Algebraic Geometry Commons^™

Toward The Salmon Prize: A Computational Certificate For The M5, M6, And M9 Equation Families Of Σ4(P3 × P3 × P3), John Trevino

Theatre Faculty Publications

We present a computational certificate for three families of explicit polyno- mial equations whose zero locus contains the fourth secant variety σ4(P3×P3×P3) inside P63. The three families are: 192 degree-5 Strassen commutation equa- tions (M5); 160 degree-6 equations lifted from the Bates–Oeding generators of σ4(P2 × P2 × P3) via the Landsberg–Manivel–Friedland theorem (M6); and 64 degree-9 Ottaviani 9 × 9 determinant equations (M9). All 416 generators are explicit polynomials in the coordinate ring Z[Zijk | i, j, k ∈ {0, 1, 2, 3}]. We verify by exact integer arithmetic that every generator vanishes on rank-4 test tensors (six independent …

Non-Redundant Rovibrational Hamiltonians By Molien Generating Functions And Gröbner-Basis Reduction, Leandro Ajo

University Libraries Undergraduate Research Award

High-resolution molecular spectroscopy requires an effective Hamiltonian whose operator content is both complete, containing every term allowed by molecular symmetry and nonredundant, free of any algebraically dependent operators that would cause ill-conditioned parameter fits. Traditional derivations based on Van Vleck contact transformations satisfy neither criterion automatically. This paper develops a rigorous, algorithmic pipeline that guarantees both properties simultaneously. Starting from the permutation– inversion (PI) group GPI of a molecule (Longuet-Higgins, 1963), we apply Molien’s theorem (Molien, 1897) to the symplectic normal-coordinate representation to obtain the vibrational generating function Φvib(t); integrate over the Haar measure of SO(3) (Weyl, 1946) to obtain …

A Zariski-Nagata Theorem For Smooth Toric Surfaces, Jordan Vincent Barrett

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the n^th symbolic power of a radical ideal I in a polynomial ring over a field in terms of the n^th ordinary powers of the maximal ideals in [affine variety]_max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with …

Navigating Geometry With Interactive Lessons: A Collection Of Resources In Geogebra That Address Frequently Held Misconceptions, Sara Vance

Honors Program: Senior Projects (Public)

Nearly every high school student in the United States is required to take a geometry class, and for many, it’s unlike any math class they’ve taken before. Additionally, there are certain topics with which past and present geometry students alike tend to struggle. This project was created to address these issues; my goal is that these materials will help students better understand these topics and clear up their frequently held misconceptions about geometry. This project was supported by interviews I conducted with veteran high school geometry teachers with over 75 years of teaching math between them.

This collection of interactive …

A Bibliographic And Topic Modeling Analysis Of The P-Adic Theory Literature Using Latent Dirichlet Allocation, Humberto Llinás, Ismael Gutiérrez, Anselmo Torresblanca, Javier De La Hoz, Brian Llinás

Computer Science Faculty Publications

P-adic analysis, introduced by Kurt Hensel in the early 20th century, has developed into a fundamental area of mathematical research with broad applications in number theory, algebraic geometry, and mathematical physics. This study aims to examine the thematic evolution and scholarly impact of p-adic research through a comprehensive topic modeling and bibliometric analysis. Using classical bibliometric techniques (e.g., performance analysis, co-authorship, and co-citation networks) combined with Latent Dirichlet Allocation (LDA), we analyzed 7388 peer-reviewed documents published between 1965 and 2024. The computational workflow was conducted using R (version 4.4.1) and VOSviewer (version 1.6.20), which enabled the identification of 20 distinct …

A Question Of Transparency, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Question 2: (a) How "long" (in meters) might such a rain shower or fog bank be?

Hint: Suppose you are looking along a linear stack of S cubes with 1-m sides (through the rain or fog). Each cube contains N drops. If p is the visibility (i.e., the fraction of …

Model-Free Organization Of Patient Reported Outcomes Data: Geometrical Rep-Resentation Of The Modified Compartmen-Talization Method, Manasi Sheth, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

There is a recent advancement in the field of mathematics and statistics to understand the geometry or connectedness of the data due to the massive amounts of data being generated. The data provided for analyses are usually very large and need to be organized and minimized in order to make it more useful and meaningful. In biostatistics or medical field, it is important for patients to have access to high-quality, safe and effective and/ or efficacious medical products. It is quite necessary to ascertain that the patients and their care-partners stay at the center of the regulatory decision-making process. In …

A Question Of Transparency: Solutions For Fermi Questions, March 2025, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Solution to Question 1: N = VI(πd³/6) = 6V/πd³ ≈ 2V/d³.

