Seminars and Colloquia by Series

Balanced truncation for Bayesian inference

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 2, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Clough Commons 125 and https://gatech.zoom.us/j/98355006347
Speaker
Elizabeth Qian – School of Aerospace Engineering and School of Computational Science and Engineering at Georgia Tech –

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a control-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions.  We define an analogous balanced truncation procedure for the Bayesian inference setting based on the trade off between prior uncertainty and data information. The resulting reduced model inherits desirable theoretical properties for both the control and inference settings: numerical demonstrations on two benchmark problems show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.

Arnold diffusion in Hamiltonian systems with small dissipation

Series
CDSNS Colloquium
Time
Monday, October 2, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 005
Speaker
Marian Gidea – Yeshiva University –

We consider a mechanical system consisting of a rotator and a pendulum, subject to a small, conformally symplectic perturbation. The resulting system has energy dissipation. We provide explicit conditions on the dissipation parameter, so that the resulting system exhibits Arnold diffusion. More precisely, we show that there are diffusing orbits along which the energy of the rotator grows by an amount independent of the smallness parameter. The fact that Arnold diffusion may play a role  in  systems with small dissipation was conjectured by Chirikov. Our system can be viewed as a simplified  model for an energy harvesting device, in which context the energy growth translates into generation of electricity.
Joint work with S.W. Akingbade and T-M. Seara.

The L^p metrics on Teichmüller space by Hannah Hoganson

Series
Geometry Topology Seminar
Time
Monday, October 2, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Hannah Hoganson – UMD

We will start by introducing the Teichmüller space of a surface, which parametrizes the possible conformal structures it supports. By defining this space analytically, we can equip it with the Lp metrics, of which the Teichmüller and Weil-Petersson metrics are special cases. We will discuss the incompleteness of the Lp metrics on Teichmüller space and what we know about their completions.

A stronger Torelli theorem for graphs

Series
Algebra Seminar
Time
Monday, October 2, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sarah Griffith – Brown University –

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.

Recent research trends have explored curious analogies between the theory of graphs and Riemann surfaces. To each graph we can associate a real metric torus, known as its Jacobian. It was previously known that isomorphisms of graph Jacobians yield isomorphisms of the associated graphic matroids, partially mirroring a classical algebraic geometry result known as the Torelli theorem. However, the result on graphs is not as strong as a direct analogue of the Riemann surface result would be, nor does it use as much data. I will discuss how the graph Torelli theorem can be refined to incorporate additional data as with Riemann surfaces, in which case it produces isomorphisms of graphs. If time permits, I will describe further recent work in this direction.

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 –
Knots and tangles play a fundamental role in the dynamics of biological and physical systems, from DNA and root networks to surgical sutures and shoelaces. Despite having been studied for centuries, the subtle interplay between topology and mechanics in tangled elastic filaments remains poorly understood. Here we investigate the dynamical rules governing the behavior of knotted and tangled matter. We first study the human-designed knots used to tie ropes together. By developing an analogy with long-range ferromagnetic spin systems, we identify simple topological counting rules to predict the relative mechanical stability of commonly used climbing and sailing knots. Secondly, we examine the complex tangling dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using ultrasound imaging datasets, we construct a minimal model that explains how the kinematics of individual active filaments determines their emergent collective topological dynamics. By identifying generic dynamical principles of topological transformations, our results can provide guidance for designing classes of self-adaptive topological metamaterials.

 

 

 

 

Maximising copies of H in K_{r+1}-free graphs

Series
Combinatorics Seminar
Time
Friday, September 29, 2023 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 308
Speaker
Natasha Morrison – University of Victoria –

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.

State Space Variance Ratio (SSVR) Test for Sequential Change Point Detection

Series
Mathematical Biology Seminar
Time
Friday, September 29, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vanja Dukic – University of Colorado - Boulder

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

Neural-ODE for PDE Solution Operators

Series
SIAM Student Seminar
Time
Friday, September 29, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan Gaby – Georgia State University –

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.

Limit results for distributed estimation of invariant subspaces in multiple networks inference and PCA

Series
Stochastics Seminar
Time
Thursday, September 28, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Minh Tang – NC State

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.

Super-Teichmueller spaces, coordinates, and applications

Series
Job Candidate Talk
Time
Thursday, September 28, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anton Zeitlin – Louisiana State University

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.

 

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