Mathematics

We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical“quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines. Building on Xue’s result, that this point is generically non-torsion, we introduce a height-theoretic notion of bigness for families of curves in the moduli space of genus four curves, and give a criterion, via dimensions of modular quotients, for when it holds. We then exhibit two concrete 3- and 4-parameter families (the bi-involutions locus and a CM example) in which the canonical point is provably big, from which we deduce the finiteness of low-height curves and non-torsion at transcendental moduli. Our methods combine adelic Arakelov intersection theory with a generic Betti-rank argument.

In this thesis, we settle an old question of William Allard's thesis from 1969, on boundary regularity of area-minimizing $m$-currents in arbitrary codimension, with higher boundary multiplicity and a convexity assumption. This setting generalizes Allard's celebrated theorem from 1975 on boundary regularity of area-minimizing currents to a higher boundary multiplicity minimizing setting. We develop a regularity theory that answers Allard's 1969 question by proving the $(m-3)$-rectifiability of the singular set. We show this regularity theory to be dimensionally sharp by constructing an example of an area-minimizing current with a boundary singularity for $m=3$. The starting point of the regularity theory is the uniqueness of the tangent cone at minimum density points, which we also establish in this thesis. The question of regularity for higher multiplicity boundaries, which we address in Allard's original form, was raised in a broader framework by Brian White in the Proceedings of the 1984 AMS Summer Institute, a famous collection of open problems in Geometric Measure Theory. Part of the regularity theory is in collaboration with Reinaldo Resende. Additionally, in collaboration with Camillo de Lellis, we generalize a theorem of Besicovitch from 1956 on classical rectifiability for curves in the two-dimensional plane to arbitrary dimension of the set and of the ambient space. As an application, we simplify the setup of the Naber-Valtorta technique, an extremely powerful and flexible tool for proving the rectifiability of singular sets in geometric analysis, which has been used for a variety of different problems. Together with Reinaldo Resende, we use this theorem to simplify part of the boundary regularity theory we develop in this thesis.

Lipshitz, Ozsváth, and Thurston extend the theory of bordered Heegaard Floer homology to compute CF minus. Like with the hat theory, their minus invariants provide a recipe to compute knot invariants associated to satellite knots. We combinatorially construct the weighted A infinity-modules associated to the (p, 1)-cable. The operations on these modules count certain classes of inductively constructed decorated planar graphs. This description of the weighted A infinity-modules provides a combinatorial proof of the A infinity structure relations for the modules. We further prove a uniqueness property for the modules we construct: any weighted extensions of the unweighted U = 0 modules have isomorphic associated type D modules.

In this thesis, we study the distribution of smallest values of polynomials over vectors of Euclideanlattices. The set consisting of these values, as we vary over the moduli space of unimodular lattices, is called the Bass Note Spectrum of the polynomial. The project of understanding the bass note spectrum of a homogeneous form dates back to Mahler’s work on star bodies in the 1940s. In the first part of this thesis, we study the bass note spectrum problem for homogeneous binary forms and show that contrary to conjectures of Mordell, the bass note spectrum of any binary form of degree at least 3 and non-vanishing discriminant is an interval. In the second part of this thesis, we study the connections between the bass note spectrum of the complex cubic norm form and one-parameter diagonal flows in SL3(Z)\SL3(R). As a consequence, we obtain new information about this spectrum, which we use to prove theorems regarding minimum representations of integer binary cubic forms of negative discriminant. Specifically, we are able to construct infinite families of pairwise GL2(Z)-inequivalent primitive integer binary cubic forms F, with smallest integer representations as large as possible: ≍ disc(F)^{1 /4} .

The asymptotics of relative characters for real Lie groups were studied forGan-Gross-Prasad pairs (G,H) by Nelson and Venkatesh. To do so, they use the orbit method to develop a theory of microlocal analysis of Lie group representations. This involves characterizing vectors by the property that they are approximate eigenvectors under the action of small subgroups. They successfully compute the asymptotics of relative characters whenever the conductor of the associated Rankin-Selberg L-function L(π⊠σ∨) lies in a stable locus, i.e. away from conductor dropping. In the non-archimedean case, although a systematic theory of coadjoint orbits remains unclear, the resulting non-archimedean microlocal analysis is much simpler. In this thesis, we focus on relative characters and present a method which overcomes the stability hypothesis and allows for significant conductor dropping. Namely, for (G,H) = (PGL2,GL1) over a non-archimedean field, this thesis computes theasymptotics of therelativecharacterand expresses themas the integral overan appropriategeometricorbit.

