Mathematics Commons^™

Lefschetz Properties Of Some Codimension Three Artinian Gorenstein Algebras, Nancy Abdallah, Nasrin Altafi, Anthony Iarrobino, Alexandra Seceleanu, Joachim Yaméogo

Department of Mathematics: Faculty Publications

Codimension two Artinian algebras A have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of …

Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper

Honors Program: Senior Projects (Public)

Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …

On The Asymptotic Behavior Of Solutions To A Structure Acoustics Model, Baowei Feng, Yanqiu Guo, Mohammad A. Rammaha

Department of Mathematics: Faculty Publications

This article concerns the long term behavior of solutions to a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain Ω ⊂ R³ which is coupled with a Berger plate equation acting on a flat portion of the boundary of . The system is influenced by several competing forces, in particular a source term acting on the wave equation which is allowed to have a supercritical exponent.

Our results build upon those obtained by Becklin and Rammaha [8]. With some re- strictions on the parameters in the system and with careful analysis involving …

The Smooth 4-Genus Of (The Rest Of) The Prime Knots Through 12 Crossings, Mark Brittenham, Susan Hermiller

Department of Mathematics: Faculty Publications

We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.

Regularity Criteria For The Kuramoto-Sivashinsky Equation In Dimensions Two And Three, Adam Larios, Mohammad Mahabubur Rahman, Kazuo Yamazaki

Department of Mathematics: Faculty Publications

We propose and prove several regularity criteria for the 2D and 3D Kuramoto-Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution ∅, the vector solution u ≜ ∇∅, as well as the divergence div(u) = Δ∅, and each component of u and ∇u. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.

Level And Gorenstein Projective Dimension, Laila Awadalla, Thomas Marley

Department of Mathematics: Faculty Publications

We investigate the relationship between the level of a bounded complex over a commutative ring with respect to the class of Gorenstein projective modules and other invariants of the complex or ring, such as projective dimension, Gorenstein projective dimension, and Krull dimension. The results build upon work done by J. Christensen [7], H. Altmann et al. [1], and Avramov et al. [4] for levels with respect to the class of finitely generated projective modules.

The concept of level in a triangulated category, first defined by Avramov, Buch- weitz, Iyengar, and Miller [4], is a measure of how many mapping cones …

The Phase Transition Of Discrepancy In Random Hypergraphs, Calum Macrury, Tomáš Masarík, Leilani Pai, Xavier Perez Gimenez

Department of Mathematics: Faculty Publications

Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on n vertices and m edges. In the first (edge-independent) model, a random hypergraph H₁ is constructed by fixing a parameter p and allowing each of the n vertices to join each of the m edges independently with probability p. In the parameter range in which pn ⟶ ∞ and pm ⟶ ∞, we show that with high probability (w.h.p.) H₁ has discrepancy at least Ω(2^-n/m √pn) when m = O(n …

Cohomological Blow Ups Of Graded Artinian Gorenstein Algebras Along Surjective Maps, Anthony Iarrobino, Pedro Macias Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are …

Bernstein-Sato Polynomials In Commutative Algebra, Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez-Betancourt

Department of Mathematics: Faculty Publications

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.

Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo

Department of Mathematics: Faculty Publications

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo

Department of Mathematics: Faculty Publications

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

Results On Nonorientable Surfaces For Knots And 2-Knots, Vincent Longo

Department of Mathematics: Dissertations, Theses, and Student Research

A classical knot is a smooth embedding of the circle into the 3-sphere. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces. First, we generalize a result of Satoh about connected sums of projective planes and twist spun knots. Specifically, we will show that for any odd natural n, the connected sum of the n-twist …

Bootstrap Percolation On Random Geometric Graphs, Alyssa Whittemore

Department of Mathematics: Dissertations, Theses, and Student Research

Bootstrap Percolation is a discrete-time process that models the spread of information or disease across the vertex set of a graph. We consider the following version of this process:

Initially, each vertex of the graph is set active with probability p or inactive otherwise. Then, at each time step, every inactive vertex with at least k active neighbors becomes active. Active vertices will always remain active. The process ends when it reaches a stationary state. If all the vertices eventually become active, then we say we achieve percolation.

