## Seminars and Colloquia by Series

Wednesday, April 10, 2019 - 13:55 , Location: Skiles 005 , , Georgia Tech , , Organizer: Josiah Park

When equiangular tight frames (ETF's), a type of structured optimal packing of lines, exist and are of size $|\Phi|=N$, $\Phi\subset\mathbb{F}^d$ (where $\mathbb{F}=\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$), for $p > 2$ the so-called $p$-frame energy $E_p(\Phi)=\sum\limits_{i\neq j} |\langle \varphi_{i}, \varphi_{j} \rangle|^p$ achieves its minimum value on an ETF over all sized $N$ collections of unit vectors. These energies have potential functions which are not positive definite when $p$ is not even. For these cases the apparent complexity of the problem of describing minimizers of these energies presents itself. While there are several open questions about the structure of these sets for fixed $N$ and fixed $p$, we focus on another question:

What structural properties are expressed by minimizing probability measures for the quantity $I_{p}(\mu)=\int\limits_{\mathbb{S}_{\mathbb{F}}^{d-1}}\int\limits_{\mathbb{S}_{\mathbb{F}}^{d-1}} |\langle x, y \rangle|^p d\mu(x) d\mu(y)$?
We collect a number of surprising observations. Whenever a tight spherical or projective $t$-design exists for the sphere $\mathbb{S}_{\mathbb{F}}^d$, equally distributing mass over it gives a minimizer of the quantity $I_{p}$ for a range of $p$ between consecutive even integers associated with the strength $t$. We show existence of discrete minimizers for several related potential functions, along with conditions which guarantee emptiness of the interior of the support of minimizers for these energies.
This talk is based on joint work with D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.

Series: PDE Seminar
Tuesday, April 9, 2019 - 15:00 , Location: Skiles 006 , Professor Gieri Simonett , Vanderbilt University , , Organizer: Xukai Yan

I will consider the motion of a rigid body with an interior cavity that is completely filled with a viscous fluid. The equilibria of the system will be characterized and their stability properties are analyzed. It will be shown that the fluid exerts a stabilizing effect, driving the system towards a state where it is moving as a rigid body with constant angular velocity. In addition, I will characterize the critical spaces for the governing evolution equation, and I will show how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity and stability properties for the system. The approach is based on the theory of Lp-Lq maximal regularity. (Joint work with G. Mazzone and J. Prüss).

Tuesday, April 9, 2019 - 14:00 , Location: Skiles 006 , , Georgia Institute of Technology , , Organizer: Andrew McCullough

We define the notion of a knot type having Legendrian large cables and
show that having this property implies that the knot type is not uniformly thick.
Moreover, there are solid tori in this knot type that do not thicken to a solid torus
with integer sloped boundary torus, and that exhibit new phenomena; specifically,
they have virtually overtwisted contact structures. We then show that there exists
an infinite family of ribbon knots that have Legendrian large cables. These knots fail
to be uniformly thick in several ways not previously seen. We also give a general
construction of ribbon knots, and show when they give similar such examples.

Tuesday, April 9, 2019 - 12:00 , Location: Skiles 005 , Guido Gentile , Universita&#039; di Roma 3 , , Organizer: Federico Bonetto

Unusual time.

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of Mercury's eccentricity. However, the mathematical model commonly used to study the problem -- sometimes called the spin-orbit model -- proved not to be entirely convincing, because of the expression used for the tidal torque. Only recently, a different model for the tidal torque has been proposed, with the advantage of both being more realistic and providing a higher probability of capture into the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and appears as a superposition of peaks, so that both analytical and numerical computations turn out to be rather delicate. We shall present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance, as predicted by the realistic model. In particular we shall provide evidence that the librations are quasi-periodic in time, so that the very concept of resonance should be revisited. The analytical results are mainly based on perturbation theory and leave several open problems, that we shall discuss.

Monday, April 8, 2019 - 14:00 , Location: Skiles 006 , Tye Lidman , NCSU , Organizer: Jennifer Hom

We will use Heegaard Floer homology to analyze maps between a certain family of three-manifolds akin to the Gromov norm/hyperbolic volume.  Along the way, we will study the Heegaard Floer homology of splices.  This is joint work with Cagri Karakurt and Eamonn Tweedy.

