Georgia Institute of Technology College of Sciences

A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results together. Our approach is based on Mazur’s famous argument and its generalization which provides a unification of all results.

Solving systems of polynomial equations is at the heart of algebraic geometry. In this talk I will discuss a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems. Our approach uses fewer homotopy continuation paths than the current leading method based on regeneration.  Moreover it is applicable in both projective and multiprojective settings. To motivate the approach I will also give some examples coming from applied algebraic geometry.
This is joint work with Tim Duff and Anton Leykin.

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that under the same condition there are four lines intersecting all the sets. We also prove a colorful version of this result under weakened conditions on the sets, improving results of Holmsen from 2013. Our proofs use the topological KKM theorem. Joint with Daniel McGinnis.

Semidefinite programming (SDP) is a very well behaved class of convex optimization problems. We will introduce this class of problems, illustrate how it allows to approximate many practical nonconvex optimization problems, and discuss the role of low rank structure.

Morse theory is a standard concept used in the study of manifolds.  PL-Morse theory is a variant of Morse theory developed by Bestvina and Brady that is used to study simplicial complexes.  We develop an extension of PL-Morse theory to simplicial complexes equipped with an action of a group G.  We will discuss some of the basic ideas in this theory and hopefully sketch proofs of some forthcoming results pertaining to the homology of the Torelli group.

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know  $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. In this setting, we can obtain conditions on $\Omega, A, l_i$ that allow the stable reconstruction of $f$. Dynamical Sampling is closely related to frame theory and has applications to wireless sensor networks among other areas. In this talk, we will discuss the Dynamical Sampling problem, its motivation, related problems inspired by it, current/future work, and open problems. 

The seminar will be held on Zoom and can be found at the link

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

Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.

One question addressed in the field of Diophantine approximation is precisely quantifying how many "good" approximations an algebraic number has by rational numbers. This is answered most soundly by a 1955 theorem of Klaus Roth. In this talk, I will cover this theorem, some related results and hint at how it can be used to bound the number of rational solutions to curves.

Traditional spacecraft trajectory optimization approaches focus on automatizing solution generation by capturing the solution space analytically, or numerically, in a single or few instances. However, critical human-computer interactions within optimization processes are almost always disregarded, and they are not well understood. In fact, human intervention spans across the entire optimization process, from the formulation of a problem that lands on known solution schemes, to the identification of an initial guess within the algorithm basin of convergence, to tuning the algorithm hyper-parameters, investigating anomalies, and parsing large databases of optimal solutions to gain insight. Vision-based interaction with sets of multi-dimensional information mitigates the complexity of several applications in astrodynamics. For example, visual-based processes are key to understanding solution space topology for orbit mechanics (e.g., Poincare’ maps), formulating higher quality initial trajectory guesses for early mission design studies, and investigating six-degree-of-freedom (6DOF) dynamics for proximity operations. The capillary diffusion of visual-based data interaction processes throughout astrodynamics has motivated the creation of virtual reality (VR) technology to facilitate scientific discovery since the advent of modern computers. The recent appearance of small, portable, and affordable devices may be a tipping point to advance astrodynamics applications via VR technology. Nonetheless, the tangible benefits for adoption of virtual reality frameworks are not yet fully characterized in the context of astrodynamics applications. What new opportunities virtual reality opens for astrodynamics? What applications benefits from virtual reality frameworks? To answer these and similar questions, our work focuses on a programmatic early assessment and exploration of VR technology for astrodynamics applications. The assessment is constructed by a review of VR literature with elements that are external to the astrodynamics community to facilitate cross-pollination of ideas. Next, the Johnson-Lindenstrauss lemma, together with a set of simplifying assumptions, is employed to analytically capture the value of projecting higher-dimensional information to a given lower dimensional space. Finally, two astrodynamics applications are presented to display solutions that are primarily enabled by virtual reality technology.

The minimum linear ordering problem (MLOP) asks to minimize the aggregated cost of a set function f with respect to some ordering \sigma of the base set. That is, MLOP asks to find a permutation \sigma that minimizes the sum \sum_{i = 1}^{|E|}f({e \in E : \sigma(e) \le i}). Many instances of MLOP have been studied in the literature, for example, minimum linear arrangement (MLA) or minimum sum vertex cover (MSVC). We will cover how graphic matroid MLOP, i.e. where f is taken to be the rank function of a graphic matroid, is NP-hard. This is achieved through a series of reductions beginning with MSVC. During these reductions, we will introduce a new problem, minimum latency vertex cover (MLVC) which we will also show has a 4/3 approximation. Finally, using the theory of principal partitions, we will show MLOP with monotone submodular function f : E \to \mathbb{R}^+ has a 2 - (1 + \ell_f)/(1 + |E|) approximation where \ell_f = f(E)/(\max_{x \in E}f({x})). As a corollary, we obtain a 2 - (1 + r(E))/(1 + |E|) approximation for matroid MLOP where r is the rank function of the matroid. We will also end with some interesting open questions.

Joint work with Majid Farhadi, Swati Gupta, Shengding Sun, and Prasad Tetali.

In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology.  More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions.  The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.