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Friday, April 12, 2019 - 15:05 ,
Location: Skiles 246 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk we will follow the paper titled "Aubry-Mather theory for homeomorphisms", in which it is developed a variational approach to study the dynamics of a homeomorphism on a compact metric space. In particular, they are described orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semidistance. This is work of Albert Fathi and Pierre Pageault.

Friday, April 12, 2019 - 12:00 ,
Location: Skiles 006 ,
Stephen McKean ,
Georgia Tech ,
mckean@math.gatech.edu ,
Organizer:

Milnor K-theory is a field invariant that originated as an attempt to study algebraic K-theory. Instead, Milnor K-theory has proved to have many other applications, including Galois cohomology computations, Voevodsky's proof of the Bloch-Kato conjecture, and Kato's higher class field theory. In this talk, we will go over the basic definitions and theorems of Milnor K-theory. We will also discuss some of these applications.

Series: Stochastics Seminar

Neural networks have led to new and state of the art approaches for image recovery. They provide a contrast to standard image processing methods based on the ideas of sparsity and wavelets. In this talk, we will study two different random neural networks. One acts as a model for a learned neural network that is trained to sample from the distribution of natural images. Another acts as an unlearned model which can be used to process natural images without any training data. In both cases we will use high dimensional concentration estimates to establish theory for the performance of random neural networks in imaging problems.

Series: Graph Theory Seminar

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.

Series: High Dimensional Seminar

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.

Wednesday, April 10, 2019 - 14:00 ,
Location: Skiles 006 ,
Hongyi Zhou ,
Georgia Institute of Technology ,
hzhou@gatech.edu ,
Organizer: Surena Hozoori

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.

Series: Analysis Seminar

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

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).

Series: Dissertation Defense

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.

Series: Math Physics Seminar

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.