Monday, April 8, 2019 - 13:50 , Location: Skiles 005 , , GT ISyE , , Organizer: Sung Ha Kang

Inference (aka predictive modeling) is in the core of many data science problems. Traditional approaches could be either statistically or computationally efficient, however not necessarily both. The existing principles in deriving these models - such as the maximal likelihood estimation principle - may have been developed decades ago, and do not take into account the new aspects of the data, such as their large volume, variety, velocity and veracity. On the other hand, many existing empirical algorithms are doing extremely well in a wide spectrum of applications, such as the deep learning framework; however they do not have the theoretical guarantee like these classical methods. We aim to develop new algorithms that are both computationally efficient and statistically optimal. Such a work is fundamental in nature, however will have significant impacts in all data science problems that one may encounter in the society. Following the aforementioned spirit, I will describe a set of my past and current projects including L1-based relaxation, fast nonlinear correlation, optimality of detectability, and nonconvex regularization. All of them integrates statistical and computational considerations to develop data analysis tools.

Monday, April 8, 2019 - 12:50 , Location: Skiles 005 , Mengyuan Zhang , University of California, Berkeley , , Organizer: Justin Chen

In this talk we discuss the following problem due to Peskine and Kollar: Let E be a flat family of rank two bundles on P^n parametrized by a smooth variety T. If E_t is isomorphic to O(a)+O(b) for general t in T, does it mean E_0 is isomorphic to O(a)+O(b) for a special point 0 in T? We construct counter-examples in over P^1 and P^2, and discuss the problem in P^3 and higher P^n.

Monday, April 8, 2019 - 12:45 , Location: Skiles 006 , Tye Lidman , NCSU , Organizer: Jennifer Hom

In this talk, we will study Seifert fibered three-manifolds. While simple to define, they comprise 6 of the 8 Thurston geometries, and are an important testing ground for many questions and invariants. We will present several constructions/definitions of these manifolds and learn how to work with them explicitly.

Monday, April 8, 2019 - 12:10 , Location: Skiles 202 , Jiangning Chen , Georgia Institute of Technology , , Organizer:

We are going talk about three topics. First of all, Principal Components Analysis (PCA) as a dimension reduction technique. We investigate how useful it is for real life problems. The problem is that, often times the spectrum of the covariance matrix is wrongly estimated due to the ratio between sample space dimension over feature space dimension not being large enough. We show how to reconstruct the spectrum of the ground truth covariance matrix, given the spectrum of the estimated covariance for multivariate normal vectors. We then present an algorithm for reconstruction the spectrum in the case of sparse matrices related to text classification.

In the second part, we concentrate on schemes of PCA estimators. Consider the problem of finding the least eigenvalue and eigenvector of ground truth covariance matrix, a famous classical estimator are due to Krasulina. We state the convergence proof of Krasulina for the least eigenvalue and corresponding eigenvector, and then find their convergence rate.

In the last part, we consider the application problem, text classification, in the supervised view with traditional Naive-Bayes method. We find out an updated Naive-Bayes method with a new loss function, which loses the unbiased property of traditional Naive-Bayes method, but obtains a smaller variance of the estimator.

Committee:  Heinrich Matzinger (Advisor); Karim Lounici (Advisor); Ionel Popescu (school of math); Federico Bonetto (school of math); Xiaoming Huo (school of ISYE);

Monday, April 8, 2019 - 11:15 , Location: Skiles 005 , M. Capinski , Jagiellonian University/Florida Atlantic University , Organizer: Rafael de la Llave

We present a topological mechanism of diffusion in a priori chaotic systems. The method leads to a proof of diffusion for an explicit range of perturbation parameters. The assumptions of our theorem can be verified using interval arithmetic numerics, leading to computer assisted proofs. As an example of application we prove diffusion in the Neptune-Triton planar elliptic restricted three body problem. Joint work with Marian Gidea.