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Recently, there has been interest in the tropicalization of the Grassmannian, known as the tropical Grassmannian. It has been shown that the tropical Grassmannian, and a related tropical space called the Dressian, can be used to construct subdivisions of matroid polytopes into matroid polytopes. There has also been interest ``nonnegative" and ``flag" versions of the tropical Grassmannian and the Dressian, which can be used to construct subdivisions of ``nonnegative" and ``flag" versions of matroid polytopes. This thesis combines these perspectives, exploring ``nonnegative flag" versions of these objects. We show that, for flags whose ranks consist of consecutive integers, many nice results that hold in the previously studied settings have ``nonnegative flag" analogues. The nonnegative flag variety of rank r=(r_1,..., r_k), defined by Lusztig, can be described as the set of rank r flags of linear subspaces in R^n which can be represented by a matrix whose minors are all positive. We show that, for flag varieties of consecutive rank, this equals the subset of the flag variety with nonnegative Plücker coordinates. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the nonnegative Grassmannian equals the subset of the Grassmannian with nonnegative Plücker coordinates, and yields a straightforward condition to determine whether a flag of consecutive rank lies in the nonnegative flag variety. A flag F in any nonnegative flag variety determines a combinatorial object called a flag positroid, defined in terms of the set of nonzero Plücker coordinates of F. We characterize flag positroids of consecutive ranks as oriented flag matroids whose constituents are positively oriented matroids. We also explore flag varieties using tropical geometry, which is the study of algebraic geometry over the (min,+) semifield. The tropical flag variety and the flag Dressian are tropical spaces parameterizing realizable and abstract flags of tropical linear spaces, respectively. In general, the flag Dressian strictly contains the tropical flag variety. However, we show that the nonnegative tropical flag variety and the nonnegative flag Dressian, which parameterize positively realizable and abstract flags of positive tropical linear spaces, respectively, are equal for flags of consecutive ranks. This generalizes the equality of the nonnegative Dressian and the nonnegative tropical Grassmannian, proven by Speyer and Williams. The flag Dressian is closely related to coherent subdivisions of flag matroid polytopes: it is a polyhedral complex whose cones parameterize coherent subdivisions of flag matroid polytopes into smaller flag matroid polytopes. We specialize this relationship to the nonnegative flag Dressian. We prove that the restriction of this parameterization to the cones of the nonnegative flag Dressian instead parameterizes coherent subdivisions of flag positroid polytopes into smaller flag positroid polytopes. In the special case of the positive (non-flag) Dressian, this recovers the description of coherent subdivisions of hypersimplices into positroid polytopes by Lukowski, Parisi, and Williams. In the special case of the nonnegative complete flag Dressian, this describes coherent subdivisions of Bruhat interval polytopes into Bruhat interval polytopes.

The goal of this dissertation is twofold. First, we give a proof of the geometric cone conjecture in relative dimension at most $2$ in the full generality of klt CY pairs. This generalizes previous work done by Kawamata in \cite{Kaw97}, Totaro in \cite{Tot10}, and Filipazzi, Hacon, and Svaldi in \cite{FHS21}. A corollary to this result is that all klt pairs $(X.\Delta)$ with $\kappa(X,\Delta)\geq \dim X - 2$ have only finitely many isomorphism classes of minimal models. In the second part, we prove that the property of Bott vanishing fails for 68 deformation families of smooth Fano threefolds in characteristic $0$. This result is sharp, as subsequent work by Totaro in \cite{Tot24} shows that Bott vanishing is satisfied by the remaining 37 deformation families.

We investigate the count of meromorphic differentials on the Riemann sphere pos-sessing a single zero, multiple poles with prescribed orders, and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed-point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties of the solution formula. This thesis is organized as follows: In Chapter 1 we give an introduction to the problem and summarize the main results obtained. In Chapter 2 we review the compactification of moduli spaces of differentials and introduce various divisor classes. In Section 2.3 we explain how to identify the universal line bundle class with the divisor class of the locus of differentials satisfying a general given residue tuple and prove Theorem 1.0.1 (i). In Section 2.4 we impose exactly one independent partial sum vanishing condition to the residues and prove Theorem 1.0.1 (ii). In Section 2.5 we give a polynomial expression in terms of the zero order for the degree of mixed products between powers of the dual tautological class and the psi-class of the zero. Finally in Chapter 3 we prove Theorem 1.0.2 for arbitrary residues and investigate combinatorial properties of the solution formula. We have also verified our formula numerically for a number of cases by using the software package [CMZ2].

