Mathematics, 1934-2024

The first part of this thesis is a short introduction to scattering amplitudes, aimed at algebraic geometers with not very much physics background. We try to emphasize simple examples and give a rough survey of topics: Feynman diagrams, tree and loop level, on-shell diagrams, the spinor-helicity formalism, BCFW recursion, the Parke-Taylor formula, and maximum helicity violating amplitudes. Next, we explain Tevelev’s attempt to apply this circle of ideas to curves. We try to give new examples and flesh out detail that is left implicit in his original paper. We work out a few amplitude forms in genus 2 and 3 and graph their probability densities. We end by discussing some possible directions for new work.

Extending the work of Sirignano and Spiliopoulos (2020), we use mean field theory to study two-layer neural networks with $l2$-regularization trained under single-sample online stochastic gradient descent. We prove that in the asymptotic regime of both infinite training steps and infinite hidden layer width, such a neural network weakly converges to a deterministic and unique solution that satisfies a partial differential equation of the gradient flow form seen elsewhere in optimal transport and physics. Moreover, we show that the parameters of said neural network, despite being interdependent throughout training, asymptotically become independent. These results are only subject to loose moment bounds at initialization. Our proofs utilize a probabilistic approach on the network's training evolution instead of studying the geometry of the loss surface. We also provide numerical simulation results consistent with our theoretical guarantees.

In this thesis, we present an adelic construction of some non-split extension of rational mixed Hodge structures arising from the cohomology of modular curves, which witnesses the order 1 vanishing of Hecke L-functions of imaginary quadratic fields at s=0, as predicted by Beilinson's Conjecture. This recasts the construction of Skinner in the adelic language. The main calculation is based on an adelic description of the cohomology of modular curves in terms of the (s,K)-cohomology of automorphic forms, an adelic description of Hecke L-functions in terms of Tate's Zeta integrals, and an adelic description of Eisenstein series in terms of Godements's flat sections.

Procedural behavior is an important part of decision making under risk. Kahneman and Tversky (1979) acknowledge this in their outline of Prospect Theory by incorporating a preliminary editing phase. This paper take this process to develop a model where procedures are central to the decision-making process, wherein complex objects are iteratively simplified until they can be directly evaluated. We impose two restrictions on the space of possible procedure: that procedures are actions take upon subsets of lotteries and that these actions serve to simplify the lottery. From this arises a natural form of procedures, similar to those analyzed by Hu (2025). We provide several additional characterization of various models of non-expected utility under this framework.

A slice knot is a knot that bounds a disk in $B4$. A ribbon knot also bounds an embedded disk in $B4$, but with certain irregularities through self intersections. Whether or not all ribbon knots are slice knots is a well known fact among knot theorists. However, the converse is still an open question, and such is called the slice ribbon conjecture. There have been certain cases where this has been proven to be true for certain subsets of knots, such as 2-bridge knots \cite{Lisca} and 3-stranded pretzel knots \cite{Greene/Jabuka}, but not all. In gathering more information about types of knots that are ribbon, Brendan Owens and Frank Swenton developed an algorithm to know whether alternating knots up to 21 crossings bounds a ribbon \cite{Owens/Swenton}. The thesis is divided into three parts 1.) providing an overview of key knot theory and topology concepts related to the research paper at hand 2.) providing a summary and contextualization of Owens' and Swenton's research paper \textit{An Algorithm to Find Ribbon Disks for Alternating Knots} with additional details that would be hopefully accessible to other math undergraduate students 3.) a series of proposals of what could be said about six 16 and one 17 crossing knots whose current status is unknown based on the Algorithm and additional methods. Furthermore, we also inspect some of simplest escapee knots to garner ideas of what could potentially be similar among them. This is particularly because it could give further information on the notion of bifacotorizability that would potentially include escapees.

Score-Based Generative Models (SGMs) generate data by evolving samples under a Stochastic Differential Equation (SDE) that may include an auxiliary momentum variable. In this thesis, we investigate the probability density functions describing the evolution of such systems with momentum when the governing SDE slowly changes. For a simple moving Gaussian potential, we show the existence of an intermediate Pseudo-Adiabatic Regime (PAR) in which the momentum variables equilibrate while the position variables continue to evolve. In this regime, we show that the forward and reverse-time SDEs are the same, potentially simplifying the generative process. Using perturbative analysis and numerical experiments, we characterize the conditions under which the PAR emerges, demonstrating that a large damping-to-mass ratio suffices. Our results offer new perspectives on SGMs, and motivate further research into a more efficient generative process for SGMs.