This process has been widely studied on many families of graphs, deterministic …

Free Complexes Over The Exterior Algebra With Small Homology, Erica Hopkins

Department of Mathematics: Dissertations, Theses, and Student Research

Let M be a graded module over a standard graded polynomial ring S. The Total Rank Conjecture by Avramov-Buchweitz predicts the total Betti number of M should be at least the total Betti number of the residue field. Walker proved this is indeed true in a large number of cases. One could then try to push this result further by generalizing this conjecture to finite free complexes which is known as the Generalized Total Rank Conjecture. However, Iyengar and Walker constructed examples to show this generalized conjecture is not always true.

In this thesis, we investigate other counterexamples of …

A Combinatorial Formula For Kazhdan–Lusztig Polynomials Of Sparse Paving Matroids, George Nasr

Department of Mathematics: Dissertations, Theses, and Student Research

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan–Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as for uniform matroids, our formula has a nice combinatorial interpretation.

Advisors: Kyungyong Lee and Jamie Radcliffe

Near-Optimal Learning Of Tree-Structured Distributions By Chow-Liu, Arnab Bhattacharyya, Sutanu Gayen, Eric Price, N. V. Vinodchandran

Department of Mathematics: Faculty Publications

We provide finite sample guarantees for the classical Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) to learn a tree-structured graphical model of a distribution. For a distribution P on Σⁿ and a tree T on n nodes, we say T is an ε-approximate tree for P if there is a T-structured distribution Q such that D(P || Q) is at most ε more than the best possible tree-structured distribution for P. We show that if P itself is tree-structured, then the Chow-Liu algorithm with the plug-in estimator for mutual information with eO (|Σ| …

Modernization Of Scienttific Mathematics Formula In Technology, Iwasan D. Kejawa Ed.D, Prof. Iwasan D. Kejawa Ed.D

Department of Mathematics: Faculty Publications

Abstract
Is it true that we solve problem using techniques in form of formula? Mathematical formulas can be derived through thinking of a problem or situation. Research has shown that we can create formulas by applying theoretical, technical, and applied knowledge. The knowledge derives from brainstorming and actual experience can be represented by formulas. It is intended that this research article is geared by an audience of average knowledge level of solving mathematics and scientific intricacies. This work details an introductory level of simple, at times complex problems in a mathematical epidermis and computability and solvability in a Computer Science. …

N-Fold Matrix Factorizations, Eric Hopkins

Department of Mathematics: Dissertations, Theses, and Student Research

The study of matrix factorizations began when they were introduced by Eisenbud; they have since been an important topic in commutative algebra. Results by Eisenbud, Buchweitz, and Yoshino relate matrix factorizations to maximal Cohen-Macaulay modules over hypersurface rings. There are many important properties of the category of matrix factorizations, as well as tensor product and hom constructions. More recently, Backelin, Herzog, Sanders, and Ulrich used a generalization of matrix factorizations -- so called N-fold matrix factorizations -- to construct Ulrich modules over arbitrary hypersurface rings. In this dissertation we build up the theory of N-fold matrix factorizations, proving analogues of …

Chudnovsky's Conjecture And The Stable Harbourne-Huneke Containment, Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên

Department of Mathematics: Faculty Publications

We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.

Demailly's Conjecture And The Containment Problem, Sankhaneel Bisui, Eloisa Grifo, Huy Tài Hà, Thái Thành Nguyên

Department of Mathematics: Faculty Publications

We investigate Demailly’s Conjecture for a general set of sufficiently many points. Demailly’s Conjecture generalizes Chudnovsky’s Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly’s bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.

