Mathematics, 1934-2026

In this thesis, we study the asymptotic behavior of the spectra of adjacency matrices of random regular multigraphs sampled from the configuration model. The Alon-Boppana theorem gives an asymptotic lower bound of 2sqrt(d-1) on the second-largest eigenvalue of the adjacency matrix of a d-regular graph as the number of vertices N goes to infinity. Friedman's theorem asserts that random regular graphs sampled from a variety of different models almost achieve this bound. For both the unconditioned configuration model and the configuration model with conditioning on the absence of a forbidden subgraph, we apply the polynomial method developed in Chen et al. (2026) to derive quantitative versions of Friedman's theorem. In the process, we study subgraph counters, compute rational formulas for the spectral statistics of adjacency matrices, and bound the error of asymptotic expansions for these spectral statistics.

Convex preferences open up roommates mechanisms to desirable properties, such as stability. For roommate problems, convex preferences are equivalent to single-peaked preferences. Assumption of individual rationality converts roommate problems from one-sided to two-sided problems. However, stability and strategyproofness are incompatible incentives for roommate problems under assumption of individual rationality and convexity. We explore the limitations that individual rationality bring to convex roommate problems before developing an individually rational, efficient, and strategyproof mechanism for convex roommate problems.

Let X be a smooth closed four manifold. The Seiberg-Witten equations yield an important invariant of the diffeomorphism type of X. In many cases these invariants are closely related to the geometry of X. For example, when X has a metric of positive scalar curvature, they vanish. When X is Kahler, they can be interpreted holomorphically as a signed count of divisors. Here we discuss similar results for the case where X is a closed symplectic 4-manifold. In particular we prove an existence result for embedded symplectic surfaces in particular homology classes when associated line bundles have non-zero Seiberg-Witten invariants. Together with a non-vanishing result, these give strong constraints on the topology of X. We treat both the cases b2+ > 1 and b2+ = 1. Using these results we prove classification theorems for topologically simple symplectic 4-manifolds. In particular we give a detailed proof of the symplectic rigidity of CP2.

Duffie, Garleanu, and Pedersen (DGP) (2005) and Vayanos and Wang (2007) develop search-based models of over-the-counter (OTC) markets in which investors meet randomly and bargain bilaterally. However, neither model accounts for dealers who match buyers and sellers in real-world OTC markets. I modify the models of DGP and Vayanos and Wang by introducing dealers who extract a share of the gains from trade via the bid-ask spread, while holding matching processes and allocation fixed. I find that under this construction, dealers reduce investor welfare by the discounted value of spreads.

Mathematicians have conjectured that the simplexes are the polytopes that maximize the isotropic ratio among all convex bodies in Rn (the Strong Slicing Conjecture). This has been established only in R2, where the triangle is the unique maximizer. In this paper we use both theoretical and computational approaches to derive explicit formulas for the first and second derivatives of the isotropic ratio of a polytope in R2 in terms of its vertices, with the aim of providing a foundation for attacking the conjecture in higher dimensions.

In this thesis, we provide an accessible exposition of a recent paper [SZZ23] on lattice Yang--Mills theories over SO(N) or SU(N) at strong coupling. Viewing these lattice Yang-Mills theories as the invariant measures of the Markov semigroups associated to certain Langevin dynamics, we obtain a collection of functional inequalities (including Poincaré and log-Sobolev inequalities) via Bakry-Émery theory. Under a suitable choice of test function adapted to the specific application, these functional inequalities imply the existence of a mass gap as well as the factorization property for Wilson loop observables. We situate the results in their broader contexts, highlighting the connections between geometry and probability.

In the last century, several models using Capture-Recapture techniques have emerged to account for the unseen members of a population. These use the basic idea that individuals’ behaviors around a capture site can be predictive of other individuals’ behaviors. While this assumption applies well in cases where rigid control over environment can reduce heterogeneity, we do not know whether these techniques have a practical application on other kinds of data. Simultaneously, the number of senators in the Roman Republic from 81-45 BC is a mystery, with a wide range of feasible values. Using the ancient orator Cicero’s letters to his friend Atticus allows us to test Capture-Recapture techniques on capture occasions that have great amounts of heterogeneity.

