Georgia Institute of Technology College of Sciences

This talk will include background information on contact structures and open book decompositions of 3-manifolds and the relationship between them. I will state the necessary definitions and include a number of concrete examples. I will also review some convex surface theory, which is the tool used in the main talk to investigate the contact structure – open book relationship.

Oriented matroids are matroids with extra sign data, and they are useful in the tropical study of real algebraic geometry. In order to study the topology of real algebraic hypersurfaces constructed from patchworking, Renaudineau and Shaw introduced an algebraically defined filtration of the tope space of an oriented matroid based on Quillen filtration. We will prove the equality between their filtration (together with the induced maps), the topologically defined Kalinin filtration, and the combinatorially defined Varchenko-Gelfand dual degree filtration over Z/2Z. We will also explain how the dual degree filtration can serve as a Z-coefficient version of the other two in this setting. This is joint work with Kris Shaw.

I will discuss recent work with K. Honda and Y. Huang on proving the Giroux correspondence between contact structures and open book decompositions. Though our work extends to all dimensions (with appropriate adjectives), this talk will focus on the 3-dimensional proof. I will first recall Giroux’s argument for existence of supporting open book decompositions, formulating it in the language adapted to our proof. The rest of the talk will be spent describing the proof of the stabilization correspondence.

In this talk, I will discuss my recent research on the asymptotic stability and inviscid damping of 2D monotone shear flows with non-constant density in inhomogeneous ideal fluids within a finite channel. More precisely, I proved that if the initial perturbations belong to the Gevrey-2- class, then linearly stable monotone shear flows in inhomogeneous ideal fluids are also nonlinear asymptotically stable. Furthermore, inviscid damping is proved to hold, meaning that the perturbed velocity converges to a shear flow as time approaches infinity.

The dynamic shortest path problem seeks to maintain the shortest paths/distances between pairs of vertices in a graph that is subject to edge insertions, deletions, or weight changes. The aim is to maintain that information more efficiently than naive recomputation via, e.g., Dijkstra's algorithm.
We present the first fully dynamic algorithm maintaining exact single source distances in unweighted graphs. This resolves open problems stated in [Demetrescu and Italiano, STOC'03], [Thorup SWAT'04], [Sankowski, COCOON 2005] and [vdBrand and Nanongkai, FOCS 2019].
In this talk, we will see how ideas from fine-grained complexity theory, computer algebra, and graph theory lead to insights for dynamic shortest paths problems.

This talk will present an overview of the behavior of the eigenvalues of the fractional Brownian matrix motion and other related matrix processes. We will do so by emphasizing the dynamics of the eigenvalues processes, the non-colliding property, the limit of the associated empirical process, as well as the free Brownian motion and the non commutative fractional Brownian motion.

A conversation with Stephen Young, 2008 GT ACO PhD and Senior Research Mathematician at Pacific Northwest National Laboratory, on opportunities for mathematicians in the unique combination of business/industry/government afforded by the DOE national labs.

(Coffee will be available at 3:30 following this discussion and before the speaker's ACO Alumni Lecture at 4pm.)

Abstract: While spectral methods provide far-ranging insights on graph structure, there remain significant challenges in their application to real data. Most notably, spectral methods do not incorporate information that maybe available beyond adjacency.  A common approach to incorporating such additional information is encode this information in an ad-hoc manner into weights associated with the edges. Not only does this have limited expressivity, but is also restricted by graph structure: if two vertices are not adjacent, then edge weights cannot capture any closeness implied by metadata.

We address this issue by introducing the inner product Hodge Laplacian for an arbitrary simplicial complex.  Within this framework we prove generalizations of foundational results in spectral graph theory, such as the Cheeger inequality and the expander mixing lemma, and show our framework recovers the usual combinatorial and normalized Laplacians as special cases. Our framework allows for the principled synthesis of combinatorial approaches in network science with more metadata driven approaches by using latent space encodings of the metadata to define an inner product both the vertices and the edges.

(Coffee will be available at 3:30 before this talk, following the speaker's Professional Development Seminar at 2:30pm.)

We study the problem of estimating the left and right singular subspaces for a collection of heterogeneous random graphs with a shared common structure. We analyze an algorithm that first estimates the orthogonal projection matrices corresponding to these subspaces for each individual graph, then computes the average of the projection matrices, and finally finds the matrices whose columns are the eigenvectors corresponding to the d largest eigenvalues of the sample averages. We show that the algorithm yields an estimate of the left and right singular vectors whose row-wise fluctuations are normally distributed around the rows of the true singular vectors. We then consider a two-sample hypothesis test for the null hypothesis that two graphs have the same edge probabilities matrices against the alternative hypothesis that their edge probabilities matrices are different. Using the limiting distributions for the singular subspaces, we present a test statistic whose limiting distribution converges to a central chi-square (resp. non-central chi-square) under the null (resp. alternative) hypothesis. Finally, we adapt the theoretical analysis for multiple networks to the setting of distributed PCA; in particular, we derive normal approximations for the rows of the estimated eigenvectors using distributed PCA when the data exhibit a spiked covariance matrix structure.

We consider a numerical method to approximate the solution operator for evolutional partial differential equations (PDEs). By employing a general reduced-order model, such as a deep neural network, we connect the evolution of a model's parameters with trajectories in a corresponding function space. Using the Neural Ordinary Differential Equations (NODE) technique we learn a vector field over the parameter space such that from any initial starting point, the resulting trajectory solves the evolutional PDE. Numerical results are presented for a number of high-dimensional problems where traditional methods fail due to the curse of dimensionality.

This talk will present a new online algorithm for sequential detection of change points in state-space models. The algorithm is computationally fast, and sensitive to changes in model parameters (including observation and evolution variances), as well as model structure. We consider change point detection in a sequential way, when observations are received one by one, or in batches, with a (possibly soft) restart after each detected change point. We provide the theoretical foundation of the algorithm, and study its performance in different state space models used to model the growth of epidemics over time, using simulated data and the recent COVID-19 dataset.  This work is joint work with Ruyu Tan.

This seminar is in a Hybrid format.  The in-person version is on campus at Georgia Tech in Skiles 005.  The virtual version will be at: https://gatech.zoom.us/j/92952024862

Let H be a graph. We show that if r is large enough as a function of H,
then the r-partite Turán graph maximizes the number of copies of H among
all Kr+1-free graphs on a given number of vertices. This confirms a
conjecture of Gerbner and Palmer.