By slicing the Heegaard diagram for a given 3-manifold in a particular way, it is possible to constructA-infinity bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. In this dissertation, we construct Koszul dual A-infinity algebras A and B, called star algebras, for a particular star-shaped class of slice, called star diagrams. Using A-infinity bimodules over A and B, we then verify the Koszul duality relation, Theorem 1.0.2. The second chapter establishes an isomorphism between endomorphism algebras from the wrappedFukaya category of the star diagrams, and the star algebras constructed in the first chapter. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that they are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.

In this thesis, we develop a bordered version of Ng's Legendrian Rational Symplectic Field Theory. Given a Legendrian knot $\Lambda \subset R3$ and a vertical line dividing the front projection of $\Lambda$ into two parts, we construct a differential graded algebra associated to each half-knot. We then show that one may obtain the commutative algebra from Legendrian Symplectic Field Theory as a pushout of the two bordered algebras. This construction extends Sivek's bordered Chekanov-Eliashberg differential graded algebra by incorporating disks with multiple positive punctures into the differential.

We study Manin's conjecture for certain families of low-degree del Pezzo surfaces, i.e. asymptotics for the number of rational points of increasing height. In general, problems about rational points on del Pezzo surfaces are considered harder as the degree gets lower. In this thesis, we resolve Manin's conjecture for all Ch^atelet surfaces over Q, which are del Pezzo surfaces of degree four, and establish the first asymptotic count for a set of rational points on a del Pezzo surface of degree one. In Chapter 2, we prove the key analytic ingredient necessary in both arguments -- a bound on the sums of the absolute values of the Hecke eigenvalues of cuspidal automorphic forms of GL_2(A_Q) along polynomial values. We use pieces of the Sato-Tate distribution to prove a logarithmic power savings over the trivial bound in the applicable cases; additionally we classify, when the polynomial is solvable, exactly when a logarithmic power savings occurs using analysis about the base change of the cuspidal representation. In Chapter 3, we study Manin's conjecture for Ch^atelet surfaces -- the proper smooth models of the affine surfaces x^2+\Delta y^2 = f(z), where f(z) is a squarefree polynomial of degree 3 or 4. Previous works handle the case of \Delta=1 (and extend to any \Delta>0 such that Q(\sqrt{-\Delta}) has class number one). We resolve the conjecture by viewing the point-count through an automorphic lens and connecting it to the sums of Chapter 2. In Chapter 4, we study rational points on the singular del Pezzo surface of degree one defined by y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3,where 4A^3-27B^2\neq 0 and Q(u,v) is a positive-definite quadratic form. The main algebraic tool used is a parameterization of the integral points on quadratic twists of elliptic curves by binary quartic forms; this parameterization was first observed by Mordell and uses syzygys of binary quartic forms. We then reduce the problem to correlation sums of binary quadratic and binary quartic forms, which are analyzed using the central analytic ingredients of Chapters 2 and 3. As a result, we establish an asymptotic count for rational points that are integral with respect to the singularity on these del Pezzo surfaces.

We study several problems in arithmetic combinatorics, loosely grouped around the themes of "combinatorial number theory" and "higher-order Fourier analysis". The problems related to the former theme are the Lonely Runner Problem (from Diophantine approximation), questions about minimal additive complements, and a conjecture of Graham about rearrangements of subsets of F_p^\times. The problems related to the latter theme concern quantitative bounds for some instances of the Bergelson--Leibman Theorem (about subsets of N^d avoiding polynomial progressions).

