Seminars and Colloquia by Series

Applications of Neural Networks with Locally Converging Inputs (NNLCI) for Classical and Quantum PDE Solvers

Series
Dissertation Defense
Time
Monday, July 7, 2025 - 11:00 for 2 hours
Location
Skiles 006
Speaker
Harris Cobb

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.

Counting cliques in graphs with excluded minors

Series
Dissertation Defense
Time
Tuesday, July 1, 2025 - 10:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Ruilin ShiGeorgia Institute of Technology

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 $t$.

 

Applications of immersed curves to the study of (1,1)-satellites

Series
Dissertation Defense
Time
Friday, April 4, 2025 - 09:00 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Weizhe ShenGeorgia Tech

This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift). By working with explicitly constructed (1,1)-diagrams and their planar analogs, we derive rank inequalities for knot Floer homology and develop a geometric algorithm for computing torsion orders. The latter, based on a depth-search procedure, translates intricate algebraic operations into tangible geometric moves on planar (1,1)-pairings, further yielding results on unknotting numbers and fusion numbers.

Geometric representation of multi-dimensional data and its applications

Series
Dissertation Defense
Time
Thursday, April 3, 2025 - 08:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 114
Speaker
Ho LawGeorgia Institute of Technology

This thesis presents several contributions to the fields of image and geometry processing. In 2D image processing, we propose a method that not only computes the relative depth of objects in bitmap format but also inpaints occluded regions using a PDE-based model and vector representation. Our approach demonstrates both qualitative and quantitative advantages over the state-of-the-art depth-aware bitmap-to-vector conversion models.

In the area of 3D point cloud processing, we introduce a method for generating a robust normal vector field that preserves first order discontinuity while being resistant to noise, supported by a degree of theoretical guarantee. This technique has potential applications in solving PDEs on point clouds, detecting sharp features, and reconstructing surfaces from incomplete and noisy data.

Additionally, we present a dedicated work on surface reconstruction from point cloud data. While many existing models can reconstruct implicit surfaces and some include denoising capabilities, a common drawback is the loss of sharp features: edges and corners are often smoothed out in the process. To address this limitation, we propose a method that not only denoises but also preserves sharp edges and corners during surface reconstruction from noisy data.

Automorphisms of the Smooth Fine Curve Graph

Series
Dissertation Defense
Time
Tuesday, March 25, 2025 - 13:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Katherine BoothGeorgia Tech

The smooth fine curve graph of a surface provides a combinatorial perspective to study the action of maps on smooth curves in the surface. It is natural to guess that the automorphism group of the smooth fine curve graph is isomorphic to the diffeomorphism group of the surface. But it has recently been shown that this is not the case. In this talk, I will give several more examples with increasingly wild behavior and give a characterization of this automorphism group for the particular case of continuously differentiable curves.

Committee:

  • Dan Margalit (advisor)
  • John Etnyre
  • Jen Hom
  • Igor Belegradek 
  • Michael Wolf

On probabilistic, planar, and descriptive graph coloring problems

Series
Dissertation Defense
Time
Tuesday, November 19, 2024 - 13:00 for 2 hours
Location
Price Gilbert 4222 and Zoom
Speaker
James AndersonGeorgia Tech

Please Note: Zoom: https://gatech.zoom.us/j/93071218913 Zoom Meeting ID: 930 7121 8913 Advisors: Dr. Anton Bernshteyn, Department of Mathematics, University of California, Los Angeles Dr. Rose McCarty, School of Computer Science and School of Mathematics, Georgia Institute of Technology Committee: Dr. Anton Bernshteyn, Department of Mathematics, University of California, Los Angeles Dr. Hemanshu Kaul, Department of Applied Mathematics, Illinois Institute of Technology Dr. Tom Kelly, School of Mathematics, Georgia Institute of Technology Dr. Rose McCarty, School of Computer Science and School of Mathematics, Georgia Institute of Technology Dr. Xingxing Yu, School of Mathematics, Georgia Institute of Technology

Graph coloring is a fundamental problem in graph theory in which the goal is to properly color a graph with the minimum number of colors in terms of some parameters (such as maximum degree). We explore this problem from the perspective of three different types of graphs: graphs with forbidden bipartite subgraphs; planar graphs; and Borel graphs that are line graphs. Each can be seen as graphs with a forbidden list of subgraphs; despite this similarity, the techniques used to study each are as varied as the results themselves.

We start with studying $F$-free graphs. We say a graph is $F$-free if it contains no subgraph isomorphic to $F$ (not necessarily induced). A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that %this conjecture holds for $F$ bipartite; moreover, if $G$ is $K_{t,t}$-free, then $\chi(G) \leq (t + o(1)) \Delta / \log\Delta$ as $\Delta \to \infty$. We improve this bound to $(1+o(1)) \Delta/\log \Delta$, making the constant factor independent of $t$. This matches the best known bound for several other class of graphs $F$, such as triangles, fans, and cycles, and lowering this bound further for nontrivial graphs is considered extremely challenging. We further extend our result to the correspondence coloring setting (also known as DP-coloring), introduced by Dvo\v{r}\'ak and Postle.

Next we study defective coloring of planar graphs. Defective coloring (also  known as relaxed or improper coloring) is a generalization of proper coloring defined as follows: for $d \in \mathbb{N}$, a coloring of a graph is $d$-defective if every vertex is colored the same as at most $d$ of its neighbors. We investigate defective coloring of planar graphs in the context of correspondence coloring. First we show there exist planar graphs that are not $3$-defective $3$-correspondable, strengthening a recent result of Cho, Choi, Kim, Park, Shan, and Zhu. Then we construct a planar graph that is $1$-defective $3$-correspondable but not $4$-correspondable, thereby extending a recent result of Ma, Xu, and Zhu from list coloring to correspondence coloring. Finally we show all outerplanar graphs are $3$-defective 2-correspondence colorable, with $3$ defects being best possible.

