Seminars and Colloquia by Series

Efficient Krylov subspace methods for uncertainty quantification

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 19, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Julianne Chung – Emory University –
Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems), for problems where computation of the square root and inverse of the prior covariance matrix are not feasible, and for hierarchical problems where the mean is not known a priori. This work exploits Krylov subspace methods to develop and analyze new techniques for large-scale uncertainty quantification in inverse problems. We assume that generalized Golub-Kahan based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography and atmospheric inverse modeling, including a case study from NASA's Orbiting Carbon Observatory 2 (OCO-2) satellite, demonstrate the effectiveness of the described approaches.

Hyperbolic models for CAT(0) spaces by Abdul Zalloum

Series
Geometry Topology Seminar
Time
Monday, September 19, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Abdalrazzaq (Abdul) Zalloum – University of Toronto

Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.

Algebraic groups, moduli spaces of matroids, and the field with one element

Series
Algebra Seminar
Time
Monday, September 19, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Matt Baker – Georgia Institute of Technology –

 I will give an introduction to Oliver Lorscheid’s theory of ordered blueprints – one of the more successful approaches to “the field of one element” – and sketch its relationship to Tits models for algebraic groups and moduli spaces of matroids. The basic idea for these applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,-1}, there is a corresponding “blue model” whose K-points (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking closed K-points of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking K-points of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,...,n}.

From walls to cube complexes by Abdul Zalloum

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 19, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Abdalrazzaq (Abdul) Zalloum – University of Toronto

A geodesic metric space is said to be CAT(0) if triangles are at most as fat as triangles in the Euclidean plane. A CAT(0) cube complex is a CAT(0) space which is built by gluing Euclidean cubes isometrically along faces. Due to their fundamental role in the resolution of the virtual Haken's conjecture, CAT(0) cube complexes have since been a central object of study in geometric group theory and their study has led to ground-breaking advances in 3–manifold theory. The class of groups admitting proper cocompact actions on CAT(0) cube complexes is very broad and it includes hyperbolic 3-manifolds, most non-geometric 3 manifold groups, small cancelation groups and many others. 

 

A revolutionary work of Sageev shows that the entire structure of a CAT(0) cube complexes is encoded in its hyperplanes and the way they interact with one another. I will discuss Sageev's theorem which provides a recipe for constructing group actions on CAT(0) cube complexes using some very simple and purely set theoretical data.

On a conjecture of Graham on the p-divisibility of central binomial coefficients

Series
Combinatorics Seminar
Time
Friday, September 16, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Ernie Croot – Georgia Institute of Technology

I will discuss an old conjecture of Ron Graham on whether there are infinitely many integers $n$ so that $\mathrm{gcd}({{2n} \choose n}, 105)=1$, as well as recent progress on a version of this problem where 105 is replaced with a product of $r$ distinct primes. This is joint work with Hamed Mousavi and Maxie Schmidt.

Smooth structures on open 4-manifolds II

Series
Geometry Topology Working Seminar
Time
Friday, September 16, 2022 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech

Please Note: One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.

When dynamics meet machine learning

Series
Time
Friday, September 16, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Molei Tao – Georgia Tech

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

Abstract:  The interaction of machine learning and dynamics can lead to both new methodology for dynamics, and deepened understanding and/or efficacious algorithms for machine learning. This talk will give examples in both directions. Specifically, I will first discuss data-driven learning and prediction of mechanical dynamics, for which I will demonstrate one strong benefit of having physics hard-wired into deep learning models; more precisely, how to make symplectic predictions, and how that probably improves the accuracy of long-time predictions. Then I will discuss how dynamics can be used to better understand the implicit biases of large learning rates in the training of machine learning models. They could lead to quantitative escapes from local minima via chaos, which is an alternative mechanism to commonly known noisy escapes due to stochastic gradients. I will also report how large learning rates bias toward flatter minimizers, which arguably generalize better.

Polynomials over ordered blueprints and tracts

Series
Algebra Student Seminar
Time
Friday, September 16, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Trevor Gunn – Georgia Tech

I will introduce the concept of an ordered blueprint and a tract and discuss some algebraic and categorical properties. I will then discuss the notion of a "tropical extension" and discuss the theory of polynomials in these contexts.

Families of Lefschetz Fibrations via Cyclic Group Actions

Series
Geometry Topology Seminar
Time
Monday, September 12, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nur Saglam – Georgia Tech –
Using various diagonal cyclic group actions on the product manifolds ΣgxΣg for g>0, we obtain some families of Lefschetz fibrations over S^2. Then, we study the monodromies of these families applying the resolution of cyclic quotient singularities. We also realize some patterns of singular fibers and study deformations of these Lefschetz fibrations. Some cases give rise to nice applications using rational blow-down operation. This is a joint work with A. Akhmedov and M. Bhupal.

 

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