Cryogenic electron microscopy (cryo-EM) produces large collections of noisy observations of biomolecules with structural heterogeneity. The central challenge is to estimate the underlying distribution of confor- mational states from these observations. In our approach, high-dimensional data are first mapped into a low-dimensional latent space, where conformational variability is more directly represented but the noise becomes heterogeneous and anisotropic. In this work, we study density estimation in this latent space using Gaussian mixture models (GMMs) as an approximation. We develop an expectation–maximization (EM) algorithm that incorporates per-observation noise covariances, allowing for probabilistic denoising when fit- ting the mixture model. We evaluate the algorithm’s effectiveness for both approximating a GMM as well as capturing the underlying conformational density in cryo-EM data. Using synthetic datasets with known ground truth, we assess the performance of the algorithm’s GMM-based estimates via quantitative error metrics. The results characterize the conditions under which GMMs provide accurate density estimates and highlight limitations using a GMM for density estimation in cryo-EM heterogeneity analysis.

Tempered perfect lattices are a generalization of Voronoi's perfect lattices that involve a lattice-sublattice pair. This paper begins by defining tempered perfect lattices and their equivalence to tempered perfect quadratic forms. Then, the paper describes and categorizes three-dimensional tempered perfect lattices.

We study the minimum dominating set MDS problem in the adjacency-list arrival streaming model (AL), in which vertices arrive one at a time, revealing their entire closed neighborhood at arrival. The MDS has been studied in the classical edge-arrival streaming model by Khanna and Konrad (2022), building on Assadi, Khanna, and Li (2016), who establish a sharp threshold at \tilde O(\sqrt{n})-approximation in semi-streaming space for any n-vertex graph G. However, the AL model is more informative and its space complexity has not yet been understood.

We show that the \tilde O(\sqrt{n}) threshold does not persist in AL and present a space-approximation tradeoff. Specifically, we prove the following: (1) For any space budget t \in [n], there is a one-pass AL algorithm using O(t \log \log n) space that outputs a dominating set of size O(n \log^2 n/t \cdot \OPT(G)). For t = n, there is a randomized one-pass semi-streaming AL algorithm which outputs a dominating set of size O(\log^2n \cdot \OPT(G)) together with a valid cover certificate. (2) Any exact one-pass AL algorithm for MDS requires \Omega(n) space.

We also develop the notion of AL-realizable reductions to prove lower bounds, and establish exact one-pass algorithms for structured graph families. Together, these results show that AL has algorithmic consequences different from other classical streaming models. The thesis presents initial results for AL space-approximation and directions for future work.

We give an exposition of the classification of minimal models of smooth complex projective surfaces, with an emphasis on the interplay between complex analytic and complex algebraic methods. We begin with an introduction to the intersection form for complex algebraic surfaces, and show that the topological intersection form on four-manifolds agrees with the algebraic intersection form on complex projective surfaces. We then discuss the structure of birational maps of smooth surfaces, where we show every birational map can be built out particularly simple birational maps called blowups. This allows us to make precise the notion of a minimal model of a surface S, a "simplest" representative of the birational equivalence class of S, and we work to characterize the structure of these minimal models and determine when a surface has a unique minimal model.

In the case of irrational ruled surfaces, we show that every minimal model is a geometrically ruled surface, and in the case of rational surfaces, we show that all minimal models are either projective space or a Hirzebruch surface. When S is not ruled or rational, we show that there is a unique minimal model in its birational equivalence class. Finally, we interpret these classification results through the lens of the Minimal Model Program and give a brief discussion on the birational classification problem in higher dimensions.

Reconstructing visual experiences from observed fMRI brain activity is a fundamental problem in computational neuroscience, though practical applications are limited by the high cost and time demands of data collection. This thesis investigates whether active learning can improve sample efficiency in fMRI-to-image decoding by strategically selecting which stimuli to present during scanning sessions, reducing the amount of per-subject data required to learn brain-to-embedding mappings for accurate image reconstruction. We evaluate a covariance-based active learning strategy for mapping voxel activity to CLIP image embeddings, benchmarking its performance against random sampling across data settings ranging from fully synthetic to fully empirical. Our findings show that in well-specified linear settings, active learning consistently outperforms random sampling, yielding the most pronounced gains near or above the interpolation threshold. However, in the fully empirical setting with real fMRI features and real CLIP responses, the gap is eliminated entirely as performance substantially degrades across both strategies. We attribute this breakdown to model misspecification: when the model fails to appropriately approximate the true underlying mapping between voxels and embeddings, uncertainty estimates become uninformative and active learning offers minimal advantage over random sampling. We conclude that the efficacy of active learning in neural decoding depends critically on model specification, and propose an extension into non-linear decoding frameworks to recover the theoretical gains of active learning for real-world stimulus selection.