Our main aim in this thesis is to study the stability properties of the family of k-self-similar solutions to the Einstein-scalar field equations. These solutions were identified by Christodoulou, and model the gravitational collapse of matter without a corresponding event horizon. As a result, they are of interest in the context of the weak cosmic censorship conjecture. In Chapter 4 we apply techniques of Rodnianski–Shlapentokh-Rothman to sharply characterize instabilities in the exterior region. Within a scale of Hölder spaces, we establish nonlinear instability for regularities slightly above BV. This extends Christodoulou's foundational result which identified a blue-shift instability in low-regularity. We then turn our attention to the interior region, where the role of the blue-shift instability is less well understood. In Chapter 5 we give a backwards construction of asymptotically k-self-similar interiors which achieve a given asymptotic profile on the light-cone incident to the singularity. In Chapter 6 we study the asymptotics of solutions to the linear wave equation in the interior, with characteristic initial data posed to the past of the singularity. The presence of finite regularity data poses considerable problems, requiring a mixture of physical-space energy methods and frequency-space scattering techniques, as well as the restriction to small-mass spacetimes. We show that sufficiently regular data leads to self-similar asymptotics, raising intriguing questions about the diminished role of the blue-shift instability in high regularity function spaces. In Chapter 7 we leave the study of k-self-similar spacetimes and consider the related topic of C1 extension principles for the spherically symmetric Einstein-scalar field system. Our result, joint with Xinliang An (NUS) and Haoyang Chen (NUS), is a modest generalization of the well-known small-μ extension principle to cover solutions which (a) globally satisfy μ < 3/8 , and (b) exhibit finite blue-shift along ingoing null-cones.

In this dissertation, we present several results related to spacelike singularities arising in Einstein’s theory of General Relativity, and their connection to the heuristics of Belinski, Khalatnikov and Lifshitz (BKL) in the physics literature. BKL proposed that spacelike singularities arising from the Einstein equations resemble an anisotropic Kasner metric, but where the rates and directions of compression and stretching depend on space and time. Further, they describe the time dependence as a cascade of Kasner-like regimes connected by BKL bounces, where the rates and directions transition rapidly from one regime to another – often termed BKL’s chaotic and oscillatory approach to singularity. Our thesis has three main parts, each of which provides support of BKL’s proposal within various contexts. In the first part, we show that a large class of spherically symmetric solutions to the Einstein–(Maxwell)–scalar field system possess a Kasner-like singularity in the above sense. Though the scalar field and symmetry assumptions suppress the above oscillations, remnants of their effects remain upon introducing the Maxwell field. This is explored further in joint work with M. Van de Moortel, where we study homogeneous black hole interiors with charged scalar hair. In the second part, we characterize a wide class of symmetric, but spatially inhomogeneous spacetimes, both in vacuum and within the Einstein–Maxwell–scalar field model, which feature BKL bounces near their singularity. These are the first inhomogeneous examples of such bounces, and although our current work allows for only one bounce rather than a chaotic cascade, our method – combining nonlinear ODE analysis with energy estimates – is robust and we outline several potential further applications. The final part analyzes the linearized Einstein–scalar field system near exact Kasner spacetimes and exhibits a scattering isomorphism between suitable Cauchy initial data and the asymptotic quantities (e.g. Kasner-like rates and directions) at the singularity. The scattering map reveals interesting gains and losses of derivatives, which depend sensitively on the anisotropy of the Kasner background.

We present three results investigating aspects of the stability of minimal hypersurfaces. First, in joint work with Otis Chodosh, Chao Li, and Paul Minter, we prove that every complete oriented stable minimal hypersurface in R^5 is a hyperplane, which settles the stable Bernstein problem in dimension 5. Some consequences of this classification are curvature estimates for stable minimal hypersurfaces in 5-dimensional Riemannian manifolds. Second, in joint work with Otis Chodosh and Chao Li, we prove a rigidity result for complete two-sided stable minimal hypersurfaces in 4-dimensional manifolds with bounded and nonnegative sectional curvature and uniformly positive scalar curvature. We use this rigidity result to prove new topological obstructions for complete noncompact 4-dimensional manifolds satisfying these curvature assumptions. Finally, we present a new localization technique to prove the existence of minimal hypersurfaces in some complete noncompact manifolds. The minimal hypersurfaces produced by these methods are not necessarily stable, but they are quantitatively almost stable (index at most one). This result can be viewed as the sharp generalization to complete noncompact manifolds of the successful min-max theory of minimal hypersurfaces in closed manifolds.