A recent series of papers, starting with the paper of Cook, Harbourne, Migliore, and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets Z of points in N-dimensional projective space. Say the complete linear system L of forms of degree d vanishing on Z has dimension t yet for any general point P the linear system of forms vanishing on Z with multiplicity m at P is nonempty. If the dimension of L is more than the expected dimension of t−r, where r is N+m−1 choose N, we say Z has unexpected hypersurfaces of degree d and multiplicity m. We extend the definition of unexpectedness to include the possibility that the unexpectedness occurs only for P on a subvariety of positive codimension. We begin our study of this stratified unexpectedness by analyzing sets of points in the plane. We are able to give a characterization for unexpectedness for sets of points which lie on a degenerate conic. We then further analyze the techniques of Faenzi and Vallés and how they were used in the 2018 paper of Cook, Harbourne, Migliore, and Nagel. In the fourth chapter, we study lower bounds on the dimension of the space L. Lastly, we study the connection between Lefschetz properties and unexpectedness. Advisor: Brian Harbourne

We compute the nef cones in codimensions 2 & 3 and the effective cones in dimensions 2 & 3 for the Hilbert scheme of three points in $\mathbb{P}^3$. This involves studying the orbits of the PGL action on the Hilbert scheme, as well as extending Mallavibarrena and Sols' bases for the Chow groups of Hilbert schemes of points on $\mathbb{P}^2$ to the case of the Hilbert scheme of 3 points in $\mathbb{P}^3$. This work builds on results of Ryan and Stathis, and utilizes results of Rossello-Llompart on intersection products in the Hilbert scheme of three points in $\mathbb{P}^3$.

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not known what other examples there can be. Furthermore, before the work in this thesis, only a few examples of geproci nontrivial non-grid non-half-grids were known and there was no known way to generate more. Here, we use geometry in the positive characteristic setting to give new methods of producing geproci half-grids and non-half-grids. We also pick up work that had been done in 2017 by Solomon Akesseh, who had proven that there are no unexpected cubics in characteristic 3 with distinct points and gave examples involving infinitely near points based on quasi-elliptic fibrations in characteristic 2. Each quasi-elliptic fibration has a Dynkin diagram. Here, in contrast, for each possible Dynkin diagram for a quasi-elliptic fibration in characteristic 3, we give an example of the fibration but show it does not give rise to an unexpected cubic. [Equations Omitted] Adviser: Brian Harbourne

Given an ordinary elliptic curve E over a field of characteristic p, there is an elliptic curve E over the Witt vectors W() for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over _p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on some conjectures of Finotti, and introduce some conjectures of our own. We also investigate the structure of rings of Witt vectors over arbitrary commutative rings and give a computationally useful isomorphism for Witt vectors over ℤ/p^αℤ [alpha].

Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. In this thesis, we study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmu ̈ller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general formulas are given. Additionally, we define a notion of Minimal and Absolute Minimal Degree Liftings modulo pn and an associated lift of points we call an Edwards Lift. We show how the Canonical Lift may be used to find examples of Absolute Minimal Degree Liftings, at least modulo p3.

We study the Hodge theory of twisted derived categories and its relation to the period-index problem. Our main contribution is the development of a theory of twisted Mukai structures for topologically trivial Brauer classes on arbitrary smooth proper varieties and in families. As applications, we construct Hodge classes whose algebraicity would imply period-index bounds; construct new counterexamples to the integral Hodge conjecture on Severi-Brauer varieties; and prove the integral Hodge conjecture for derived categories of Deligne-Mumford surfaces. Finally, we solve the period-index problem for the complex-analytic Brauer group of a general complex torus of dimension at least three.

Morihiko Saito’s theory of Hodge modules have made an incredible impact in the study of singularities. So far, the strongest results have been obtained in the case of hypersurface singularities, using the strong properties of the V -filtration along hypersurfaces which is built into the theory of Hodge modules. This thesis extends two important tools from the case of hypersurfaces. The first is a compatibility property between the Hodge filtration of a mixed Hodge module and the V - filtration along a higher codimension subvariety. The second is a formula explaining how to restrict to a smooth subvariety of higher codimension using the V -filtration along that subvariety. The main tool at work in proving these theorems is the blow-up along the smooth subvariety. There are two main applications of these theorems: the first is to analyze the Hodge and weight filtration on the local cohomology module along a singular locally complete intersection subvariety. We define the minimal exponent of a locally complete intersection variety and show that its value dictates when the Hodge filtration on local cohomology is equal to the pole order filtration. This shows that the minimal exponent understands information about k-du Bois singularities, and it turns out that the minimal exponent also understands k-rational singularities, by its relation to the weight filtration on local cohomology. The second application is to the study of the Fourier-Laplace transform of monodromic mixed Hodge modules. These modules naturally arise through Verdier’s specialization construction. We explicitly write out the Hodge and weight filtrations for such modules.

A projective, normal variety is called a Mori dream space when its Cox ring is finitely generated. These spaces are desirable to have, as they behave nicely under the Minimal Model Program, but no complete classification of them yet exists. Some early work identified that all toric varieties are examples of Mori dream spaces, as their Cox rings are polynomial rings. Therefore, a natural next step is to investigate projectivized toric vector bundles. These spaces still carry much of the combinatorial data as toric varieties, but have more variable behavior that means that they aren't as straightforward as Mori dream spaces. Expanding on Gonzalez's 2012 result that all rank 2 projectivized toric vector bundles are Mori dream spaces, we give a combinatorial sufficient condition for when a rank r bundle is Mori dream, using Kaveh and Manon's description of a toric vector bundle by a linear ideal and an integral matrix. We then address the question: if a toric vector bundle projectivizes to a Mori dream space, when is the projectivization of the direct sum of that bundle with itself a Mori dream space? Expanding on the nice families of bundles found, we compute the positivity-related cones for these bundles and provide a description of additional classes of toric vector bundles that uphold the Fujita conjectures. Finally, we conclude with the subduction and KM algorithms, two Macaulay2-implemented algorithms that allow us to produce finite presentations of Cox rings of projectivized toric vector bundles, provided they exist, allowing for future work in the study of these bundles as Mori dream spaces.