Both synchronous and asynchronous phase analysis patterns have been identified in mice at different cell-specific layers of the iso-cortex. While a strong pipeline exists to understand these patterns on a functional level, there is little information about how this same brain activity might be seen structurally. On a biological level, there should be the same wave patterns shown through structural and functional methods. This paper attempts to confirm this assertion by utilizing the Wilson-Cowan equations to better understand the correlation between functional and structural data. These equations model excitatory and inhibitory neurons in the cortical layer through eigenvalues and eigenvectors computed from a mouse brain correlation matrix. By manipulating the parameters to change whether we have a stable or unstable balance of excitatory and inhibitory neurons, we can see if the same regional patterns found through functional data are present in the coupled equations.

Computer vision models have been shown to exhibit and amplify biases across a wide array of datasets and tasks. Existing methods for quantifying bias in classification models primarily focus on dataset distribution and model performance on subgroups, overlooking the internal workings of a model. We introduce the Attention-IoU (Attention Intersection over Union) metric and related scores, which use attention maps to reveal biases within a model's internal representations and identify image features potentially causing the biases. First, we validate Attention-IoU on the synthetic Waterbirds dataset, showing that the metric accurately measures model bias. We then investigate the CelebA dataset, finding that Attention-IoU uncovers correlations beyond accuracy disparities. Through an investigation of individual attributes through the protected attribute of Male, we examine the distinct ways biases are represented in CelebA. We furthermore explore contextual biases with Attention-IoU through the COCO dataset, highlighting the challenges in measuring contextual bias. Lastly, we analyze distribution shifts in iWildCam, revealing the impact of background environments in camera trap images on model performance. Altogether, Attention-IoU reveals aspects of biases beyond dataset labels and model accuracies, enabling us to gain deeper insights into the representations of bias within computer vision models and develop better debiasing methods and fairer models.

This thesis partially contains materials to be presented at the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) at Nashville, Tennessee, USA in June 2025 as "Attention IoU: Examining Biases in CelebA using Attention Maps", in collaboration with Tyler Zhu, Olga Russakovsky, and Vikram V. Ramaswamy. Our code and data are available at https://github.com/aaronserianni/attention-iou.

This thesis uses projective geometry over both the real numbers and finite fields to explore the structure of locally symmetric spaces, following the 1989 work of MacPherson and McConnell. We detail fundamental projective geometry concepts such as cross ratios, harmonic quadruples, and Desargues' theorem before extending these ideas to the finite field setting. The heart of the exposition centers on the construction of a cell complex $W\subset X$, is a deformation retract of a modular variety $\Gamma (p)\setminus X$ associated with the arithmetic group $\Gamma (p)$.

The Frobenius map is a known and well-studied homomorphism on commutative rings of finite characteristic. In this paper, we examine the splitting behavior of this homomorphism when restricted to certain finitely generated algebras. We utilize a criterion of Fedder to categorize several of these algebras based on the polynomials which we use to define them, developing heuristics for determining which polynomials define Frobenius-split rings. We conclude by examining elliptic curves specifically, using our heuristics to determine Frobenius-splitness and using this property to estimate the number of solutions to a given elliptic curve over a finite field.

We show that subdivisions of binary trees satisfy a localized version of the Erdős–Pósa property. Our first result is that if a graph G contains no two disjoint subdivisions of a given binary tree B, then there exists a subgraph H of G isomorphic to a subdivision of B and a set X⊆V(H) such that G- X contains no subgraph isomorphic to a subdivision of B, and the size of X is bounded by an exponential function of |V(B)|. We then generalize this result to settings where G does not contain k vertex-disjoint subgraphs each isomorphic to a subdivision of B. In this case, we demonstrate the existence of a set X whose size depends on both |V(B)| and k, so that G- X is B-minor-free.

This thesis explores two independent problems, both of which aim to describe graph and directed graph (digraph) properties via sets of forbidden obstructions. Taking inspiration from Kuratowski’s Theorem for planar graphs, the first chapter investigates structural characterizations of strongly connected digraphs whose underlying undirected graphs contain specific subdivisions or minors. In particular, we discuss strongly connected digraphs which are outerplanar, series-parallel, planar, and which contain subdivisions or minors of various wheel graphs and the triangular prism graph. In the second chapter, we expand upon a theorem of Thomassen on acyclic digraphs embedded in the closed disc by proving a structural result on acyclic digraphs embedded in the closed annulus.

Earth's surface topography exists in a long-term dynamic feedback system with climate and geologic processes, the implications of which extend from landslides to agriculture to carbon cycling. High-order models have been constructed to understand the intricacies of the landscape evolution system, but sacrifice mechanistic, process-level understanding of the characteristic feedbacks and nonlinearities. Here, we study a minimalist landscape evolution model (LEM) that is able to isolate and capture the core modes of topographic variability. We characterize the behavior of this LEM on circular domains under a range of conditions to improve both quantitative and qualitative understanding of soil diffusion, channel inception, and landscape self-similarity. Our results serve as a critical first step in quantifying the length scales over which transitions between different landscape behaviors occur, and highlight the need for further study of the effects of boundary conditions and parameter choice on modeled topography.