This thesis consists of two parts: the study of long-time dynamics of differential equations derived from 1) physics, and 2) machine learning. In the first part, we study the cubic nonlinear Schr"odinger (NLS) equation with periodic boundary conditions defined on $[0,L]$. By proving a vanishing property of the normal form transformation and applying it to the quintic resonance interactions, we obtain a description of the dynamics for a time up to $T=L2/\epsilon 4$, where $ϵ$ is the size of the initial data. Since $T$ is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of the 1D cubic NLS. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively. Notably, our approach can be adapted to other integrable systems with minimal difficulty. In the second part, we consider differential equations in machine learning algorithms. We develop a novel approach to model discrete-time machine learning optimization algorithms as continuous-time dynamics, including stochastic gradient descent (SGD) and its variants. We propose the stochastic gradient process consists in a gradient flow minimizing an indexed target function that is coupled with a continuous-time index process determining the index. Index processes are, e.g., reflected diffusions, pure jump processes, or other L'evy processes on compact spaces. We analyze the approximation properties of the stochastic gradient process and study its long-time behavior and ergodicity under constant and decreasing learning rates. We illustrate the applicability of the stochastic gradient process in a polynomial regression problem with noisy functional data, as well as in a physics-informed neural network.

The Ising model is one of the most extensively studied particle system models thatoriginated from Statistical physics. With the help of the Gibbs measure and the language of the Markov chain, several questions regarding the mixing times and cutoff phenomena have been dealt with. The Ising type models vary in terms of the underlying graph and the dynamics of how to evolve the given model. Similar to the general card shuffling problems, we can imagine a particle system in which each particle updates in a certain order but with the same rule as the original dynamics. This systematic scan dynamics pose another question, which is often challenging due to the lack of symmetry. We apply the systematic scan idea to the Ising type model. We study the mixing time and the existence of cutoffs of the systematic scan Glauber dynamics Ising model on the complete graph. There is a cutoff in the high-temperature regime, while there is not in the critical and the low-temperature regime. We provide not only the mixing times for each regime but also where exactly the cutoff happens and the window size for the high-temperature regime, in the last chapter of this dissertation. The upper bound can be achieved from the coupling argument, and the lower bound comes by computing the drift of the magnetization.

In this thesis we construct 3-parameter families G(p, q, r) of embedded arcs with fixed boundary in a 4-manifold. We then analyze these elements of pi_3Emb(I, M) using embedding calculus by studying the induced map from the embedding space to “Taylor approximations” T_kEmb(I, M). We develop a diagrammatic framework inspired by cubical ω-groupoids to depict G(p, q, r) and related homotopies. We use this framework extensively in Chapter 4 to show explicitly that G(p, q, r) is trivial inpi_3T3Emb(I, M) (however, we conjecture that it is non-trivial in pi_3T_4Emb(I, M)). In Chapter 5 we use the Bousfield-Kan spectral sequence for homotopy groups of cosimplicial spaces to show that the rational homotopy group pi_3Emb(I, S1 × B3) is Q. This thesis extends work by Budney and Gabai in [BG21] which proves analogous results for pi_2Emb(I, M).

In this thesis, we discuss aspects of lower-order statistical behavior of coefficients of GL2 automorphic forms. First, we establish two cases of recently observed correlation phenomena, referred to as “murmurations,” between root numbers and L-function coefficients. The first is for the family of weight k modular cuspidal newforms. In that case, we show that averages of P-th Fourier coefficients correlated against the root number in a family of forms of conductor ∼ N converge to a function of P/N. In the second case (from joint work with Booker, Lee, Lowry-Duda, and Seymour-Howell), we prove an analogous result for the family of weight 0 level 1 Maass forms. In this case, additional averaging on P is required, and the answer is given by a measure evaluated on the interval of P-averaging. Finally, in joint work with Sarnak, we give new rates of convergence to the Plancherel measure for coefficients of holomorphic forms of weight 2 and bound the number of d-dimensional factors of the Jacobian of the modular curve.