In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of the pipe dream complex of the identity element. We examine pipe dream complexes in the case a permutation is a product of commuting adjacent transpositions. Finally, we consider pattern avoidance results. For 132-avoiding permutations, the Rothe diagram forms a Young diagram. In the case a permutation w has exactly one 132-pattern, the associated pipe dream complex is an m-dimensional simplex, where m = n choose 2 − l(w) − 1 and l(w) is the length of w. In the case of exactly two 132 patterns, there are three possible configurations. We include generalizations of these cases.

Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map to be surjective on a cubic graph. The Wahl map on a cubic graph has two parts, one map onto the vertices and another onto the edges. We found that if the cubic graph is 3-edge-connected and non-planar, then the map on the vertices is surjective. Surjectivity of the map on the edges is not as clear. However, if the Wahl map is surjective on a cubic graph, the girth of the graph must be at least 5. Finally, based on data collected, we have observed that as the size of cubic graphs of girth at least 5 increases, the probability that the Wahl map is surjective on those graphs appears to approach 1.

This thesis sets up a framework for planar affine geometry with an algebraic approach. We start with points and affine combination as the only operation on points. This allows us to develop objects such as lines and vectors, and operations such as 1-combination and 0-combination; and derive basic properties such as parallelism and signed area. Such foundation is then further enhanced with barycentric coordinates of lines to elevate lines as a subject of interest in triangle geometry. A closer look at Ceva’s and Menelaus’ theorems leads to the finding of Ceva/Menelaus triples. The area principle and parametrisation of a non-degenerate general conic are then introduced as tools for studies in affine geometry. As a direct application, Hoehn’s theorem is further investigated for its general application on general n-gons, resulting in the n-gon Proportion Product and Conic n-gon Proportions theorems.

There are two ways that certain Fano fourfolds (for example, cubic fourfoldsand Gushel-Mukai fourfolds) can be associated with K3 surfaces. On the one hand, we can associate a K3 surface to the fourfold Hodge-theoretically, meaning the middle cohomology of the fourfold contains the middle cohomology of the K3 as a sub-Hodge structure; on the other hand, we can homologically associate a K3 to the fourfold, which is to require the Kuznetsov component of the fourfolds to be equivalent to the bounded derived of the K3. Conjecturally, one expect such K3 associations detect rationality of the fourfolds. It has been proved that for cubic fourfolds, these two types of K3 associations are equivalent, whereas for Gushel-Mukai fourfolds, Hodge-association of a K3 strictly implies homological association of a K3. We continue this line of study, instead of comparing fourfolds- K3 association we consider fourfolds-fourfolds association. We prove that, at least for a generic Gushel-Mukai fourfold in the Hodge-special loci, if it admits Hodgeassociated cubic fourfolds, then it admits a homological-associated one, and vice versa.

We prove a twisting theorem for nodal classes in permutation-equivariant quantum K-theory, and combine it with existing theorems of Givental to obtain a twisting a theorem for general characteristic classes of the virtual tangent bundle of the moduli space of stable maps. Using this result, we develop complex cobordism-valued Gromov-Witten invariants defined via K-theory, and relate those invariants to K-theoretic ones via the quantization of suitable symplectic transformations. This procedure is a K-theoretic analogue of the quantum cobordism theory developed by Givental and Coates. Using the universality of cobordism theory, we give an example of these results in the context of ``Hirzebruch K-theory", which is the cohomology theory determined by the Hirzebruch y-genus.We then introduce Euler-theoretic Gromov-Witten invariants, which are based on the ordinary topological Euler characteristics of loci of curves. These invariants have useful enumerative properties. They are integer valued, there is a reliable way to remove boundary contributions, and when the moduli spaces are smooth orbifolds, have a concrete geometric intepretation. In this case, the invariants also encompass the ordinary (i.e. integer) topological Euler characteristic of the moduli space of stable maps and its compactification. We give a Wick-type formula computing these invariants in terms of twisted cohomological Gromov-Witten invariants, and we also show that they can be regarded as the limit as y approaches 1 of the Hirzebruch-theoretic invariants introduced earlier in this work.Specializing to the case of a point target space, our formulas yield similar results to the computation of the Euler characteristics of the moduli spaces of n-pointed curves, given by Bini-Harer.

