Test
- Series
- ACO Alumni Lecture
- Time
- Thursday, January 15, 2026 - 08:00 for
- Location
- Speaker
TBA
Please Note: zoom link: https://gatech.zoom.us/j/99430137245
We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell’s equations with perfectly matched‐layer boundary conditions for light–PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well‐defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson–Nernst–Planck ion‐channel model. Building on these results, we applied NNLCI to multi‐asset cash‐or‐nothing options under Black–Scholes. By correcting coarse‐ and fine‐mesh ADI solutions, NNLCI reduced RMSE by factors of 4–12 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far‐field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI’s first application to quantum algorithms by improving variational quantum‐algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ‐device noise. Although noisy VQA solutions deviate from classical finite‐difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high‐accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well‐defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data‐efficient tool for accelerating solvers in classical and quantum settings.
This thesis explores Turán-type extremal problems in graphs that exclude certain minors, focusing on the maximum number of $k$-cliques such graphs can contain. The first part of the thesis studies planar graphs, which forbid $K_5$ and $K_{3,3}$ as minors. We determine the maximum number of edges is in a planar graph that contains no cycle of length k, and establish a general upper bound for the number of edges in a planar graph avoiding $C_k$ for any $k\ge 11$.
The second part addresses the maximum number of $k$-cliques in $K_t$-minor-free graphs. We show essentially sharp bounds on the maximum possible number of cliques of order $k$ in a $K_t$-minor-free graph on $n$ vertices. More precisely, we determine a function $C(k, t)$ such that for each $k < t$ with $t - k \gg \log_2 t$, every $K_t$-minor-free graph on $n$ vertices has at most $n \cdot C(k, t)^{1 + o_t(1)}$ cliques of order $k$. We also show that this bound is sharp by constructing a $K_t$-minor-free graph on $n$ vertices with $C(k, t) n$ cliques of order $k$. This result answers a question of Wood and Fox–Wei asymptotically up to an $o_t(1)$ factor in the exponent, except in the extreme case where $k$ is very close to $t$.
The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Finally, we will discuss the statistical sides of group synchronization by quantifying the uncertainty of both MLE and spectral estimators.
Computational methods have long inspired conjecture and counterexamples in mathematics, and in recent years they appear more frequently in proofs of interesting mathematical theorems like the four color problem, Kepler's optimal sphere packing problem, and the proof of the Feigenbaum conjectures in nonlinear dynamics. In this talk I'll discuss recent work of R. Calleja, C. Garcia, O. Henot, J.P. Lessard and myself on choreographic solutions of the gravitational N-body problem. After reviewing some history and motivation, I'll explain the role of the digital computer in the (quite constructive) proofs of the theorems.
https://gatech.zoom.us/my/rkuske7?pwd=aHlLUFBFc2JndXlTelV1d3NlOEJBdz09
Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, natural language processing, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there is much less theoretical understanding about the modelling, approximation or generalization abilities of deep learning models with network architectures. An important family of structured deep neural networks is deep convolutional neural networks (CNNs) induced by convolutions. The convolutional architecture gives essential differences between deep CNNs and fully-connected neural networks, and the classical approximation theory for fully-connected networks developed around 30 years ago does not apply. This talk describes approximation and generalization analysis of deep CNNs and related structured deep neural networks.
Using the theory of total linear stability for Dynkin quivers and an interplay between the Bruhat order and the noncrossing partition lattice, we define a family of triangulations of the permutohedron indexed by Coxeter elements. Each triangulation is constructed to give an explicit homotopy between two complexes (the Salvetti complex and the Bessis--Brady--Watt complex) associated to two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). Our triangulations have several notable combinatorial properties. In addition, they refine similar Bruhat interval polytope decompositions of Knutson, Sanchez, and Sherman-Bennett. This is based on joint work with Melissa Sherman-Bennett and Nathan Williams.
The incipient infinite cluster was first proposed by physicists in the 1970s as a canonical example of a two-dimensional medium on which random walk is subdiffusive. It is the measure obtained in critical percolation by conditioning on the existence of an infinite cluster, which is a probability zero event. Kesten presented the first rigorous two-dimensional construction of this object as a weak limit of the one-arm event. In high dimensions, van der Hofstad and Jarai constructed the IIC as a weak limit of the two-point connection using the lace expansion. Our work presents a new high-dimensional construction which is "robust", establishing that the weak limit is independent of the choice of conditioning. The main tools used are Kesten's original two-dimensional construction combined with Kozma and Nachmias' regularity method. Our robustness allows for several applications, such as the explicit computation of the limiting distribution of the chemical distance, which forms the content of our upcoming project. This is joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2502.10882.
Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugated by a homeomorphism $h$. It was proved by de la Llave in 1992 that the conjugacy $h$ is automatically $C^{1+}$ if and only if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all periodic orbits. We prove that if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$, then $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a a $C^{1+\alpha}$ diffeomorphism $\overline{h}_N$ that is exponentially close to $h$ in the $C^0$ norm, and such that $f$ and $f_N:=\overline{h}_N^{-1}\circ g\circ \overline{h}_N$ is exponentially close to $f$ in the $C^1$ norm.
Zoom link -
https://gatech.zoom.us/j/5506889191?pwd=jIjsRmRrKjUWYANogxZ2Jp1SYdaejU.1
Meeting ID: 550 688 9191
Passcode: 604975