Georgia Institute of Technology College of Sciences

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.

 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}.

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.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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.

Abstract: In the field of structural dynamics, engineers heavily rely on high-fidelity models of the structure at hand to predict its dynamic response and identify potential threats to its integrity.

The structure under investigation, be it an aircraft wing or a MEMS device, is typically discretised with finite elements, giving rise to a very large system of nonlinear ODEs. Due to the high dimensionality, the solution of such systems is very expensive in terms of computational time. For this reason, a large amount of literature in this field is devoted to the development of reduced order models of much lower dimensionality, able to accurately reproduce the original system’s dynamics. Not only the lower dimensionality increases the computational speed, but also provides engineers with interpretable and manageable models of complex systems, which can be easily coupled with data and uncertainty quantification, and whose parameter space can be easily explored. Slow invariant manifolds prove to be the perfect candidate for dimensionality reduction, however their computation for large scale systems has only been proposed in recent years (see Gonzalez et al. (2019), Haller et al. (2020), AV et al. (2019)).

In this talk, the Direct Parametrisation of Invariant Manifolds method (DPIM) will be presented. The theoretical basis of the method is provided by the results of Cabré, Fontich and de la Llave and its algorithmic implementation relies on the parametrisation method for invariant manifolds proposed by Haro et al.. The idea is to parametrise the invariant manifold around a fixed point through a power series expansion which can be solved recursively for each monomial in the reduced coordinates. The main limitation of the original algorithm is the necessity to operate in diagonal representation, which is unfeasible for large finite element systems as it would require the computation of the whole eigenspectrum. The main novelty of the proposed method lies in the expression of the normal homological equation directly in physical coordinates, which is the key aspect that permits its application to large scale systems.

The talk will focus on problems in structural dynamics in both autonomous and nonautonomous settings. The accuracy of the reduction will be shown on several examples, covering phenomena like internal resonances and parametric resonances. Finally, the current limitations and future developments of the method will be discussed.