Georgia Institute of Technology College of Sciences

In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the Stone-Weierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in L-infty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible least-squares polynomial approximants of $f$ which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function $f$ at these points, in order to recover the totality of the local minima of $f$ over the domain.

Elliptic surfaces are some of the most well-behaved families of smooth, simply-connected four-manifolds from the geometric and analytic perspective. Many of their smooth invariants are easily computable and they carry a fibration structure which makes it possible to modify them by various surgical operations. However, elliptic surfaces have large Euler characteristics which means even their simplest handle-decompositions appear to be quite complicated. In this seminar, we will learn how to draw several different handle diagrams of elliptic surfaces which show explicitly many of their nice properties. This will allow us to see many useful properties of elliptic surfaces combinatorially, and gives insight into the constructions of their exotic smooth structures. 

A computationally expensive physical simulation is a huge bottleneck to advance in science and technology. Fortunately, many data-driven approaches have emerged to accelerate those simulations, thanks to the recent advancements in machine learning (ML) and artificial intelligence. For example, a well-trained 2D convolutional deep neural network can predict the solution of the complex Richtmyer–Meshkov instability problem with a speed-up of 100,000x [1]. However, the traditional black-box ML models do not incorporate existing governing equations, which embed underlying physics, such as conservation of mass, momentum, and energy. Therefore, the black-box ML models often violate important physics laws, which greatly concern physicists, and require big data to compensate for the missing physics information. Additionally, it comes with other disadvantages, such as non-structure-preserving, computationally expensive training phase, non-interpretability, and vulnerability in extrapolation. To resolve these issues, we can bring physics into the data-driven framework. Physics can be incorporated into different stages of data-driven modeling, i.e., the sampling stage and model-building stage. Physics-informed greedy sampling procedure minimizes the number of required training data for a target accuracy [2]. Physics-guided data-driven model better preserves the physical structure and is more robust in extrapolation than traditional black-box ML models. Numerical results, e.g., hydrodynamics [3,4], particle transport [5], plasma physics, and 3D printing, will be shown to demonstrate the performance of the data-driven approaches. The benefits of the data-driven approaches will also be illustrated in multi-query decision-making applications, such as design optimization [6,7].

Reference
[1] Jekel, Charles F., Dane M. Sterbentz, Sylvie Aubry, Youngsoo Choi, Daniel A. White, and Jonathan L. Belof. “Using Conservation Laws to Infer Deep Learning Model Accuracy of Richtmyer-meshkov Instabilities.” arXiv preprint arXiv:2208.11477 (2022).
[2] He, Xiaolong, Youngsoo Choi, William D. Fries, Jon Belof, and Jiun-Shyan Chen. “gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification.” arXiv preprint arXiv:2204.12005 (2022).
[3] Copeland, Dylan Matthew, Siu Wun Cheung, Kevin Huynh, and Youngsoo Choi. “Reduced order models for Lagrangian hydrodynamics.” Computer Methods in Applied Mechanics and Engineering 388 (2022): 114259.
[4] Kim, Youngkyu, Youngsoo Choi, David Widemann, and Tarek Zohdi. “A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder.” Journal of Computational Physics 451 (2022): 110841.
[5] Choi, Youngsoo, Peter Brown, William Arrighi, Robert Anderson, and Kevin Huynh. “Space–time reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems.” Journal of Computational Physics 424 (2021): 109845.
[6] McBane, Sean, and Youngsoo Choi. “Component-wise reduced order model lattice-type structure design.” Computer methods in applied mechanics and engineering 381 (2021): 113813.
[7] Choi, Youngsoo, Gabriele Boncoraglio, Spenser Anderson, David Amsallem, and Charbel Farhat. “Gradient-based constrained optimization using a database of linear reduced-order models.” Journal of Computational Physics 423 (2020): 109787.

