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ItemRandom subsequences problems : asymptotics, variance, and quantum statistics.(Georgia Institute of Technology, 2024-05-07) Deslandes, ClementThis work considers some random words combinatorial problems and their applications. The starting point of this endeavor is the following question : given two random words, ”how much do they have in common” ? Even if this question has emerged independently in various fields, including computer science, biology, linguistics, it remains mostly unsolved. Firstly, we study the asymptotic distribution of the length of the longest common and increasing subsequences. There we consider a totally ordered alphabet with an order, say 1,...,m, and the subsequences are simply made of a block of 1’s, followed by a block of 2’s, ... and so on (such a subsequence is increasing, but not strictly). Secondly, we deal with the problem of the variance of the LCS. By introducing a general framework going beyond this problem, partial results in this direction are presented, and various upper and lower variance bounds are revisited in diverse settings. Lastly, we consider the Longest Increasing Subsequences (LIS) of one random word, and the surprising connection with quantum statistics.
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ItemTopology, Geometry, and Combinatorics of Fine Curve Graphs(Georgia Institute of Technology, 2024-05-02) Shapiro, RobertaThe goal of this thesis 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, which has its vertices essential simple closed curves on the surface and whose edges connect pairs of curves that are disjoint. We will prove various geometric, topological, and combinatorial results about these curve graph variants, including hyperbolicity (or lack thereof), contractibility of induced flag complexes, automorphism groups, and admissible induced subgraphs.
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ItemToric and Tropical Geometry: Positivity and Completion(Georgia Institute of Technology, 2024-04-28) Cai, MayThis thesis is divided into the introductory chapter and then three main chapters. The the introductory first chapter is devoted to providing some preliminary background on polytopes, tropical geometry, and tropical linear algebra. The second chapter consists of a result on the probability completion of log-linear models. In particular, given partial entry-wise information about a point in some toric variety intersected with the set of probability vectors, we describe the number of completions of that partial point into an actual point in the semi-algebraic set, as well as necessary conditions to be a valid partial point. A preprint of the content of this chapter can be found online at https://arxiv.org/abs/2312.15154, and was joint work with Cecilie Olesen Recke and Thomas Yahl. The third chapter is concerned with the tropical variety of symmetric tropical rank at most 2. We discuss a refinement of the fan of the variety that gives a tree characterization of the variety, as in Markwig and Yu, and from this we deduce the shellability for the tropical variety as well as a condition to verify independence in the algebraic matroid of this variety. This chapter was joint work with Kisun Lee and Josephine Yu, and a preprint of the content can be found online at https://arxiv.org/abs/2404.08121. The fourth chapter focuses on applying notions of tropical positivity developed by Brandenburg, Loho, and Sinn to tropical symmetric determinantal varieties. It describes the ``positive'' and ``really positive'' parts of the tropical varieties of rank 2 symmetric matrices, using the tree characterization established in the third chapter, as well as of the symmetric tropical determinantal hypersurface. We also re-prove certain results about the really positive part of the tropical varieties of rank 2 usual tropical matrix and of the usual tropical determinantal hypersurface. This chapter was joint work with Abeer al Ahmadieh and Josephine Yu.
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ItemThe Second Rational Homology of the Torelli Group(Georgia Institute of Technology, 2024-04-27) Minahan, Daniel SullaWe show that the second rational homology of the Torelli group of a closed, orientable, connected surface is finite dimensional for all surfaces of sufficiently high genus.
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ItemAlgebraic Methods in Optimization(Georgia Institute of Technology, 2024-04-26) Shu, KevinThis thesis broadly concerns the usage of techniques from algebra, the study of higher order structures in mathematics, toward understanding difficult optimization problems. Of particular interest will be optimization problems related to systems of polynomial equations, algebraic invariants of topological spaces, and algebraic structures in convex optimization. We will discuss various concrete examples of these kinds of problems. Firstly, we will describe new constructions for a class of polynomials known as hyperbolic polynomials which have connections to convex optimization. Secondly, we will describe how we can use ideas from algebraic geometry, notably the study of Stanley-Reisner varieties to study sparse structures in semidefinite programming. This will lead to quantitative bounds on some approximations for sparse problems and concrete connections to sparse linear regression and sparse PCA. Thirdly, we will use methods from algebraic topology to show that certain optimization problems on nonconvex topological spaces can be turned into convex problems due to a phenomenon known as `hidden convexity'. Specifically, we give a sufficient condition for the image of a topological space under a continuous map to be convex, and give a number of examples of this phenomena with practical importance. This unifies and generalizes a number of existing results. Finally, we will describe how to use techniques inspired by the sum of squares method to find new variants of gradient descent which converge faster than typical gradient descent on smooth convex problems.