Abstract: The topological recursion is a construction in algebraic geometry that takes in the data of a so-called spectral curve, $\\mathcal{S}=\\left(\\Sigma,x,y\\right)$ where $\\Sigma$ is a Riemann surface and $x,y:\\Sigma\\to\\mathbb{C}_\\infty$ are meromorphic, and recursively constructs correlators which, in applications, are then interpreted as generating functions. In many of these applications, for example the $r$-spin Hurwitz case $\\mathcal{S}=\\left(\\mathbb{C},x(z)=z{\\rm e}^{-z^r},y(z)={\\rm e}^{z^r}\\right)$, $x$ has essential singularities when the underlying Riemann surface is compactified. Previously, these essential singularities have been ignored and the topological recursion considered on the non-compact surface. Here we argue that it is more natural to include the essential singularities as ramification points and give the corresponding definition for topological recursion; that is, a topological recursion for \\emph{transalgebraic} spectral curves rather than \\emph{algebraic} spectral curves. We use this definition to shed light on the TR/QC connection, Hurwitz theory, the Gromov-Witten invariants of $\\mathbb{CP}^1$, and mirror curves.

We have codified the algebraic fundamental group of anabelian geometry as a multi-sorted logical structure so as to use model-theoretic ideas, analogies, and language to go further with the study of hyperbolic curves over number fields. Consequently, a definability analysis is now possible on smooth quasi-projective schemes and their algebraic fundamental groups in characteristic zero. We form a connection between the algebraic fundamental group and the Lascar group of a stable first-order theory of covering spaces. We then provide a formulation of a Grothendieck-type section conjecture in terms of pure stability. One such use-case, for finitely generated k, is the application of geometric stability theory to use elimination of imaginaries to construct k-rational points on hyperbolic k-curves.

We investigate how the coefficients of a sparse polynomial system influence the sum, or the trace, of its solutions. We discuss an extension of the classical trace test in numerical algebraic geometry to sparse polynomial systems. Two known methods for identifying a trace affine linear subset of the support of a sparse polynomial system use sparse resultants and polyhedral geometry, respectively. We introduce a new approach which provides more precise classifications of trace affine linear sets than was previously known. For this new approach, we developed software in Macaulay2.

For any $G((t))$-category, we define a filtration which provides a categorification of the notion of depth studied by Moy and Prasad in the context of representations of a p-adic group. We also prove a generation statement which is the analogue of their theory of unramified minimal K-types and use it to compute our filtration on a number of categories of interest to geometric Langlands. Finally, joint with Sam Raskin, we apply this theory to prove an affine Beilinson-Bernstein localization statement conjectured by Frenkel and Gaitsgory.

W pracy zajmujemy się kontrakcjami typu III pewnych trójwymiarowych rozmaitości Calabi-Yau. Opisujemy warunki, które muszą być spełnione aby rozmaitość po kontrakcji była wygładzalna. Opisujemy zmianę liczb Hodge'a rozmaitości po kontrakcji i po wygładzeniu. Konstruujemy nowe przykłady rozmaitości Calabi-Yau otrzymane w ten sposób.

Varieties with group action have been of interest to algebraic geometers for centuries. In particular, toric varieties have proven useful both theoretically and in practical applications. A rich theory blending algebraic geometry and polyhedral geometry has been developed for T-varieties which are natural generalizations of toric varieties. The first results discussed in this dissertation study the relationship between torus actions and the well-poised property. In particular, I show that the well-poised property is preserved under a geometric invariant theory quotient by a (quasi-)torus. Conversely, I argue that T-varieties built on a well-poised base preserve the well-poised property when the base satisfies certain degree conditions. The second half of this dissertation covers two projects in algebraic statistics. The first studies level-1 phylogenetic network models which model evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. In particular, I found the quadratic invariants of the Cavendar-Farris-Neyman model for level-1 networks and conjecture these generate the corresponding ideal. In the final project, I study a class of statistical models known as binary hierarchical models. Hierarchical models are known to be log-linear; thus, the joint probability distributions of the random variables naturally lie on a toric variety. For many applications such as testing normality of the model and finding a maximum likelihood estimate, a H-description of the marginal polytope is needed to drastically speed up computations. Here I provide an alternative polytope isomorphic to the marginal polytope in the binary case. This polytope is known as the generalized cut polytope, and I compute H-descriptions for all binary hierarchical models whose underlying simplicial complex is pure and of codimension 1.

Stetson and Baranovsky provided an algorithm with Mathematica to compute the zeta function of projective hypersurface over $\mathbb{F}_{p}$ with isolated ordinary double points. In this thesis, I extend this algorithm to hypersurfaces with ADE singularities over $\mathbb{P}^{3}$. In the process of doing so, I characterize the Jacobian ideal as a zero set of differential operators.

The necessary conditions for a quantization of a module over an algebra on a symplectic manifold to exist are investigated. Considered is a symplectic algebraic variety $M$ with a fixed deformation quantization $\mathcal{O}_{\hbar}$ of its sheaf of regular functions, and a vector bundle $E$ on M with a deformation quantization of order $k$ (as a module over $\mathcal{O}_{\hbar}$). It is found that range of cohomology classes must vanish if this order admits an extension to quantization of order $\ell>k$. For $\ell

Secondary effects are key to adding fluidity and style to animation. This thesis introduces the idea of “Acceleration Skinning” following a recent well-received technique, Velocity Skinning, to automatically create secondary motion in character animation by modifying the standard pipeline for skeletal rig skinning. These effects, which animators may refer to as squash and stretch or drag, attempt to create an illusion of inertia. In this thesis, I extend the Velocity Skinning technique to include acceleration for creating a wider gamut of cartoon effects. I explore three new deformers that make use of this Acceleration Skinning framework: followthrough, centripetal stretch, and centripetal lift deformers. The followthrough deformer aims at recreating this classic effect defined in the fundamental principles of animation. The centripetal stretch and centripetal lift deformers use rotational motion to create radial stretching and lifting effects, as the names suggest. I explore the use of effect-specific time filtering when combining these various deformations together, allowing for more stylized and aesthetic results. I finally conclude with a production evaluation, exploring possible ways in which these techniques can be used to enhance the work of an animator without losing the essence of their art.