Fintushel and Stern’s knot surgery constructions has been a central source of exotic 4-manifolds since its introduction in 1997. In the simply connected setting, it is known that there are also embedded corks in knot-surgered manifolds whose twists undo the knot surgery. This has been known abstractly since the construction was first given, but the explicit corks and embeddings have remained elusive. We will give an algorithmic process for transforming a generic Kirby diagram of a simply-connected knot surgered 4-manifold into one which contains an explicit cork whose twist undoes the surgery: answering the question. Along the way we will discuss $S^2\times S^2$-stable diffeomorphisms of knot-surgered 4-manifolds, and their relationship to the existence of corks.

Many data in real-world applications are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. I will talk about machine learning tasks like regression and classification, as well as PDE simulations. We consider deep learning as a tool to solve these problems. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. Our results demonstrate that deep neural networks can utilize low-dimensional geometric structures of data in machine learning and PDE simulations.

Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4. Morse theory has been adapted to study complex manifolds, and even algebraic varieties over more general fields, but the underlying principles remain the same. In this talk, we will define the basic notions of Morse theory and describe some of the fundamental results. Then we will define Morse homology and discuss some important corollaries and applications. 

We study the Schur-Siegel-Smyth trace problem. We introduce a new linear programming problem that inclues Smyths' constraints, and we give an exact solution to it. This improves the best known lower bound on the Siegel trace problem which is based on Smyths' method. In a special case, we recover Siegel's original upper bound.  Our method unifies Siegel's and Smyth's work under the same framework. This is joint work with Bryce Orloski.

We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell_{p}$ norm. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p\ge 2$, there exists a critical inverse temperature $\beta_{c}(p)$ such that for $\beta<\beta_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. On the other hand, for $\beta>\beta_{c}(p)$, the magnetization is not concentrated at zero similar to the classical case. We further generalize the model for general symmetric spin distributions and prove similar phase transition. In this talk, we discuss a brief overview of classical Curie-Weiss model, a generalized Hubbard-Stratonovich transforms, and how we apply the transform to Curie-Weiss model under $\ell^p$ constraint. This is based on joint work with Partha Dey.

Atlanta Combinatorics Colloquium Hosted by Georgia Tech

A k-block in a graph is a set of k vertices every two of which are joined by k vertex disjoint paths. By a result of Weissauer, graphs with no k-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions to bounded tree width, a lot can be said about graphs with no k-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a k-block in a graph. We will discuss this phenomenon and its consequences on the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.

In this talk I will develop a parallel between the classical field theory of electromagnetism and geometric mechanics of animal locomotion. I will illustrate this parallel using some informative examples from the two disciplines. In the realm of electromagnetism, we will investigate the magnetic monopole, as classically as possible. In the realm of animal locomotion, we will investigate the aphorism that a cat dropped (from a safe height) upside-down always lands on her feet. It turns out that both of these phenomena are caused by the presence of non-trivial topology.

No prior knowledge of classical field theory will be assumed, and this talk may continue into a second session at a later date.

We develop nanopteron solutions for a coupled system of singularly perturbed ordinary differential equations.  To leading order, one equation governs the traveling wave profile for the Korteweg-de Vries (KdV) equation, while the other models a simple harmonic oscillator whose small mass is the problem’s natural small parameter.  A nanopteron solution consists of the superposition of an exponentially localized term and a small-amplitude periodic term.  We construct two families of nanopterons.  In the first, the periodic amplitude is fixed to be exponentially small but nonzero, and an auxiliary phase shift is introduced in the periodic term to meet a hidden solvability condition lurking within the problem.  In the second, the phase shift is fixed as a (more or less) arbitrary value, and now the periodic amplitude is selected to satisfy the solvability condition.  These constructions adapt different techniques due to Beale and Lombardi for related systems and is intended as the first step in a broader program uniting the flexible framework of Beale’s methods with the precision of Lombardi’s for applications to various problems in lattice dynamical systems.  As a more immediate application, we use the results for the model problem to solve a system of coupled KdV-KdV equations that models the propagation of certain surface water waves.