Georgia Institute of Technology College of Sciences

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.

We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is efficient in that it exploits of the low dimension of the invariant manifold, and takes time steps of size dependent on the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them. 

Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts. In my talk, I will report on joint work with Renzo Cavalieri and Hannah Markwig, in which we define tropical psi classes and study relations between them. I will explain how some of the expected identities cannot be recovered from a purely tropical perspective, whereas others can, revealing the tropical nature they have been of in the first place.

This talk is motivated by the Erdős–Szekeres theorem on monotone subsequences: given a sequence of $rs+1$ distinct numbers, there is either a subsequence of $r+1$ of them in increasing order, or a subsequence of $s+1$ of them in decreasing order.

We'll consider many related questions with an algorithmic flavor, such as: if we want to find one of the subsequences promised, how many comparisons do we need to make? What if we have to pre-register our comparisons ahead of time? Does it help if we search a longer sequence instead?

Some of these questions are still open; some of them have answers. The results I will discuss are joint work with Jozsef Balogh, Felix Clemen, and Emily Heath at UIUC.

Hybridization plays an important role during the evolutionary process of some species. In such cases, phylogenetic trees are sometimes insufficient to describe species-level relationships. We show that most topological features of a level-1 species network (a network with no interlocking cycles) are identifiable under the network multi-species coalescent model using the logDet distance between aligned DNA sequences of concatenated genes. 

Different aspects of q-calculus are widely used in number theory, combinatorics, orthogonal polynomials, to name a few. In this talk we introduce q-calculus and consider its applications  to the Stirling numbers.

A general theme in studying manifolds is understanding lower dimensional submanifolds that encode information. For contact manifolds, these are Legendrians. I will discuss some low and high dimensional examples of Legendrians, their invariants, and how they are applied to understand manifolds. I will also talk about the Legendrian Low Conjecture, which says that understanding linking of certain Legendrians is the key to understanding causal relations between events in a globally hyperbolic spacetime.

This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday.  Years ago, John wrote a text "Real Variable and Integration", published in 1976.  This was not the text that I first learned real analysis from, but it became an important reference for me.  A later revision and expansion by John and Wojtek Czaja appeared in 2009.  Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory.  My goal was that each proof was to be both rigorous and enlightening.  I failed (in the chapters on differentiation and absolute continuity).  I will discuss the real analysis theorem whose proof I find the most difficult and unenlightening.  But I will also present the Banach-Zaretsky Theorem, which I first learned from John's text.  This is an elegant but often overlooked result, and by using it I (re)discovered enlightening proofs of theorems whose standard proofs are technical and difficult.  This talk will be a tour of the absolutely fundamental concept of absolute continuity from the viewpoint of the Banach-Zaretsky Theorem.

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

Zoom link: https://us02web.zoom.us/j/84598656431?pwd=UGN5QmJZdnE2MktpM005bFZFK29Gdz09

By a theorem of Ben-Yaacov, Melleray, and Tsankov, whenever $G$ is a Polish group with metrizable universal minimal flow $M(G)$, then $M(G)$ must contain a comeager orbit. This has the following peculiar consequence: If $G$ is a Polish group and $X$ is some minimal metrizable $G$-flow with all orbits meager, then there must exist some non-metrizable minimal $G$-flow. So given such an $X$, can we use $X$ directly in order to construct a non-metrizable minimal $G$-flow? This talk will discuss such a construction, thus providing a new proof of the Generic Point Problem.

Semidefinite programming is a useful type of convex optimization, which has applications in both graph theory and industrial engineering. Many semidefinite programs exhibit a kind of structured sparsity, which we can hope to exploit to reduce the memory requirements of solving such semidefinite programs. We will discuss an interesting relaxation of such sparse semidefinite programs, and a measurement of how well this relaxation approximates a true semidefinite program. We'll also discuss how these approximations relate to graph theory and the theory of sum-of-squares and nonnegative polynomials. This talk will not assume any background on semidefinite programming.

Donaldson’s Diagonalization Theorem has been used extensively over the past 15 years as an obstructive tool. For example, it has been used to obstruct: rational homology 3-spheres from bounding rational homology 4-balls; knots from being (smoothly) slice; and knots from bounding (smooth) Mobius bands in the 4-ball. In this multi-part series, we will see how this obstruction works, while getting into the weeds with concrete calculations that are usually swept under the rug during research talks.