This paper will provide an overview on the Mermin-Wagner Style shift methodology used by Cipolloni, Peled, Schenker, and Shapiro in their proof of Anderson Localization for the Wegner W-orbital model at localization length << W^4. It provides insights into how this technique can be adapted towards other RBM systems including two alternative models. In doing so, the paper suggests a new shift methodology that can potentially show localization at stricter lengths for some generalized systems.

The neutral rate of interest, or R star, is the monetary policy rate at which economic growth will neither accelerate nor slow. As a theoretical concept, and not a directly observable or measurable number, we are only able to estimate it with models rather than measure it directly. We review econometric models (the Holston-Laubach-Williams Kalman filter, and the Del Negro et al. vector autoregression) in Chapter 2, macroeconomic theory models (the New York Fed DSGE, and Carvalho et al's aging demographics model) in Chapter 3, and a financial model (Rungcharoenkitkul and Winkler's ``hall of mirrors'') in Chapter 4. In Chapter 3 we also modify the demographics model to simulate a transient high-immigration period, which resulted in a positive but mostly negligible movement in R star. We also propose a Markov chain model in Chapter 4 to demonstrate that the zero lower bound alone can introduce a wedge between R star and the long-run average of short rates, which is often suggested as a way of measuring long run R star (at least as well as such an average can be captured by long Treasury rates). Finally, in Chapter 5 we analyze the link between fiscal policy and R star by plotting the quarterly changes in the US debt to GDP ratio against the quarterly change in the HLW model's estimate of R star. In this analysis, we found a weak or even negative relationship, which does not support the common assumption in economics that higher government deficits raise R star. Our ultimate conclusion, given the results of other models as well as our own work, is that R star is likely to remain low for the foreseeable future.

This thesis models the high-frequency trading arms race as a strategic competition through the lens of all-pay auctions, where traders incur costs to gain execution priority in a continuous-time financial market. Building on the conceptual foundation of Budish, Cramton, and Shim (2015), but departing from their model's general framework, we consider a setting in which firms compete to access prized arbitrage opportunities by investing in capital to execute trades before others. Using tools from auction theory and game theory, we characterize equilibrium behavior in both symmetric and asymmetric cases, proving conditions under which pure or mixed strategy equilibria exist. The analysis reveals the scale of the social surplus lost in the current financial market, and how small asymmetries can lead to disproportionate advantages for some firms. These results provide a theoretical basis for understanding the inefficiencies inherent in continuous-time markets and inform discussions of market design.

This thesis is an exploration into the concept of a uniform space using algebraic and category-theoretic techniques usually applied to topological spaces. The hope is that we can help future research to know more about the nature of uniform continuity from this perspective.

The point-free perspective of the lattice of open sets of a topological space is applied to give another lattice, of entourages of the uniform space. A binary operation of entourage addition or composition is fundamental to the definition of a uniform space, and equipping the lattice of entourages with this binary operation yields an algebraic structure known by several names, one of which being a Heyting Algebra, but which we denote by the name “uniform frame” for consistency in terminology.

In the first two chapters, we establish some definitions and concepts that will be of use throughout the paper. In the third chapter, we prove a precise condition for the metrizability of a uniform space, which is equivalent to a known condition, but given in a more order-theoretic and category-theoretic language. In the fourth chapter, the relation between a uniform space and its induced topology is defined and explored. We stack the uniform frame of covers on top of the frame of open sets and observe the presence of a canonical projection morphism from the former to the latter, and this morphism is denoted as a uniform setting. We conjecture that every uniform space can be reconstructed from its uniform setting.

Mathematicians long to observe and analyze all sorts of patterns across a range of different fields. This sense of regularity is illustrated by linguists throughout the syntax of an incredibly large amount of human languages, however, becomes blurred by non-configurational languages- that is languages who do not seem to abide by a rigid word order on the surface level. Although, this is not at all a pleasing result, therefore this paper exemplifies through the use of spectral graph theory and in particular the concept of algebraic connectivity that these so called non-configurational languages do indeed display preference for a particular word order that obeys existing patterns in the syntax of human languages. This paper explores the intersection of spectral graph theory and the syntax of non-configurational languages, with a particular focus on the Nyulnyulan language, Bardi. The goal of this paper is to observe the nuances in the syntax of Bardi that are revealed by the application of spectral graph theory. The intersection of spectral graph theory and syntax offers a novel perspective on understanding the structure and properties of syntactic representations. By modeling syntactic trees and dependency structures as graphs, we can apply spectral techniques to analyze relationships between constituents and their hierarchical organization. The eigenvalues and eigenvectors of adjacency and Laplacian matrices derived from these syntactic graphs can reveal insights into grammatical patterns, such as language similarity, syntactic complexity, and the connectivity of linguistic constructs. As such, spectral graph theory serves as a powerful tool for linguists, providing quantitative methods to investigate the intricacies of language structure.