Algebraic Geometry Commons^™

Generalization Of Bi-Canonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos

Publications and Research

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers–the type of R, the reduction number of ω–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings …

The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell

Dissertations, Theses, and Capstone Projects

The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be …

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

Prime Factors: America’S Prioritization Of Literacy Over Numeracy And Its Relationship To Systemic Inequity, Troy Smith

Dissertations, Theses, and Capstone Projects

For much of American history, literacy has been prioritized in K-12 education and society, at large, at the expense of numeracy. This lack of numerical emphasis has established innumeracy as an American cultural norm that has resulted in America not producing a sufficient number of numerate citizens, and ranking poorly on mathematical performance in international comparisons. This paper investigates the decisions and circumstances that led to this under prioritization, along with the public and cultural impact of said actions. Toward this end, literature regarding contemporary and historical influences on American mathematics education (e.g., civic, policy, and parental) was reviewed. The …

Fano 3-Folds, Reflexive Polytopes And Brane Brick Models, Sebastían Franco, Rak-Kyeong Seong

Publications and Research

Reflexive polytopes in n dimensions have attracted much attention both in mathematics and theoretical physics due to their connection to Fano n-folds and mirror symmetry. This work focuses on the 18 regular reflexive polytopes corresponding to smooth Fano 3-folds. For the first time, we show that all 18 regular reflexive polytopes have corresponding 2d (0, 2) gauge theories realized by brane brick models. These 2d gauge theories can be considered as the worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities whose toric diagrams are given by the associated regular reflexive polytopes. The generators of the mesonic moduli …

Spectral Sequences For Almost Complex Manifolds, Qian Chen

Dissertations, Theses, and Capstone Projects

In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [CW18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Frolicher spectral sequence. …

A Differential-Algebraic Criterion For Obtaining A Small Maximal Cohen-Macaulay Module, Hans Schoutens

Publications and Research

We show how for a three-dimensional complete local ring in positive characteristic, the existence of an F-invariant, differentiable derivation implies Hochster’s small MCM conjecture. As an application we show that any three-dimensional pseudo-graded ring in positive characteristic satisfies Hochster’s small MCM conjecture.

On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

Dissertations, Theses, and Capstone Projects

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential …

Graded Quivers And B-Branes At Calabi-Yau Singularities, Cyril Closset, Sebastían Franco, Jirui Guo, Azeem Hasan

Publications and Research

A graded quiver with superpotential is a quiver whose arrows are assigned degrees c ∈ {0, 1, · · · , m}, for some integer m ≥ 0, with relations generated by a superpotential of degree m − 1. Ordinary quivers (m = 1) often describe the open string sector of D-brane systems; in particular, they capture the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type IIB string theory, in the guise of 4d N = 1 supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with m = 2 and m …

Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag

Dissertations, Theses, and Capstone Projects

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these …

Higher Cluster Categories And Qft Dualities, Sebastián Franco, Gregg Musiker

Publications and Research

We introduce a unified mathematical framework that elegantly describes minimally supersymmetry gauge theories in even dimensions, ranging from six dimensions to zero dimensions, and their dualities. This approach combines and extends recent developments on graded quivers with potentials, higher Ginzburg algebras, and higher cluster categories (also known as m-cluster categories). Quiver mutations studied in the context of mathematics precisely correspond to the order-(m + 1) dualities of the gauge theories. Our work indicates that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities.

Counting Rational Points, Integral Points, Fields, And Hypersurfaces, Joseph Gunther

Dissertations, Theses, and Capstone Projects

This thesis comes in four parts, which can be read independently of each other.

In the first chapter, we prove a generalization of Poonen's finite field Bertini theorem, and use this to show that the obvious obstruction to embedding a curve in some smooth surface is the only obstruction over perfect fields, extending a result of Altman and Kleiman. We also prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

In the second chapter, for a fixed base curve over a finite field of characteristic at least 5, we …

On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim

Dissertations, Theses, and Capstone Projects

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight monoids of smooth affine spherical varieties, using …