Seminars and Colloquia by Series

Fractional coloring with local demands

Series
Graph Theory Seminar
Time
Thursday, April 11, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tom Kelly – University of Waterloo

In a fractional coloring, vertices of a graph are assigned subsets of the [0, 1]-interval such that adjacent vertices receive disjoint subsets. The fractional chromatic number of a graph is at most k if it admits a fractional coloring in which the amount of "color" assigned to each vertex is at least 1/k. We investigate fractional colorings where vertices "demand" different amounts of color, determined by local parameters such as the degree of a vertex. Many well-known results concerning the fractional chromatic number and independence number have natural generalizations in this new paradigm. We discuss several such results as well as open problems. In particular, we will sketch a proof of a "local demands" version of Brooks' Theorem that considerably generalizes the Caro-Wei Theorem and implies new bounds on the independence number. Joint work with Luke Postle.

Optimal estimation of smooth functionals of high-dimensional parameters

Series
High Dimensional Seminar
Time
Wednesday, April 10, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vladimir Koltchinskii – Georgia Tech –

Please Note: We discuss a general approach to a problem of estimation of a smooth function $f(\theta)$ of a high-dimensional parameter $\theta$ of statistical models. In particular, in the case of $n$ i.i.d. Gaussian observations $X_1,\doot, X_n$ with mean $\mu$ and covariance matrix $\Sigma,$ the unknown parameter is $\theta = (\mu, \Sigma)$ and our approach yields an estimator of $f(\theta)$ for a function $f$ of smoothness $s>0$ with mean squared error of the order $(\frac{1}{n} \vee (\frac{d}{n})^s) \wedge 1$ (provided that the Euclidean norm of $\mu$ and operator norms of $\Sigma,\Sigma^{-1}$ are uniformly bounded), with the error rate being minimax optimal up to a log factor (joint result with Mayya Zhilova). The construction of optimal estimators crucially relies on a new bias reduction method in high-dimensional problems and the bounds on the mean squared error are based on controlling finite differences of smooth functions along certain Markov chains in high-dimensional parameter spaces as well as on concentration inequalities.

Definition of Casson Invariant

Series
Geometry Topology Student Seminar
Time
Wednesday, April 10, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hongyi Zhou – Georgia Institute of Technology –

Casson invariant is defined for the class of oriented integral homology 3-spheres. It satisfies certain properties, and reduce to Rohlin invariant after mod 2. We will define Casson invariant as half of the algebraic intersection number of irreducible representation spaces (space consists of representations of fundamental group to SU(2)), and then prove this definition satisfies the expected properties.

Energy on Spheres and Discreteness of Minimizing Measures

Series
Analysis Seminar
Time
Wednesday, April 10, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josiah Park – Georgia Tech –

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.

On the motion of a rigid body with a cavity filled with a viscous liquid

Series
PDE Seminar
Time
Tuesday, April 9, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Gieri Simonett – Vanderbilt University –
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).

Legendrian Large Cables

Series
Dissertation Defense
Time
Tuesday, April 9, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew McCullough – Georgia Institute of Technology –

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.

Periodic and quasi-periodic attractors of the spin-orbit dynamics of Mercury

Series
Math Physics Seminar
Time
Tuesday, April 9, 2019 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guido Gentile – Universita' di Roma 3 –

Please Note: 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.

Heegaard Floer homology and non-zero degree maps

Series
Geometry Topology Seminar
Time
Monday, April 8, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tye Lidman – NCSU

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.

Interface of statistics and computing

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 8, 2019 - 13:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Xiaoming Huo – GT ISyE –

 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.

 

Limits of split rank two bundles on P^n

Series
Algebra Seminar
Time
Monday, April 8, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mengyuan Zhang – University of California, Berkeley –

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.

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