The cross-sectional area A of each drop is πd²/4 ≈ 3d²/4, so the total area blocked off (assuming no overlapping drops—so this is an upper bound) is NA ≈ 1.5V/d, so the area blocked off is inversely proportional to …

Deterministic Global 3d Fractal Cloud Model For Synthetic Scene Generation, Aaron M. Schinder, Shannon R. Young, Bryan J. Steward, Michael L. Dexter, Andrew Kondrath, Stephen Hinton, Ricardo Davila

Faculty Publications

This paper describes the creation of a fast, deterministic, 3D fractal cloud renderer for the AFIT Sensor and Scene Emulation Tool (ASSET). The renderer generates 3D clouds by ray marching through a volume and sampling the level-set of a fractal function. The fractal function is distorted by a displacement map, which is generated using horizontal wind data from a Global Forecast System (GFS) weather file. The vertical windspeed and relative humidity are used to mask the creation of clouds to match realistic large-scale weather patterns over the Earth. Small-scale detail is provided by the fractal functions which are tuned to …

Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel

Mathematics, Physics, and Computer Science Faculty Articles and Research

Both acoustics and electromagnetism represent measurable fields in terms of dynamical potential fields. Electromagnetic force-fields form a spacetime bivector that is represented by a dynamical energy–momentum 4-vector potential field. Acoustic pressure and velocity fields form an energy–momentum density 4-vector field that is represented by a dynamical action scalar potential field. Surprisingly, standard field theory analyses of spin angular momentum based on these traditional potential representations contradict recent experiments, which motivates a careful reassessment of both theories. We analyze extensions of both theories that use the full geometric structure of spacetime to respect essential symmetries enforced by vacuum wave propagation. The …

Prospective Teachers' Conceptions Of Area, Betsy Mcneal, Sayonita Ghosh Hajra, Michael Battista, Ayse Ozturk

Teaching & Learning Faculty Publications

This study explored 10 prospective teachers’ (PTs’) understanding of the area of a rectangular region using square and non-square rectangular area-units. In an hour-long interview, each PT was first asked to explain how they would find the area of a given rectangular region in terms of a non-square notecard. For several PTs, this task prompted discussion of a square unit defined by the edge of the notecard. In a second task, PTs were presented with a rectangular array of squares and were asked to explain to a fictional child why multiplying length times width does not count the top left …

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 …

Unexpectedness Stratified By Codimension, Frank Zimmitti

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

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 …

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 (x₀, y₀) 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, 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 …

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 …

Bridging The Mathematics Gap Through The Use Of Mathematical Apps, Ma. Louise Antonette N. De Las Penas, Debbie Marie B. Verzosa, Maria Alva Q. Aberin, Agnes D. Garciano, Juan Carlo F. Mallari, Jumela F. Sarmiento, Mark Anthony C. Tolentino

Mathematics Faculty Publications

During the COVID-19 pandemic, school campuses worldwide were forced to close, and students had to learn primarily from home. This sudden disruption is estimated to have caused significant learning loss among learners. This paper reports the use of mathematical applications (apps) to bridge the mathematical learning gaps in Grades 1 to 11 in the Philippines after the pandemic, as part of a project funded by a national government agency. The apps include those that strengthen foundational concepts in number and fraction sense in grade school mathematics, develop proving skills in geometry, promote mastery in algebraic and trigonometry through drill and …

More Properties Of Optimal Polynomial Approximants In Hardy Spaces, Raymond Cheng, Christopher Felder

Mathematics & Statistics Faculty Publications

We study optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, H^p (1 < p < ∞). For fixed f ∈ H^p and n ∈ N, the OPA of degree n associated to f is the polynomial which minimizes the quantity ∥qf −1∥_p over all complex polynomials q of degree less than or equal to n. We begin with some examples which illustrate, when p ≠ 2, how the Banach space geometry makes the above minimization problem interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions …

Another Angle On Perspective: Solutions For Fermi Questions, May 2023, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Ellipses, Leaves, And Solar Crescents, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

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 …

Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, Bill Hemphill

STEM Guitar Project’s BBT Acoustic Kit

This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are
(a) a one-piece base for the standard kit's (Martin-style) bracing,
(b) 277 Ladder-style bracing, and
(c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.