In this dissertation, we investigate extremal black holes in general relativity. Extremal black holes are exceptional solutions of Einstein’s equations which have absolute zero temperature in the celebrated thermodynamic analogy of black hole mechanics. Our first main result is a definitive disproof of the “third law of black hole thermodynamics.” We construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein–Maxwell-charged scalar field system which are exactly isometric to extremal Reissner–Nordström after a finite advanced time along the event horizon. Moreover, in each of these examples the apparent horizon of the black hole coincides with that of a Schwarzschild solution at earlier advanced times. We also prove similar black hole formation results for very slowly rotating Kerr black holes in vacuum.Our second main result is a proof that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein–Maxwell–Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal Reissner–Nordström black hole. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity. The above mentioned results constitute Part I of this dissertation and were all obtained in joint work with Christoph Kehle. In Part II of this dissertation, we study extensions of the celebrated positive mass theorem to a very general class of initial data, including extremal black holes. These results were obtained in collaboration with Dan A. Lee, Martin Lesourd, and Shing-Tung Yau. We provide a resolution of the spacetime positive mass theorem on manifolds with boundary, a resolution of the remaining cases of Schoen and Yau’s Liouville conjecture for locally conformally flat manifolds, and demonstrate a novel scalar curvature shielding phenomenon for the ADM mass.

This dissertation studies Euler Systems and their arithmetic applications. Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of $L$-functions. However, despite their importance in foundational conjectures in number theory like the Bloch--Kato conjecture, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle in constructing Euler Systems lies in producing candidate Galois cohomology classes. This thesis presents a method to overcome this obstacle without relying on rare motivic classes. In joint work with C. Skinner, I use Eisenstein classes on Siegel threefolds to construct a cyclotomic Euler System for the adjoint of an elliptic modular form. I also construct integral absolute 'etale classes on modular curves associated to theta series at inert primes. These theta classes should give new insight into the Iwasawa main conjecture for modular forms over quadratic imaginary fields at inert primes.

In the first part of this thesis, we study the local fundamental group of low-dimensional log canonical singularities. In dimensions 2 and 3 we establish some constraints on the possible local fundamental groups, while in dimensions 3 and 4 we construct examples of interesting groups that can appear. In dimension 2, by classifying all the possible singularities, we can prove that the local fundamental groups are virtually solvable. Moreover, we give a bound on the number of generators and relations of the group, along with identifying the circumstances under which this maximum is achieved. In dimension 3, we show that free groups with 2 or more generators do not appear as local fundamental groups of log canonical isolated singularities. In dimensions 3 and 4, we construct the fundamental groups of 2-dimensional closed manifolds and some special 3-dimensional manifolds, respectively, as the local fundamental groups of isolated log canonical singularities. One notable example includes the connected sum of copies of S1 × S2, leading to free groups appearing in dimension 4. In the second part, we study the orbifold fundamental groups of the smooth locus of Calabi-Yau type pairs of low coregularity. Here, we establish that the fundamental group of pairs with low coregularity exhibits similar behavior to log Calabi-Yau pairs of low dimension, specifically being virtually abelian of bounded rank. Furthermore, we prove in the case of virtually nilpotency, there are effective bounds on the index and length depending only on the dimension and coregularity.

Questions about the existence of minimal submanifolds in compact Riemannian manifolds occupy a prominent place in the theory of geometric variational problems, and have been addressed using a variety of tools coming from topology, partial differential equations, and the calculus of variations.Geometric measure theory methods have proved particularly successful in providing a weak formulation of this variational problem, and proving the existence of weak solutions under very general assumptions. Thanks to geometric measure theory techniques, a variety of regularity results have also been obtained, showing that in many important situations these weak solutions are indeed smooth minimal submanifolds. In this thesis, we will discuss two regularity results with applications to the theory of minimal hypersurfaces. In Part I, which is based on joint work with Douglas Stryker, we develop a sharp regularity theory for minimisers of the prescribed mean curvature functional in isotopy classes of surfaces in 3-manifolds, which enables us to construct prescribed mean curvature embedded spheres in $S3$. Minimisation over isotopies also plays a central role in min-max constructions of minimal surfaces with prescribed (or controlled) topology. In Part II, which is based on joint work with Martin Li and Davide Parise, we describe a regularity theory for singular limits of solutions of the Allen--Cahn equation with a homogeneous Neumann boundary condition on a compact manifold with boundary. In particular, we show that these limit-interfaces are measure-theoretic free boundary minimal hypersurfaces, opening up the possibility of developing a min-max construction of free boundary minimal hypersurfaces based on the Allen--Cahn equation.