Seminars and Colloquia by Series

Embeddings of lens spaces and rational homology balls in complex projective space

Series
Geometry Topology Working Seminar
Time
Friday, September 23, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva RoyGeorgia Tech

Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. In this talk, we will consider the background necessary for an approach to this problem. Specifically, we will survey some essential notions and terminology related to low-dimensional contact and symplectic topology. These will involve Dehn surgery, tightness, overtwistedness, concave and convex symplectic fillings, and open book decompositions. We will also look at some results about these and mention some research trends.

Determinant Maximization via Matroid Intersection Algorithms

Series
ACO Student Seminar
Time
Friday, September 23, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Aditi Laddha

Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics, convex geometry, fair allocations, combinatorics, spectral graph theory, network design, and random processes. In an instance of a determinant maximization problem, we are given a collection of vectors $U = {v_1, \ldots, v_n}$ in $d$ dimensions, and a goal is to pick a subset $S$ of given vectors to maximize the determinant of the matrix $\sum_{i \in S} v_i v_i^T$. Often, the set $S$ of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint ($|S| \leq k$) or matroid constraint ($S$ is a basis of a matroid defined on the vectors). In this talk, we give a polynomial-time deterministic algorithm that returns an $r^{O(r)}$-approximation for any matroid of rank $r \leq d$. Our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improves any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the determinant function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.

 

This talk is based on joint work with Adam Brown, Madhusudhan Pittu, Mohit Singh, and Prasad Tetali.

Efficient parameterization of invariant manifolds using deep neural networks

Series
Time
Friday, September 23, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Shane KepleyVU

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

Spectral methods are the gold standard for parameterizing manifolds of solutions for ODEs because of their high precision and amenability to computer assisted proofs. However, these methods suffer from several drawbacks. In particular, the parameterizations are costly to compute and time-stepping is far more complicated than other methods. In this talk we demonstrate how computing these parameterizations and accurately time-stepping can be reduced to a related manifold learning problem. The latter problem is solved by training a deep neural network to interpolate charts for a low dimensional manifold embedded in a high dimensional Euclidean space. This training is highly parallelizable and need only be performed once. Once the neural network is trained, it is capable of parameterizing invariant manifolds for the ODE and time-stepping with remarkable efficiency and precision.

Sparse Quadratic Programs via Polynomial Roots

Series
Algebra Student Seminar
Time
Friday, September 23, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin ShuGeorgia Institute of Technology

We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.

Formation of small scales in passive scalar advection

Series
Math Physics Seminar
Time
Thursday, September 22, 2022 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex BlumenthalSchool of Mathematics

I will describe my recent joint work with Jacob Bedrossian and Sam Punshon-Smith on the formation of small scales in passively-advected scalars being mixed by a fluid evolving by the Navier-Stokes equation. Our main result is a confirmation of Batchelor's law, a power-law for the spectral density of a passively advected scalar in the so-called Batchelor regime of infinite Schmidt number. Along the way I will describe how this small-scale formation is intimately connected with dynamical questions, such as the connection between shear-straining in the fluid and sensitive dependence on initial conditions (Lyapunov exponents). Time-permitting I will describe some work-in-progress as well as interesting open problems in the area.

BEAUTY Powered BEAST

Series
Stochastics Seminar
Time
Thursday, September 22, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Kai ZhangUNC Chapel Hill

Link to the online seminar: https://gatech.zoom.us/j/94538442915

We study nonparametric dependence detection with the proposed binary expansion approximation of uniformity (BEAUTY) approach, which generalizes the celebrated Euler's formula, and approximates the characteristic function of any copula with a linear combination of expectations of binary interactions from marginal binary expansions. This novel theory enables a unification of many important tests through approximations from some quadratic forms of symmetry statistics, where the deterministic weight matrix characterizes the power properties of each test. To achieve a robust power, we study test statistics with data-adaptive weights, referred to as the binary expansion adaptive symmetry test (BEAST). By utilizing the properties of the binary expansion filtration, we show that the Neyman-Pearson test of uniformity can be approximated by an oracle weighted sum of symmetry statistics. The BEAST with this oracle provides a benchmark of feasible power against any alternative by leading all existing tests with a substantial margin. To approach this oracle power, we develop the BEAST through a regularized resampling approximation of the oracle test. The BEAST improves the empirical power of many existing tests against a wide spectrum of common alternatives and provides clear interpretation of the form of dependency when significant. This is joint work with Zhigen Zhao and Wen Zhou.

Recurrent solutions and dynamics of turbulent flows

Series
Colloquia
Time
Thursday, September 22, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Predrag CvitanovićSchool of Physics, Georgia Tech

In the world of moderate, everyday turbulence of fluids flowing across planes and down pipes, a quiet revolution is taking place. Applied mathematicians can today compute 'exact coherent structures', i.e. numerically precise 3D, fully nonlinear Navier-Stokes solutions: unstable equilibria, traveling waves, and (relative) periodic orbits. Experiments carried out at Georgia Tech today yield measurements as detailed as the numerical simulations; our experimentalists measure 'exact coherent structures' and trace out their unstable manifolds. What emerges is a dynamical systems theory of low-Reynolds turbulence as a walk among sets of weakly unstable invariant solutions.

 

We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the talk is aimed at anyone who had ever wondered why - if no cloud is ever seen twice - we know a cloud when we see one? And how do we turn that into mathematics?

Affine spheres over Polygons, Extremal length and a new classical minimal surface: a problem I can do and two I cannot

Series
Analysis Seminar
Time
Wednesday, September 21, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael WolfGeorgia Tech

In this introductory talk, we describe an older result (with David Dumas) that relates hyperbolic affine spheres over polygons to polynomial Pick differentials in the plane. All the definitions will be developed.  In the last few minutes, I will quickly introduce two analytic problems in other directions that I struggle with.

New lift matroids for gain graphs

Series
Graph Theory Seminar
Time
Tuesday, September 20, 2022 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zach WalshGeorgia Tech

Given a graph G with edges labeled by a group, a construction of Zaslavsky gives a rank-1 lift of the graphic matroid M(G) that respects the group-labeling. For which finite groups can we construct a rank-t lift of M(G) with t > 1 that respects the group-labeling? We show that this is possible if and only if the group is the additive subgroup of a non-prime finite field. We assume no knowledge of matroid theory.

Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations

Series
PDE Seminar
Time
Tuesday, September 20, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lukas WesselsGeorgia Tech and Technische Universität Berlin

In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs. 

As another application of our characterization of the second order adjoint state, we derive additional necessary optimality conditions in terms of the value function. These results generalize a classical relationship between the adjoint states and the derivatives of the value function to the case of viscosity differentials.

The last part of the talk is devoted to sufficient optimality conditions. We show how the necessary conditions lead us directly to a non-smooth version of the classical verification theorem in the framework of viscosity solutions.

This talk is based on joint work with Wilhelm Stannat:  W. Stannat, L. Wessels, Peng's maximum principle for stochastic partial differential equations, SIAM J. Control Optim., 59 (2021), pp. 3552–3573 and W. Stannat, L. Wessels, Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations, https://arxiv.org/abs/2112.09639, 2022.

Pages