Topological dynamics of knotted and tangled matter
- Series
- CDSNS Colloquium
- Time
- Friday, September 29, 2023 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 249
- Speaker
- Vishal Patil – Stanford – vppatil@stanford.edu
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.
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
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.
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.
Zoom link: https://gatech.zoom.us/j/94868589860
The Teichmueller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various contexts of mathematics and physics. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmüller spaces extend this notion to appropriate higher rank classical Lie groups G. Other generalizations are given by the super-Teichmueller spaces, describing Riemann surfaces enhanced by odd or anti-commuting coordinates (known as super Riemann surfaces). The super-Teichmueller spaces arise naturally as higher Teichmueller spaces, corresponding to supergroups, which play an important role in geometric topology, algebraic geometry, and mathematical physics, where the anti-commuting variables correspond to Fermions.
After introducing these spaces, I will explain the solution to the long-standing problem of describing the counterpart of Penner coordinates on the super-Teichmueller space and its higher analogues. The importance of these coordinates is justified by two remarkable properties: the action of the mapping class group is rational, and the Weil-Petersson form is given by a simple explicit formula. From the algebraic and combinatorial perspectives, their transformations lead to an important generalization of cluster algebras.
In the end, I will discuss some recent applications of this construction.
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.)
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.)
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.