Unexpectedness Stratified By Codimension, 2023 University of Nebraska-Lincoln

#### Unexpectedness Stratified By Codimension, Frank Zimmitti

*Department of Mathematics: Dissertations, Theses, and Student Research*

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 …

Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, 2023 University of Massachusetts Amherst

#### Intersection Cohomology Of Rank One Local Systems For Arrangement Schubert Varieties, Shuo Lin

*Doctoral Dissertations*

In this thesis we study the intersection cohomology of arrangement Schubert varieties with coefficients in a rank one local system on a hyperplane arrangement complement. We prove that the intersection cohomology can be computed recursively in terms of certain polynomials, if a local system has only $\pm 1$ monodromies. In the case where the hyperplane arrangement is generic central or equivalently the associated matroid is uniform and the local system has only $\pm 1$ monodromies, we prove that the intersection cohomology is a combinatorial invariant. In particular when the hyperplane arrangement is associated to the uniform matroid of rank $n-1$ …

Semi-Infinite Flags And Zastava Spaces, 2023 University of Massachusetts Amherst

#### Semi-Infinite Flags And Zastava Spaces, Andreas Hayash

*Doctoral Dissertations*

ABSTRACT SEMI-INFINITE FLAGS AND ZASTAVA SPACES SEPTEMBER 2023 ANDREAS HAYASH, B.A., HAMPSHIRE COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D, UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Ivan Mirković We give an interpretation of Dennis Gaitsgory’s semi-infinite intersection cohomol- ogy sheaf associated to a semisimple simply-connected algebraic group in terms of finite-dimensional geometry. Specifically, we construct machinery to build factoriza- tion spaces over the Ran space from factorization spaces over the configuration space, and show that under this procedure the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology sheaf in the Beilinson-Drinfeld Grassmannian. We also construct …

Msis-Kadelka: Algebraic Methods For Inferring Discrete Models Of Biological Networks, 2023 Southern Methodist University

#### Msis-Kadelka: Algebraic Methods For Inferring Discrete Models Of Biological Networks, Brandilyn Stigler

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Differential Calculus: From Practice To Theory, 2023 Pennsylvania State University

#### Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers

*Milne Open Textbooks*

*Differential Calculus: From Practice to Theory* covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the …

Interpolation Problems And The Characterization Of The Hilbert Function, 2023 University of Arkansas, Fayetteville

#### Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

*Mathematical Sciences Undergraduate Honors Theses*

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, 2023 Chapman University

#### A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x_{0}, y_{0}) such that A nontrivially intersects curves y − y0 = a(x −x0)^{β} for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …

On The Superabundance Of Singular Varieties In Positive Characteristic, 2023 University of Nebraska-Lincoln

#### On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger

*Department of Mathematics: Dissertations, Theses, and Student Research*

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 …

Invariants Of 3-Braid And 4-Braid Links, 2023 United Arab Emirates University

#### Invariants Of 3-Braid And 4-Braid Links, Mark Essa Sukaiti

*Theses*

In this study, we established a connection between the Chebyshev polynomial of the first kind and the Jones polynomial of generalized weaving knots of type *W*(3,n,m).

Through our analysis, we demonstrated that the coefficients of the Jones polynomial of weaving knots are essentially the Whitney numbers of Lucas lattices which allowed us to find an explicit formula for the Alexander polynomial of weaving knots of type*W*(3,n).

In addition to confirming Fox’s trapezoidal conjecture, we also discussed the zeroes of the Alexander Polynomial of weaving knots of type *W*(3,n) as they relate to Hoste’s conjecture. In addition, …

Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, 2023 University of Tennessee, Knoxville

#### Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein

*Doctoral Dissertations*

Given an ordinary elliptic curve *E* over a field 𝕜 of characteristic *p*, there is an elliptic curve **E** over the Witt vectors **W**(𝕜) 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 𝔽_*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 …

Brill--Noether Theory Via K3 Surfaces, 2023 Dartmouth College

#### Brill--Noether Theory Via K3 Surfaces, Richard Haburcak

*Dartmouth College Ph.D Dissertations*

Brill--Noether theory studies the different projective embeddings that an algebraic curve admits. For a curve with a given projective embedding, we study the question of what other projective embeddings the curve can admit. Our techniques use curves on K3 surfaces. Lazarsfeld's proof of the Gieseker--Petri theorem solidified the role of K3 surfaces in the Brill--Noether theory of curves. In this thesis, we further the study of the Brill--Noether theory of curves on K3 surfaces.