Buchberger’s algorithm is the classical algorithm for computing a Gröbner basis, and optimized implementations are crucial for many computer algebra systems. In this thesis we introduce a new approach to Buchberger’s algorithm that uses deep reinforcement learning agents to perform S-pair selection, a key choice in the algorithm. We first study how the difficulty of the problem depends on several random distributions of polynomial ideals, about which little is known. Next, we train a policy model using proximal policy optimization to learn S-pair selection strategies for random systems of binomial equations. In certain domains the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of critical algorithms in symbolic computation. Finally, we apply these techniques to Bigatti’s algorithm for computing a Hilbert series, and consider extending these results with neural network value models and tree search.

We study singularities of algebraic varieties, in particular those arising in birational geometry, from several points of view. The first is that of arc schemes: arc schemes parametrize “infinitesimal curves” on a variety, and their geometry reflects properties of singularities. We show that morphisms of arc schemes (more precisely, of “local” arc schemes) can detect local isomorphisms of varieties. More precisely, we use the triviality of a certain ideal-closure operation to show that if a morphism induces an isomorphism of local arc schemes then it must be an isomorphism on local rings. We then use arc schemes, in conjunction with the theory of determinantal rings, to verify the semicontinuity conjecture for the behavior of the minimal log discrepancy (a subtle invariant of singularities) in the case of determinantal varieties. In particular, we calculate the Nash ideal of a generic square determinantal variety, which then allows us to give an explicit formula for the minimal log discrepancies of pairs of determinantal varieties and determinantal subvarieties. This allows us to verify the semicontinuity conjecture for such pairs. We then take another point of view, via the study of differential operators on singular rings. At least since [Levasseur and Stafford 1989], the question had been asked of whether one can characterize singularities of rings via certain properties of their rings of differential operators. In particular, one question is whether a ring with mild singularities is a simple module under the action of its ring of differential operators. While an answer in characteristic p had been provided by [Smith 1995], no answer had been forthcoming in characteristic 0. We provide a counterexample showing that the expected connection does not exist, through the study of the global geometry of Fano varieties. More specifically, we show that certain del Pezzo surfaces do not have big tangent bundles, and thus their homogeneous coordinate rings are not simple under the action of their rings of differential operators, despite having “mild” singularities.

The geometry of a curve can be analyzed in many ways. One way of doing this is to study the set of all divisors on a curve of prescribed rank and degree, known as a Brill-Noether variety. A sequence of results, starting in the 1980s, answered several fundamental questions about these varieties for general curves. However, many of these questions are still unanswered if we restrict to special families of curves. This dissertation has three main goals. First, we examine Brill-Noether varieties for these special families and provide combinatorial descriptions of their irreducible components. Second, we provide a natural generalization of Brill-Noether varieties, known as Splitting-Type varieties, that parameterize this decomposition. Lastly, we provide purely combinatorial descriptions of these Splitting-Type varieties and explore the geometric consequences of these descriptions. These results are based upon and extend tools and techniques from tropical geometry.

The research presented in this manuscript investigates computational methods and integrative approaches for planar and spatial strut-and-tie modeling. This study is formulated in response to the limitations of the existing computational techniques in generating practical truss models and the lack of useful developments and applications of computational graphical statics in the current structural engineering core research and practice. Stress-based methods like finite element analysis or topology optimization use detailed stress analysis to help create truss models. Since these methods focus on analysis rather than design, they deliver less freedom to create practical models or consider modification scenarios. Alternatively, geometry-based techniques like the load path method or graphic statics are design tools that aid in the procedural creation of strut-and-tie models. So, they provide more control and creativity, but at the same time, no pre-analysis of the structure is provided; therefore, more relevant expertise is required to create a model using these techniques. Furthermore, the latest advancements in graphic statics still require considerable improvements to find practical applications for computational strut-and-tie modeling. Three key research objectives are established by examining the potentials and limitations of the existing techniques: (i) exploring applications and extensions of algebraic graphic statics and layout optimization in creating truss models, (ii) investigating strategies to implement modifications to the initiated models using graphic statics, and (iii) addressing the challenges for spatial cases and develop required algorithmic extensions. Through the development of integrative algorithms and expansion of existing formulations, the methodologies of this research demonstrated the potential to bridge the gap between analytical and graphical approaches and contribute to the evolving nature of computational graphic statics in structural design. The presented algorithms of this study can be incorporated into the various parametric design, structural analysis, and digital fabrication workflows, proceeding one step further towards the materialization of the tension-compression networks.