The CADD data set for each of the three (3) top bracing designs includes …

The Local Cohomology Spectral Sequence For Topological Modular Forms, Robert Bruner, John Greenlees, John Rognes

Mathematics Faculty Research Publications

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald–Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

Bbt Side Mold Assy, Bill Hemphill

STEM Guitar Project’s BBT Acoustic Kit

This electronic document file set covers the design and fabrication information of the ETSU Guitar Building Project’s BBT (OM-sized) Side Mold Assy for use with the STEM Guitar Project’s standard acoustic guitar kit. The extended 'as built' data set contains an overview file and companion video, the 'parent' CADD drawing, CADD data for laser etching and cutting a drill &/or layout template, CADD drawings in AutoCAD .DWG and .DXF R12 formats of the centerline tool paths for creating the mold assembly pieces on an AXYZ CNC router, and support documentation for CAM applications including router bit specifications, feeds, speed, multi-pass …

The Adams Spectral Sequence For The Image-Of-J Spectrum, Robert R. Bruner, John Rognes

Mathematics Faculty Research Publications

We show that if we factor the long exact sequence in cohomology of a cofiber sequence of spectra into short exact sequences, then the d_2-differential in the Adams spectral sequence of any one term is related in a precise way to Yoneda composition with the 2-extension given by the complementary terms in the long exact sequence. We use this to give a complete analysis of the Adams spectral sequence for the connective image-of-J spectrum, finishing a calculation that was begun by D. Davis [Bol. Soc. Mat. Mexicana (2) 20 (1975), pp. 6–11].

Lines On A Smooth Projective Surface, Jordan Demoura

Mathematics & Computer Science Student Scholarship

Jordan DeMoura ’22
Major: Mathematics
Faculty Mentor: Dr. Su-Jeong Kang, Math

This research is to investigate lines on a smooth projective surface. A quadric surface contains two families of planes that provide a ruling of the surface. A cubic surface contains twenty-seven lines, and we provide a complete description of these lines for a Fermat cubic surface. Furthermore, under the Plucker embedding, we show that each family of the lines on a quadric surface corresponds to plane conic curves lying on complementary planes in the projective space of dimension five.

The Cohomology Of The Mod 2 Steenrod Algebra, Robert R. Bruner, John Rognes

Open Data at Wayne State

The dataset contains a minimal resolution of the mod 2 Steenrod algebra in the range 0 < = s < = 128, 0 < = t < = 200, together with chain maps for each cocycle in that range and for the squaring operation Sq^0 in the cohomology of the Steenrod algebra. The included document CohomA2.pdf explains the contents and usage of the dataset in detail (also available as supplemental material in this record).

Dataset is also available at the NIRD Research Data Archive, https://doi.org/10.11582/2021.00077; Data Description also available at arXiv.org, https://doi.org/10.48550/arXiv.2109.13117.

Nidus Idearum. Scilogs, X: Via Neutrosophica, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this tenth book of scilogs – called via neutrosophica (the neutrosophic way) –, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY, but also MULTISPACE, with miscellaneous addition of topics on Physics, Mathematics, or Sociology – email messages to research colleagues, or replies, notes about authors, articles, or books, spontaneous ideas, and so on.

Exchanging ideas with A. Elhassouny, Junhui Kim, Jeong Gon Lee, Kul Hur, Hojjatollah Farahani, W. B. Vasantha Kandasamy, Said Broumi, Mumtaz Ali, Mohamed Abdel-Basset, Ozen Ozer, Madad Khan, Gheorghe Săvoiu, John Mordeson, Adesina Agboola, Waldyr Rodrigues, Ajay Sharma, Stephen …

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 …

ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka

Mathematics Faculty Research Publications

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.