Constructing Non-Proxy Small Test Modules For The Complete Intersection Property, Benjamin Briggs, Eloísa Grifo, Josh Pollitz

Department of Mathematics: Faculty Publications

A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a …

Frobenius And Homological Dimensions Of Complexes, Taran Funk

Department of Mathematics: Dissertations, Theses, and Student Research

Much work has been done showing how one can use a commutative Noetherian local ring R of prime characteristic, viewed as algebra over itself via the Frobenius endomorphism, as a test for flatness or projectivity of a finitely generated module M over R. Work on this dates back to the famous results of Peskine and Szpiro and also that of Kunz. Here I discuss what work has been done to push this theory into modules which are not necessarily finitely generated, and display my work done to weaken the assumptions needed to obtain these results.

Adviser: Tom Marley

Free Semigroupoid Algebras From Categories Of Paths, Juliana Bukoski

Department of Mathematics: Dissertations, Theses, and Student Research

Given a directed graph G, we can define a Hilbert space H_G with basis indexed by the path space of the graph, then represent the vertices of the graph as projections on H_G and the edges of the graph as partial isometries on H_G. The weak operator topology closed algebra generated by these projections and partial isometries is called the free semigroupoid algebra for G. Kribs and Power showed that these algebras are reflexive, and that they are semisimple if and only if each path in the graph lies on a cycle. We extend …

Gauge-Invariant Uniqueness And Reductions Of Ordered Groups, Robert Huben

Department of Mathematics: Dissertations, Theses, and Student Research

A reduction φ of an ordered group (G,P) to another ordered group is an order homomorphism which maps each interval [1, p] bijectively onto [1, φ(p)]. We show that if (G,P) is weakly quasi-lattice ordered and reduces to an amenable ordered group, then there is a gauge-invariant uniqueness theorem for P -graph algebras. We also consider the class of ordered groups which reduce to an amenable ordered group, and show this class contains all amenable ordered groups and is closed under direct products, free products, and hereditary subgroups.

Adviser: Mark Brittenham and David Pitts

On The Bounded Negativity Conjecture And Singular Plane Curves, Alexandru Dimca, Brian Harbourne, Gabriel Sticlaru

Department of Mathematics: Faculty Publications

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the H-constant of a rational curve C with at most 9 singular points satisfies H(C) > -2 regardless of the characteristic.

Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu

Department of Mathematics: Faculty Publications

We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.

Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu

Department of Mathematics: Faculty Publications

We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.

A Stable Version Of Harbourne's Conjecture And The Containment Problem For Space Monomial Curves, Eloísa Grifo

Department of Mathematics: Faculty Publications

The symbolic powers I(ⁿ⁾ of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(ⁿ⁾⊆Im holds. When I is an ideal of height 2 in a regular ring, I(³⁾⊆I2 may fail, but we …

Diagrams Of ⋆-Trisections, José Román Aranda, Jesse Moeller

Department of Mathematics: Faculty Publications

In this note we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we describe how to perform 1-surgery on closed trisections. The insight gained from this description leads us to the classification of an infinite family of genus three trisections. We include an appendix where we extend two classic results for relative trisections for the case when the trisection surface is closed.

Trisections Of Flat Surface Bundles Over Surfaces, Marla Williams

Department of Mathematics: Dissertations, Theses, and Student Research

A trisection of a smooth 4-manifold is a decomposition into three simple pieces with nice intersection properties. Work by Gay and Kirby shows that every smooth, connected, orientable 4-manifold can be trisected. Natural problems in trisection theory are to exhibit trisections of certain classes of 4-manifolds and to determine the minimal trisection genus of a particular 4-manifold.

Let $\Sigma_g$ denote the closed, connected, orientable surface of genus $g$. In this thesis, we show that the direct product $\Sigma_g\times\Sigma_h$ has a $((2g+1)(2h+1)+1;2g+2h)$-trisection, and that these parameters are minimal. We provide a description of the trisection, and an algorithm to generate a …