The Knop Conjecture, which was proven by Losev in [Los09a], states that smooth affinespherical varieties are classified up to equivariant isomorphism by their weight monoids. This is in contrast with the standard classification of spherical varieties, which involves combinatorial invariants related to divisors and valuations. In this thesis, we prove that some of these combinatorial invariants are also determined by weight monoids in the smooth projective case. This results in certain partial analogs of the Knop Conjecture for smooth projective spherical varieties. We provide counterexamples to demonstrate that these partial analogs are relatively optimal.Our results indicate that weight monoids of smooth projective spherical varieties are closelyrelated to the data of certain divisors on these varieties. In analogy with the total coordinate ring discussed in [Bri07], we develop methods for comparing the data of weight monoids with the data of divisors, even without smoothness hypotheses. We then show that, under mild hypotheses, the data provided by weight monoids is equivalent to the data provided by divisors on projective spherical varieties.

We study the problem of whether repeated normalized Nash blowups resolve toric singularities. We first review results of Gonzalez-Sprinberg, describing the normalized Nash blowup combinatorially and proving resolution for toric surfaces. We then investigate the problem in higher dimensions, letting $X$ be an affine toric variety of dimension at least $3$ and considering each invariant divisor $Y$ of its normalized Nash blowup $\text{NNB}(X)$. For the case where $Y$ is the strict transform of an invariant divisor $Y_0$ of $X$, we show that the birational map from $Y$ to $\text{NNB}(Y_0)$ is a morphism. For the case where $X$ is $3$-dimensional and $Y$ is any invariant divisor of $\text{NNB}(X)$, we prove the following result. Let $\tau$ be the ray corresponding to $Y$ in the fan of $\text{NNB}(X)$, and let $Z$ be the surface corresponding to $\tau$ in the stellar subdivision (also known as the star subdivision) of the fan of $X$ along $\tau$. Let $\text{MR\textsuperscript{+}}(Z)$ be the surface obtained from the minimal resolution of $Z$ by blowing up all invariant points. Then the birational map from $\text{MR\textsuperscript{+}}(Z)$ to $Y$ is a morphism. For the case where $X$ is a $3$-dimensional quotient singularity and $Y$ is the strict transform of an invariant divisor $Y_0$ of $X$, we deduce that the singularities of $Y$ are milder than those of $Y_0$.

Abstract: In this thesis, we study some aspects of algebraic geometry that have had a significant influx of ideas from physics. The first part focuses on the Eynard- Orantin topological recursion and its variants as a theory of enumerative ge- ometry. We investigate the conjectural relationship between the topological recursion and quantum curves in the case of elliptic curves. We show that the perturbative wave-function is not the solution to a quantum curve, while the non-perturbative one is (up to certain order in ħ). We define the formalism of Higher Airy Structures (HAS), which quantized higher order Lagrangians in a symplectic vector space. By showing that the Bouchard-Eynard topological recursion is a HAS, we find a necessary and sufficient condition on the spectral curve for producing symmetric correlators ω g,n . We construct numerous other examples, some of which are (sometimes conjecturally) related to FJRW theory or open (r-spin) intersection theory. In this thesis, we study some aspects of algebraic geometry that have had a significant influx of ideas from physics. The first part focuses on the Eynard-Orantin topological recursion and its variants as a theory of enumerative geometry. We investigate the conjectural relationship between the topological recursion and quantum curves in the case of elliptic curves. We show that the perturbative wave-function is not the solution to a quantum curve, while the non-perturbative one is (up to certain order in $ \hbar $). We define the formalism of Higher Airy Structures (HAS), which quantized higher order Lagrangians in a symplectic vector space. By showing that the Bouchard-Eynard topological recursion is a HAS, we find a necessary and sufficient condition on the spectral curve for producing symmetric correlators $ \omega_{g,n} $. We construct numerous other examples, some of which are (sometimes conjecturally) related to FJRW theory or open ($ r $-spin) intersection theory. In the second part of this thesis, we focus on the study of derived categories and its connections to birational geometry; in particular, we are interested in a conjecture of Bondal and Orlov about flops. Using the presentation of a flop as a variation of geometric invariant theory (VGIT) problem, Ballard, Diemer and Favero proposed a Fourier-Mukai kernel and conjectured that it induces a derived equivalence. We verify this conjecture in the case of Grassmann flops first. Then, we tackle the case of singular VGIT problems using semi-free commutative dg-algebra resolutions, and prove that we obtain derived equivalences of dg-schemes, under some conditions.

This thesis develops an algebraic analog of psuedo-Riemannian geometry for relative schemes whose cotangent sheaf is finite locally free. It is a generalization of the algebraic differential calculus proposed by Dr. Ernst Kunz in an unpublished manuscript to the non-affine case. These analogs include the psuedo-Riemannian metric, Levi-Civit´a connection, curvature, and various existence theorems.

This work develops a method to canonically compactify mirror families for positive pairs (Y,D), where Y is a projective rational surface and D is an anti-canonical cycle of rational curves. The compactified families can be identified with universal families associated to generalized marked pairs, a notion that is also developed here. In order to make this identification, period integrals for the mirror family are required. To carry this out, recent techniques from mirror symmetry and tropical geometry are used.