This thesis studies an optimal trading problem for commodities under predictable returns, transaction costs, and stochastic volatility. Extending the work of Chan, Sircar, and Zimbidis in equities, I adapt the model to a commodity setting by introducing price dynamics via the two-factor Schwartz model. This modification makes the associated Hamilton-Jacobi-Bellman equation nonlinear, so the problem is no longer tractable in closed form.

I formulate the problem in a stochastic optimal control framework and reduce the HJB equation through a quadratic ansatz in the inventory variable. The resulting system is solved numerically to converge to the viscosity solution of the HJB using a finite-difference scheme with backward Euler time stepping and upwind discretization. I then use Monte Carlo simulation to evaluate the performance of the strategy. In the empirical implementation, I calibrate a reduced one-factor stochastic-volatility model using WTI crude oil market data and the OVX volatility index. The numerical results indicate that stochastic volatility materially affects the optimal trading strategy. Compared to a constant-volatility benchmark, the stochastic-volatility strategy delivers higher expected profit and loss, but also a wider distribution of outcomes with positive skewness.

The code and data may be accessed in GitHub: https://github.com/aidenkaufman/Senior_Thesis/tree/main.

In this article, we discuss a proof of the modularity theorem for semistable elliptic curves with a focus on the patching arguments, and on highlighting the overall structure of the original arguments of Wiles and Taylor-Wiles. We start by introducing necessary background concerning the deformation theory of Galois representations. In particular, we discuss explicit constructions of universal deformation rings following an argument of Faltings, and explain the relationship between tangent spaces of certain universal deformation rings and Selmer groups. From there, we review the construction of Galois representations arising from cuspidal Hecke eigenforms, as well as several structural results on Hecke Algebras. The last chapter is devoted to the proof of the modularity theorem for semistable elliptic curves; here we discuss residual modularity via the Langlands-Tunnell theorem, a variant of the Taylor-Wiles patching argument due to Diamond, and a numerical freeness criterion also due to Diamond. The variants of Taylor-Wiles patching and Wiles' numerical coincidence due to Diamond simplify the proof, in the sense that the deep "strong multiplicity 1 theorems" are no longer needed as input into the patching argument, but rather are obtained as a byproduct of the patching construction itself. Finally, we complete the proof with an explanation of the 3-5 switch.

We study a continuous-time repeated averaging process on n coordinates under a mean-zero initialization. At logarithmic time scales, the empirical distribution of appropriately normalized coordinates converges in probability to a Gaussian mixture, identifying the asymptotic behavior of the process beyond the L1 cutoff.

Donaldson’s introduction of gauge-theoretic methods into 4-manifold topology in the early 1980s revealed a sharp divergence between the topological and smooth classifications in dimension four. This thesis develops the analytic and topological foundations of four-dimensional gauge theory, culminating in a proof of Donaldson’s diagonalization theorem on definite intersection forms of smooth 4-manifolds. On the topological side, we introduce smooth 4-manifolds and their intersection forms, classify unimodular symmetric bilinear forms, and develop the homotopy classification of vector bundles via classifying spaces. On the analytic side, we review the differential geometry of connections and curvature, develop Chern–Weil theory, and introduce the Sobolev spaces and infinite-dimensional manifold framework needed for the analysis of the space of connections. We then study anti-self-dual connections and the topology of their moduli space, which provides the cobordism underlying Donaldson’s theorem. Throughout, we aim for a self-contained and accessible exposition of the machinery that four-dimensional gauge theory requires.

To a pointed matched circle, we introduce an associative, graded algebra related to bordered $HF-$ and the wrapped Fukaya category of genus $g$ surfaces. We prove existence and uniqueness of a (graded) $A\infty$ and weighted $A\infty$ deformation of the associative algebra. We prove these results using Hochschild cohomology and Koszul duality results but describe a concrete (combinatorial) realization of these algebras as well.