Georgia Institute of Technology College of Sciences

I'm interested in the smooth mapping class group of S^4, i.e. pi_0(Diff^+(S^4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S^4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory. Most of what works in Cerf theory works in dimension 5 and higher, but with a little digging one discovers statements that work in dimension 4 as well. Putting all this stuff together we can show that there is a surjective homomorphism from (a certain limit of) fundamental groups of spaces of embeddings of 2-spheres in connected sums of S^2XS^2 onto this smooth mapping class group of S^4. Furthermore, we can identify two natural, and in some sense complementary, subgroups of this fundamental group, one in the kernel of this homomorphism and one whose image we can understand explicitly in terms of Dehn twist-like diffeomorphisms supported near pairs of embedded S^2's in S^4 (Montesinos twins).

We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning.  By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved.  Connections to geometric analysis, stochastic processes, and deep learning are emphasized.  The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points.  Applications to image processing and bioinformatics will be shown, demonstrating state-of-the-art empirical performance.  Extensions to active learning, generative modeling, and computational geometry will be discussed.

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. We show that the conjecture of Glock, Kühn and Osthus is affirmative.

We introduce the averages $K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) f(x+n)$, where $d(n)$ denotes the divisor function and $D(N) = \sum _{n=1} ^N d(n) $. We shall see that these averages satisfy a uniform, scale free, $\ell^p$-improving estimate for $p \in (1,2)$, that is

$$ \Bigl( \frac{1}{N} \sum |K_Nf|^{p'} \Bigl)^{1/p'}  \leq C  \Bigl(\frac{1}{N} \sum |f|^p \Bigl)^{1/p} $$

as long as $f$ is supported on the interval $[0,N]$.

We will also see that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse bounds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

The seminar will be held on Zoom, and can be accessed by the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, together with M. Loss and others, we worked to extend the analysis to more "realistic" versions of the original Kac model. I will give a brief overview of kinetic theory, introduce the Kac model and explain the standard results on it. Finally I will present to new papers with M. Loss and R. Han and with J. Beck.

At each site of Z^d, initially there is a car with probability p or a vacant parking spot with probability (1-p), and the choice is independent for all sites. Cars perform independent simple, symmetric random walks, which do not interact directly with one another, and parking spots do not move. When a car enters a site that contains a vacant spot, then the car parks at the spot and the spot is filled – both the car and the spot are removed from the system, and other cars can move freely through the site. This model exhibits a phase transition at p=1/2: all cars park almost surely if and only if p\le 1/2, and all vacant spots are filled almost surely if and only if p \ge 1/2. We study the rates of decay of cars and vacant spots at, below and above p=1/2. In many cases these rates agree with earlier findings of Bramson—Lebowitz for two-type annihilating systems wherein both particle types perform random walks at equal speeds, though we identify significantly different behavior when p<1/2. Based on joint works with Damron, Gravner, Johnson, Junge and Lyu.

Online at https://bluejeans.com/129119189 

A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree. 
In 1981, Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. 
We prove an approximate version of this conjecture: for every $\epsilon>0$ there exists $n_0$ such that every diregular bipartite tournament on $2n>n_0$  vertices contains a collection of $(1/2-\epsilon)n$ cycles of length at least $(2-\epsilon)n$. 
Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every $c>1/2$ and $\epsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n>n_0$ vertices contains $(1-\epsilon)cn$ edge-disjoint Hamilton cycles.

Base on joint work with Yanitsa Pehova, see https://arxiv.org/abs/1907.08479

Please note the special time/day: Thursday 6pm

Consider the three body problem with masses $m_0,m_1,m_2>0$. Take units such that $m_0+m_1+m_2 = 1$. In 1922 Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of them are what is known as oscillatory motions, that is, solutions of the three body problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy
\[
\liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad 
\limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty.
\] At the time of Chazy, all types of final motions were known, except the oscillatory ones. We prove that, if all three masses $m_0,m_1,m_2>0$ are not equal to $1/3$, such motions exist. In fact, we prove more, since our result is based on the construction of a hyperbolic invariant set whose dynamics is conjugated to the Bernouilli shift of infinite symbols, we prove (if all masses are not all three equals to $1/3$) 1) the existence of chaotic motions and positive topological entropy for the three body problem, 2) the existence of periodic orbits of arbitrarily large period in the 3BP. Reversing time, Chazy's classification describes ``starting'' motions and then, the question if starting and final motions need to coincide or may be different arises.  We also prove that one can construct solutions of the three body problem whose starting and final motions are of different type.

Suppose that n sample genomes are collected from the same population. The expected sample frequency spectrum (SFS) is the vector of probabilities that a mutation chosen at random will appear in exactly k out of the n individuals. This vector is known to be highly dependent on the population size history (demography); for this reason, geneticists have used it for demographic inference. What does the set of all possible vectors generated by demographies look like? What if we specify that the demography has to be piecewise-constant with a fixed number of pieces? We will draw on tools from convex and algebraic geometry to answer these and related questions.

Meeting Link: https://gatech.bluejeans.com/348270750

The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space $\ell_1$. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space $\ell_1^n$. We prove that for any $N \geq n$, FUNTFs of length $N$ exist for real $\ell_1^n$. To show the existence of FUNTFs, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements. However, a different method of frame constructions allows us to prove the existence of FUNTFs for real $\ell_1^n$ of lengths $2n-1$ and $2n-2$ that do not have repeated elements.

The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space $\ell_1$. A Schauder frame provides a reconstruction formula for elements in the space, but need not be associated with a frame inequality. Our main theorem on this topic establishes a set of conditions under which an $\ell_1$-type of frame inequality is applicable towards unconditional Schauder frames. A primary motivation for choosing this set of hypotheses involves appropriate modifications of the Rademacher system, a version of which we prove to be an unconditional Schauder frame that does not satisfy an $\ell_1$-type of frame inequality.

Bluejeans link to meeting: https://bluejeans.com/544995272