A construction of the supermoduli space $\mnr$ of super Riemann surfaces of genus zero with $\nr$ Ramond punctures as a quotient Deligne-Mumford superstack of dimension $(\nr-3\, | \, \nr/2-2)$ is presented.

A tropical variety is a weighted polyhedral complex whose maximal dimensional cells are pure dimensional, rational, connected, and balanced weighted around every vertex. It is known that every irreducible algebraic variety can be tropicalized, that is, there is a way one can derive a tropical variety from an algebraic one. However, there exist tropical varieties that are not the tropicalization of algebraic varieties. The goal of this work is to answer whether a given tropical curve (a 1−dimensional tropical variety) in R^3 is the tropicalization of an algebraic curve in R^3 .

One area of interest in studying plane curves is intersection. Namely, given two plane curves, we are interested in understanding how they intersect. In this paper, we will build the machinery necessary to describe this intersection. Our discussion will include developing algebraic tools, describing how two curves intersect at a given point, and accounting for points at infinity by way of projective space. With all these tools, we will prove Bézout’s theorem, a robust description of the intersection between two curves relating the degrees of the defining polynomials to the number of points in the intersection.

The aim of this paper is to prove a wall-crossing formula at the level of virtual fundamental classes of weighted stable quasimaps to a class of GIT quotients via localization. The idea of the proof is to use virtual localization on a larger moduli space, referred to as the ``master space'', the fixed-point loci of which are in one-to-one correspondence with the terms in the wall-crossing formula. As an immediate application, we obtain the modified string and dilaton equations for stable quasimaps to GIT quotients.

We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends Brinkmann's theorem that free-by-cyclic groups are word-hyperbolic if and only if they have no Z2 subgroups. To get started on our main theorem, we first prove a structure theorem for injective but nonsurjective endomorphisms of free groups. With the decomposition of the free group given by this structure theorem, we (more or less) construct representatives for nonsurjective endomorphisms that are expanding immersions relative to a homotopy equivalence. This structure theorem initializes the development of (relative) train track theory for nonsurjective endomorphisms.

We show how equivariant cohomology can be applied to enumerative geometry in three different settings: orbits of plane curves, strata of points on a line, and effective divisors on the moduli space of curves. We first give a brief introduction to equivariant cohomology. Then, we include three different applications that are essentially unchanged from their published versions and contain joint work with Mitchell Lee, Anand Patel, and Hunter Spink. We conclude with a short section with unpublished observations and conjectures stemming from a concrete connection between counting singularities and equivariant cohomology.

This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points. We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' passing through a fixed set of points Z. We show in certain cases these can be characterized via combinatorial data and geometric data from the Hyperplane Arrangement dual to Z. This generalizes earlier results on unexpected curves in the plane due to Faenzi, Valles, Cook, Harbourne, Migliore and Nagel.

In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.

We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and compute these invariants for a much-studied family of tropical curves. Our examples highlight many parallels between the classical and tropical theories, but also point to some substantive distinctions.

Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow for increased security. Finally, we develop a family of twisted Hermitian codes that meets the criteria set forth for security.

Algebraic geometry is a classical subject which studies shapes arising as zero sets of polynomial equations. Such objects, called varieties, may be quite complicated but many aspects of their geometry are governed by discrete data. In turn, combinatorial structures arising from particularly meaningful varieties, such as moduli spaces, are interesting in their own right. In recent years tropical geometry has emerged as a robust tool for studying varieties. As a result, many rich connections between algebraic geometry and combinatorics have developed. Tropical geometry associates polyhedral complexes, like tropical varieties and skeletons, to algebraic varieties. These encode information about the variety or the equations they came from, providing insight in to the underlying combinatorial structure.In this thesis, I develop tropical geometry of curves from the perspectives of divisors, moduli spaces, computation of skeletons, and enumeration. Already, the realm of curves is rich to explore using the tools of tropical geometry. This thesis is divided into five chapters, each focusing on different aspects of the tropical geometry of curves.I begin by introducing algebraic curves, tropical curves, and non-Archimedean curves. The ways in which these objects interact will be a common theme throughout the thesis. The picture on the previous page expresses the idea that tropical curves are projections of non-Archimedean curves. Berkovich analytic spaces are heavenly abstract objects which can be viewed by earthly beings through their tropical shadows.In the second chapter, I develop divisors on tropical curves and tropicalize algebraic divisors. Many constructions for classical curves related to divisors carry over to the tropical world. This will include a tropical Jacobian, a tropical version of the Riemann-Roch theorem, and a tropical Abel-Jacobi map. I first define and compute these objects. Then, I focus on the symmetric power of a curve, because this functions as a moduli space for effective divisors on the curve. I prove that the non-Archimedean skeleton of the symmetric power of a curve is equal to the symmetric power of the non-Archimedean skeleton of the curve. Using this, I prove a realizable version of the tropical Riemann-Roch Theorem.In the third chapter I focus on moduli spaces. A recurring phenomenon in tropical geometry is that the non-Archimedean skeleton of an algebraic moduli space gives a tropical one. I will develop detailed examples of this. Then, I define a divisorial motivic zeta function for marked stable curves, and prove that it is rational.In the fourth chapter I compute abstract tropicalizations or non-Archimedean skeletons of a curve. In genus one and two there are known methods for computing these tropicalizations. I develop an algorithm for computing the abstract tropicalizations of hyperelliptic and superelliptic curves. In higher genus, these are the only known results for computing abstract tropicalizations of curves.In the final chapter I study enumerative problems. Tropical geometry has proven to be a very useful tool in counting curves in the plane. I turn my attention to surfaces in space, and develop tropical counting techniques in this domain. This leads to a preliminary count of binodal cubic surfaces.

