The generalization of these results in higher dimensions is the first known results achieving tightness based on curvature conditions.

Series: Algebra Seminar

In this talk we will discuss an arithmetic analogue of the gonality of a nice curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is understood when this invariant is 1, 2, or 3; by work of Debarre-Fahlaoui these criteria do not generalize. We will focus on scenarios under which we can guarantee that this invariant is actually equal to the gonality using the auxiliary geometry of a surface containing the curve. This is joint work with Geoffrey Smith.

Series: ACO Student Seminar

In this talk we introduce two different random graph models that produce
sparse graphs with overlapping community structure and discuss
community detection in each context. The Random Overlapping Community
(ROC) model produces a sparse graph by constructing many Erdos Renyi
random graphs (communities) on small randomly selected subsets of
vertices. By varying the size and density of these communities, ROC
graphs can be tuned to exhibit a wide range normalized of closed walk
count vectors, including those of hypercubes. This is joint work with
Santosh Vempala. In the second half of the talk, we introduce the
Community Configuration Model (CCM), a variant of the configuration
model in which half-edges are assigned colors and pair according to a
matching rule on the colors. The model is a generalization of models in
the statistical physics literature and is a natural finite analog for
classes of graphexes. We describe a hypothesis testing algorithm that
determines whether a graph came from a community configuration model or a
traditional configuration model. This is joint work with Christian
Borgs, Jennifer Chayes, Souvik Dhara, and Subhabrata Sen.

Series: Stochastics Seminar

Heavy tailed distributions have been shown to be
consistent with data in a variety of systems with multiple time
scales. Recently, increasing attention has appeared in different
phenomena related to climate. For example, correlated additive and
multiplicative (CAM) Gaussian noise, with infinite variance or heavy
tails in certain parameter regimes, has received increased attention in
the context of atmosphere and ocean dynamics. We discuss how CAM noise
can appear generically in many reduced models. Then we show how reduced
models for systems driven by fast linear CAM noise processes can be
connected with the stochastic averaging for multiple scales systems
driven by alpha-stable processes. We identify the conditions under
which the approximation of a CAM noise process is valid in the averaged
system, and illustrate methods using effectively equivalent fast,
infinite-variance processes. These applications motivate new
stochastic averaging results for systems with fast processes driven by
heavy-tailed noise. We develop these results for the case of
alpha-stable noise, and discuss open problems for identifying
appropriate heavy tailed distributions for these multiple scale systems.
This is joint work with Prof. Adam Monahan (U Victoria) and Dr. Will
Thompson (UBC/NMi Metrology and Gaming).

Series: Job Candidate Talk

The cellular cytoskeleton ensures the dynamic transport, localization, and anchoring of various proteins and vesicles. In the development of egg cells into embryos, messenger RNA (mRNA) molecules bind and unbind to and from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters.We consider these models in the framework of partial differential equations as well as stochastic processes and derive the effective velocity and diffusivity of cargo at large time for a general class of problems. Including the geometry of the microtubule filaments allows for better prediction of particle localization and for investigation of potential anchoring mechanisms. Our numerical studies incorporating model microtubule structures suggest that anchoring of mRNA-molecular motor complexes may be necessary in localization, to promote healthy development of oocytes into embryos. I will also briefly go over other ongoing projects and applications related to intracellular transport.

Series: Other Talks

Oral Comprehensive Exam

<p>The purpose of this work is approximation of generic Hamiltonian dynamical systems by those with a finite number of islands. In this work, we will consider a Lemon billiard as our Hamiltonian dynamical system apparently with an infinitely many islands. Then, we try to construct a Hamiltonian dynamical system by deforming the boundary of our lemon billiard to have a finite number of islands which are the same or sub-islands of our original system. Moreover, we want to show elsewhere in the phase space of the constructed billiard is a chaotic sea. In this way, we will have a dynamical system which preserves some properties of our lemon billiards while it has much simpler structure.</p>

Series: Graph Theory Working Seminar

Continuing
on the theme mentioned in my recent research horizons lecture, I will
illustrate two techniques by deriving upper and lower bounds on the
number of independent sets in bipartite and triangle-free graphs.

Wednesday, November 28, 2018 - 14:00 ,
Location: Skiles 006 ,
Sidhanth Raman ,
Georgia Tech ,
Organizer: Sudipta Kolay

The Archimedes Hatbox Theorem is a wonderful little theorem about the
sphere and a circumscribed cylinder having the same surface area, but
the sphere can potentially still be characterized by inverting the
statement. There shall be a discussion of approaches
to prove the claim so far, and a review of a weaker inversion of the
Hatbox Theorem by Herbert Knothe and discussion of a related problem in
measure theory that would imply the spheres uniqueness in this property.

Series: Analysis Seminar

Recently Bourgain and Dyatlov proved a fractal uncertainty principle
(FUP), which roughly speaking says a function in $L^2(\mathbb{R})$ and
its Fourier transform can not be simultaneously localized in
$\delta$-dimensional fractal
sets, $0<\delta<1$. In this talk, I will discuss a joint work
with Schlag, where we obtained a higher dimensional version of the FUP.
Our method combines the original approach by Bourgain and Dyatlov, in
the more quantitative rendition by Jin and Zhang, with
Cantan set techniques.

Series: High Dimensional Seminar

In this talk I will describe those linear subspaces of $\mathbf{R}^d$ which can be formed by taking the linear span of lattice points in a half-open parallelepiped. I will draw some connections between this problem and Keith Ball's cube slicing theorem, which states that the volume of any slice of the unit cube $[0,1]^d$ by a codimension-$k$ subspace is at most $2^{k/2}$.