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ItemNumerical Methods for Optimal Transport Problems(Georgia Institute of Technology, 2024-04-17) Omarov, DaniyarIn my work, I present numerical methods for solving the optimal transport (OT) problems in three settings. Firstly, I discuss discrete OT problems from the perspective of linear programming and assignment problems. Additionally, I provide a solution for a discrete problem with an obstacle in the domain. Next, I consider and compare several different numerical methods to solve the classic continuous OT problem with the squared Euclidean cost function. I compare two numerical methods used for the fluid dynamics formulation with a direct discretization of the Monge-Ampère PDE. Furthermore, I 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
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ItemImproving Foundation Models(Georgia Institute of Technology, 2023-12-10) Komatsuzaki, AranFoundation models are the family of models (e.g. GPT-4, CLIP) that are trained on a massive dataset and perform various down-streaming tasks, usually with either zero- or few-shot learning, optionally after fine-tuning. This dissertation presents a wide range of important measures we have made to make foundation models more efficient, performant and versatile. In particular, we focus on three points of improvement: architecture, dataset and training. We first present our findings on how to optimally scale language models, which leads to significant performance improvement. We then present GPT-J, one of the earliest open-source large language models. We then show that the performance of ViT and T5, both Transformer-based foundation models, can be greatly improved for a given compute budget using Sparse Upcycling, which is to resume training a sparsely gated model made out of pretrained dense models. We also briefly discuss LAION datasets, massive open-source datasets with around one billion pairs of text and image that are used to train various state-of-the-art multimodal models, and ARB benchmark, a highly challenging benchmark to measure the state-of-the-art LLMs such as GPT-4. On the theoretical side, we prove that feedforward layers of a transformer cannot be compressed without information loss, which may explain the power of sparsely gated models such as mixture-of-experts.
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ItemMathematical Approaches to Identification problems -- Counting, RNA folding, and PDE identification(Georgia Institute of Technology, 2023-11-28) Tang Rajchel, MengyiMathematical algorithms have become an essential tool in uncovering hidden patterns and unraveling dynamic behaviors within complex datasets, aiding in gaining deeper insights and making informed choices in an era driven by data-driven decision-making. This thesis introduces several numerical algorithms addressing identification problems derived from mathematical models. These works place a specific emphasis on identifying and predicting structures and patterns within various types of datasets while also offering the capacity to forecast the behavior of future data. Our contributions include StemP, a novel algorithm using graph notations predicting RNA sequence secondary structures with simplicity and being deterministic, without a training process. Additionally, our work Counting Objects by Diffused Index(CODI) efficiently counts objects in digital images using a diffusion algorithm with an operator-splitting approach and the alternating direction minimization method inspired by color inpainting, delivering results within seconds.Furthermore, our works WeakIdent and FourierIdent focus on identifying differential equations in the physical and frequency domains, respectively. WeakIdent provides a general and robust framework for identifying differential equations, enhancing accuracy with proposed innovative mechanisms of narrow-fit and trimming. FourierIdent explores the benefits and challenges of frequency domain utilization in differential equation identification, presenting comprehensive experiments to demonstrate their benefits in robustness over state-of-the-art methods.
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ItemSet Images and Convexity Properties of Convolutions for Sum Sets and Difference Sets(Georgia Institute of Technology, 2023-07-31) Lee, Chi-NuoMany recent breakthroughs in additive combinatorics, such as results relating to Roth's theorem or inverse sum set theorems, utilize a combination of Fourier analytical and physical methods. Physical methods refer to results relating to the physical space, such as almost-periodicity results regarding convolutions. This thesis will focus on the properties of convolutions. Given an abelian group $G$ and sets $A \subseteq G,$ we study the properties of the convolution for sum sets and difference sets, $1_A*1_A$ and $1_A*1_{-A}.$ Given $\bm{x} \in G^n,$ we consider its corresponding \emph{set image} of the sum set, the image of $f(A):= 1_A*1_A(\bm{x}),$ and the similarly defined set image of the difference set. We break down the study of set images into two cases, when $\bm{x}$ is independent, and when $\bm{x}$ is an arithmetic progression. In both cases, we provide some convexity result for the set image of both the sum set and difference set. For the case of the arithmetic progression, we prove convexity by first showing a recurrence relation for the distribution of the convolution. Finally, we prove a smoothness property regarding 4-fold convolutions $1_A*1_A*1_A*1_A.$ We then construct different examples to better understand possible bounds for the smoothness property in the case of 2-fold convolutions $1_A*1_A.$
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ItemUnfoldings of Convex Polyhedra(Georgia Institute of Technology, 2023-07-28) Barvinok, NicholasA pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability criterion for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Durer's problem is false for pseudo-edge unfoldings. We then use the Maxwell-Cremona Correspondence to present evidence both for and against Durer's problem.