Finally, we study Borel graphs. We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the 9 finite graphs from the classical result of Beineke together with a 10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman--Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

This includes work coauthored with Anton Bernshteyn and Abhishek Dhawan.

Quantitative convergence analysis of dynamical processes in machine learning

Series
Dissertation Defense
Time
Tuesday, June 25, 2024 - 10:30 for 2 hours
Location
Skiles 006 and online
Speaker
Yuqing WangGeorgia Tech

Zoom link: https://gatech.zoom.us/j/6681416875?pwd=eEc2WEpxeUpCRUFiWXJUM2tPN1MvUT09

This talk focuses on analyzing the quantitative convergence of selected important machine learning processes, from a dynamical perspective, in order to understand and guide machine learning practices. More precisely, it consists of four parts: 1) I will illustrate the effect of large learning rates on optimization dynamics in a specific setup, which often correlates with improved generalization. 2) The theory from part 1 will be extended to a unified mechanism of several implicit biases in optimization, including edge of stability, balancing, and catapult. 3) I will concentrate on diffusion models, which is a concrete and important real-world application, and theoretically demonstrate how to choose its hyperparameters for good performance through the convergence analysis of the full generation process, including optimization and sampling. 4) The generalization performance of different architectures, namely deep residual networks (ResNets) and deep feedforward networks (FFNets), will be discussed.

On Extremal, Algorithmic, and Inferential Problems in Graph Theory

Series
Dissertation Defense
Time
Thursday, May 30, 2024 - 13:00 for 2 hours
Location
Skiles 005 and Online: https://gatech.zoom.us/j/6125656239
Speaker
Abhishek DhawanGeorgia Tech Math

In this dissertation we study a variety of graph-theoretic problems lying at the intersection of mathematics, computer science, and statistics. This work consists of three parts, all of which use probabilistic techniques. 

In Part 1, we consider structurally constrained graphs and hypergraphs. We examine a celebrated conjecture of Alon, Krivelevich, and Sudakov regarding vertex coloring. Our results provide improved bounds in all known cases for which the conjecture holds. We introduce a generalized notion of local sparsity and study the independence and chromatic numbers of graphs satisfying this property. We also consider multipartite hypergraphs, a natural extension of bipartite graphs. We show how certain probabilistic techniques for problems on bipartite graphs can be adapted to multipartite hypergraphs, and are therefore able to extend and generalize a number of results.

In Part 2, we investigate edge coloring from an algorithmic standpoint. We focus on multigraphs of bounded maximum degree, i.e., $\Delta(G) = O(1)$. Following the so-called augmenting subgraph approach, we design deterministic and randomized algorithms using a near-optimal number of colors in the sequential setting as well as in the LOCAL model of distributed computing. Additionally, we study list-edge-coloring for list assignments satisfying certain local constraints, and describe a polynomial-time algorithm to compute such a coloring.

Finally, in Part 3, we explore a number of statistical inference problems in random hypergraph models. Specifically, we consider the statistical-computational gap for finding large independent sets in sparse random hypergraphs, and the computational threshold for the detection of planted dense subhypergraphs (a generalization of the classical planted clique problem). We explore the power and limitations of low-degree polynomial algorithms, a powerful class of algorithms which includes the class of local algorithms as well as approximate message passing and power iteration.

Numerical Methods for Optimal Transport Problems

Series
Dissertation Defense
Time
Friday, April 12, 2024 - 13:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 268
Speaker
Daniyar OmarovSchool of Mathematics, Georgia Tech

I will present numerical methods for solving the optimal transport (OT) problems in three settings. Firstly, I will discuss discrete OT problems from the perspective of linear programming and assignment problems. Additionally, I will provide a solution for a discrete problem with an obstacle in the domain.

Next, I will consider and compare several different numerical methods to solve the classic continuous OT problem with the squared Euclidean cost function. I will compare two numerical methods used for the fluid dynamics formulation with a direct discretization of the Monge-Ampère PDE. Furthermore, I will introduce a new class of problems called separable, for which very accurate methods can be devised. 

Lastly, I propose a novel implementation of Newton's method for solving semi-discrete OT problems for cost functions that are a positive combination of $p$-norms, $1

Geometry, topology, and combinatorics of fine curve graph variants

Series
Dissertation Defense
Time
Friday, April 5, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Roberta ShapiroGeorgia Tech

The goal of this talk is to explore curve graphs, which are combinatorial tools that encode topological information about surfaces. We focus on variants of the fine curve graph of a surface. The fine curve graph has its vertices essential simple closed curves on the surface and its edges connect pairs of curves that are disjoint. We will mention a sampling of related theorems which were proven in collaboration with various coauthors and then prove several results regarding the finitary curve graph, which has as its vertices essential simple closed curves while its edges connect pairs of curves that intersect at finitely many points.

In this talk, we will prove that the finitary curve graph has diameter 2 (geometry), that the flag complex induced by the finitary curve graph is contractible (topology), and that the automorphism group of the finitary curve graph is naturally isomorphic to the homeomorphism group of the surface (combinatorics).

Work mentioned in the talk will be a subset of independent work and of collaborations with Katherine Booth, Ryan Dickmann, Dan Minahan, and Alex Nolte. The talk will be aimed at a non-expert audience.

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