This thesis is devoted to the study of two central objects in algebraic geometry - algebraic curves and vector bundles - through the unifying lens of liaison theory. The liaison theory of curves originated in the work of M. Noether in the late XIX century, and has since become an instrumental tool in the classification of space curves and the study of Hilbert schemes in general. In Chapter I, we develop a biliaison theory of sheaves and prove several main results in analogy to those from the liaison theory of codimension two subvarieties. In Chapter II, we use the liaison of curves on surfaces with ordinary singularities to prove multiple results about homological invariants of general projections of curves into projective three-space. In Chapter III, we apply biliaison of sheaves to the study of vector bundles. A general program is outlined to describe the moduli of bundles on a projective variety of irregularity zero through a stratification approach. We take the first steps by classifying bundles in the biliaison class of the zero sheaf on projective spaces and describing the moduli. In particular, our results give a description of the moduli of bundles on the projective plane.

Geometric Invariant Theory (GIT) is a powerful theory for constructing and studying the geometry of moduli spaces in algebraic geometry, as quotients of a parameter space by a linear group action. However GIT only applies to classification problems involving reductive group actions, and only produces a moduli space for the subclass of "stable" objects. Non-Reductive GIT was developed over the last ten years to overcome these limitations. The aim of this thesis is firstly to describe a systematic way of constructing and studying the geometry of quotients using both classical and Non-Reductive GIT, and secondly to show how these methods can be applied to the classification problem for Higgs bundles. In the first part we describe an algorithm which, given the linear action of a linear algebraic group with so-called "internally graded unipotent radical" on a projective variety, uses classical and Non-Reductive GIT to produce a non-empty open subset of the variety admitting a quasi-projective geometric quotient, together with an explicit projective completion of this quotient. Assuming the initial variety is smooth, we describe a procedure for calculating the Poincaré series of the resulting projective completion, which first requires showing that it can have at worst finite quotient singularities. We then obtain as a result a formula which can be viewed as the non-reductive analogue of the formula for the Poincaré series of the partial desingularisation of classical GIT quotients. In the second part we show how Non-Reductive GIT can be used to refine two instability stratifications of the stack of Higgs bundles: the Higgs Harder-Narasimhan and the Harder-Narasimhan stratifications. These refined stratifications satisfy the property that each stratum admits a quasi-projective coarse moduli space with an explicit projective completion. In the rank 2 case we provide a complete moduli-theoretic interpretation of these refined stratifications and study the geometry of the corresponding moduli spaces.

Proteins are an important component of living organisms, composed of one or more polypeptide chains, each containing hundreds or even thousands of amino acids of 20 standard types. The structure of a protein from the sequence determines crucial functions of proteins such as initiating metabolic reactions, DNA replication, cell signaling, and transporting molecules. In the past, proteins were considered to always have a well-defined stable shape (structured proteins), however, it has recently been shown that there exist intrinsically disordered proteins (IDPs), which lack a fixed or ordered 3D structure, have dynamic characteristics and therefore, exist in multiple states. Based on this, we extend the mapping of protein sequence not only to a fixed stable structure but also to an ensemble of protein conformations, which help us explain the complex interaction within a cell that was otherwise obscured. The objective of this dissertation is to develop effective ab initio methods and tools for protein un/structure prediction by developing effective statistical energy function, conformational search method, and disulfide connectivity patterns predictor. The key outcomes of this dissertation research are: i) a sequence and structure-based energy function for structured proteins that includes energetic terms extracted from hydrophobic-hydrophilic properties, accessible surface area, torsion angles, and ubiquitously computed dihedral angles uPhi and uPsi, ii) an ab initio protein structure predictor that combines optimal energy function derived from sequence and structure-based properties of proteins and an effective conformational search method which includes angular rotation and segment translation strategies, iii) an SVM with RBF kernel-based framework to predict disulfide connectivity pattern, iv) a hydrophobic-hydrophilic property based energy function for unstructured proteins, and v) an ab initio conformational ensemble generator that combines energy function and conformational search method for unstructured proteins which can help understand the biological systems involving IDPs and assist in rational drugs design to cure critical diseases such as cancer or cardiovascular diseases caused by challenging states of IDPs.

This thesis describes [the] action on the Koszul dual ∞-category in terms of the shifted Poisson structure on T [superscript *] X[2]: The main result is that the action is the exponential of the Poisson bracket