Georgia Institute of Technology College of Sciences

For $x$ real, let $ \{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor $).  The lonely runner conjecture can be stated as follows: for any $n$ positive integers $ v_1 < v_2 < \dots < v_n $ there exists a real number $t$ such that $ 1/(n+1) \le \{ v_i t\} \le n/(n+1) $ for $ i = 1, \dots, n$.  In this paper we prove that if $ \epsilon >0 $ and $n$ is sufficiently large (relative to $\epsilon$) then such a $t$ exists for any collection of positive integers $ v_1 < v_2 < \dots < v_n$ such that $ v_n < (2-\epsilon)n$.  This is an approximate version of a natural next step for the study of the lonely runner conjecture suggested by Tao.  
    
The key ingredient in our proof is a result on coprime mappings.  Let $A$ and $B$ be sets of integers.  A bijection $ f:A \to B$ is a coprime mapping if $ a $ and $f(a)$ are coprime for every $ a \in A$. We show that if $A,B \subset [n]$ are intervals of length $2m$ where $ m = e^{ \Omega({(\log\log n)}^2)}$ then there exists a coprime mapping from $A$ to $B$. 

We study the effect of time-periodic forcing on the edge state of the semi-infinite Su–Schrieffer–Heeger (SSH) model, a 1D tight-binding model. Numerical simulations and an asymptotic expansion demonstrate that if the frequency of forcing is in resonance with the continuous spectrum of the unforced Hamiltonian, then on a time scale proportional to the inverse square of the forcing amplitude, the edge state decays in amplitude due to the radiation of its energy into the bulk. A proof is work in progress, and makes use of a new dispersive decay estimate for the time-evolution induced by the Hamiltonian. 

The Ozsvath-Szabo tau-invariant is a concordance invariant coming from knot Floer homology. The tools of bordered Heegaard Floer homology provide a way to study the knot Floer homology of satellite knots, and for many patterns have given formulas for the behavior of tau under satelliting. We give formulas for tau and epsilon of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing, and we do this without the use of bordered Heegaard Floer homology. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller-Lewark-Lobb. We show that all braided patterns are squeezed, and we give a tau formula for squeezed patterns as well. Also, towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.

This dissertation develops algorithms and theory for optimization, sampling, and generative modeling on manifolds, with Lie groups as a central object of study. Lie groups are manifolds with additional group structure; when endowed with a left-invariant Riemannian metric they become Riemannian manifolds, a setting that plays a central role throughout this work.

We first consider optimization on the Stiefel manifold $\mathrm{St}(n,d)$, the set of $n\times d$ matrices with orthonormal columns. By deriving a variational principle on this manifold, we construct the Momentum Stiefel Optimizer, a momentum-based algorithm that exactly preserves the orthogonality constraint at every iteration. The optimizer is applied to suitably-orthogonal attention in transformers and to optimal transport problems, achieving consistent improvements over existing methods.

We then establish quantitative convergence guarantees for momentum optimizers on Lie groups equipped with a left-invariant metric. Using the left-trivialization technique, which maps the curved dynamics to a flat Euclidean space for the momentum variable, we prove the first explicit convergence rates for both Heavy-Ball and Nesterov Accelerated methods on compact Lie groups, with rates that match Euclidean theory in terms of the smoothness and strong-convexity constants.

Next, we develop gauge-equivariant accelerated methods for optimization over the Grassmannian $\mathrm{Gr}(n,d)$, the set of $d$-dimensional subspaces of $\mathbb{R}^n$. Because each subspace has infinitely many orthonormal representatives related by an $\mathrm{O}(d)$ rotation, a naive lift of Stiefel algorithms to the Grassmannian is not gauge-equivariant. We introduce a gauge-fixing pipeline that converts any Stiefel optimizer into a gauge-equivariant Grassmann algorithm, yielding Grassmann Anderson Acceleration and Grassmann NAG, validated on density functional theory and low-rank matrix completion.

We then turn to sampling on Lie groups. By adding tractable noise to the left-trivialized momentum dynamics, we construct the first kinetic (momentum) Langevin Monte Carlo sampler on Lie groups with rigorous nonasymptotic convergence guarantees. The sampler preserves the group structure exactly at every step. Exponential convergence in $W_2$ distance is proved under only compactness of the Lie group and geodesic smoothness of the potential, without any convexity or isoperimetric assumption.

Finally, we address score-based generative modeling on general Riemannian manifolds. The standard denoising score matching framework requires a tractable forward-process transition kernel, which is unavailable on general manifolds because the heat kernel is intractable. We propose splitting diffusion, which lifts the dynamics to the tangent bundle and alternates closed-form stochastic momentum updates in the Euclidean tangent space with deterministic geodesic transport. The resulting transition kernel is closed-form, enabling denoising score matching training on general Riemannian manifolds requiring only exponential map as oracle.