We prove results concerning lifting line bundles from curves to K3 surfaces. Via an analysis of the stability of Lazarsfeld--Mukai bundles, we deduce a bounded version …

Area Activity, 2023 NJIT CSLA

#### Area Activity, Admin Stem For Success

*STEM for Success Showcase*

Lesson plan to teach students about area including an activity plan, activity description, activity video, and additional activity materials

Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, 2023 University of Missouri-St. Louis

#### Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, Adam Wagenknecht

*Dissertations*

Convolutional Neural Networks (CNNs) have become one of the most commonly used tools for performing image classification. Unfortunately, as with most machine learning algorithms, CNNs suffer from a lack of interpretability. CNNs are trained by using a training data set and a loss function to tune a set of parameters known as the layer weights. This tuning process is based on the classical method of gradient descent, but it relies on a strong stochastic component, which makes the weight behavior during training difficult to understand. However, since CNNs are governed largely by the weights that make up each of the …

On The Cohomology Of Solvable Leibniz Algebras, 2023 University of South Alabama

#### On The Cohomology Of Solvable Leibniz Algebras, Jorg Feldvoss, Friedrich Wagemann

*University Faculty and Staff Publications*

This paper is a sequel to J. Feldvoss and F. Wagemann: On Leibniz cohomology, (2021), where we mainly consider semisimple Leibniz algebras. It turns out that the analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to many ideals, and therefore this spectral sequence seems not to be applicable for computing the cohomology of non-semi-simple Leibniz algebras. The main idea of the present paper is to use similar tools as developed by Farnsteiner for Hochschild cohomology (see R. Farnsteiner: On the cohomology of associative algebras and Lie algebras (1987) and R. Farnsteiner: On the vanishing of homology …

Elliptic Curves Over Finite Fields, 2023 Colby College

#### Elliptic Curves Over Finite Fields, Christopher S. Calger

*Honors Theses*

The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of **F**_{q}-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over **Q**. In particular, we approach the topic largely through the use …

Surjectivity Of The Wahl Map On Cubic Graphs, 2023 University of Kentucky

#### Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson

*Theses and Dissertations--Mathematics*

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 …

Toric Bundles As Mori Dream Spaces, 2023 University of Kentucky

#### Toric Bundles As Mori Dream Spaces, Courtney George

*Theses and Dissertations--Mathematics*

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 …

Geometry Of Pipe Dream Complexes, 2023 University of Kentucky

#### Geometry Of Pipe Dream Complexes, Benjamin Reese

*Theses and Dissertations--Mathematics*

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 …

Explorations In Well-Rounded Lattices, 2023 Claremont Colleges

#### Explorations In Well-Rounded Lattices, Tanis Nielsen

*HMC Senior Theses*

Lattices are discrete subgroups of Euclidean spaces. Analogously to vector spaces, they can be described as spans of collections of linearly independent vectors, but with integer (instead of real) coefficients. Lattices have many fascinating geometric properties and numerous applications, and lattice theory is a rich and active field of theoretical work. In this thesis, we present an introduction to the theory of Euclidean lattices, along with an overview of some major unsolved problems, such as sphere packing. We then describe several more specialized topics, including prior work on well-rounded ideal lattices and some preliminary results on the study of planar …

Studying Extended Sets From Young Tableaux, 2022 Butler University

#### Studying Extended Sets From Young Tableaux, Eric Nofziger

*Rose-Hulman Undergraduate Mathematics Journal*

Young tableaux are combinatorial objects related to the partitions of an integer and have various applications in representation theory. They are particularly useful in the study of the fibers arising from the Springer resolution. In recent work of Graham-Precup-Russell, an association has been made between a given row-strict tableau and three disjoint subsets of {1,2,...,n}. These subsets are then used in the study of extended Springer fibers, so we call them extended sets. In this project, we use combinatorial techniques to classify which of these extended sets correlate to a valid row-strict or standard tableau and give bounds on the …