The Conference will be virtual this year due to the high number of Covid cases and safety concerns. Check the schedule. This year, we also have satellite conferences, please visit here for more information.

GRAD SCHOOL Q&A

This panel is targeted towards our undergraduate audience who are interested in applying to graduate school.

Organizers: Dr. Kalina Mincheva, Thai Nguyen, Vaishavi Sharma

Undergraduate organizers: Isabelle Haines, Maya Ross

Aalliyah Celestine

Permutations of k-color Parking Functions

With this project, we explore a new variant of the classical combinatorial objects known as parking functions, first introduced by Konheim and Weiss. This variant, called exact k-color parking functions, are tuples describing the parking of M cars on a street with M parking spots. We assign each car one of k ordered colors, while m_i denotes the number of cars of color i, hence M = m_i + m_2 + ... + m_k. The set of exact k-color parking functions consist of tuples A = (m_1; P_2,P_3,...,P_k) where P_i = (p_1^{(i)}, p_2^{(i)},..., p_{m_i}^{(i)}). We begin by establishing that every k-color parking function results in a unique parking configuration C, which indicates the ordering of the parked cars. We also show that given a k-CPF, all permutations of the preference list(s) P_i, generate a disjoint subset of k-color parking functions corresponding to a unique parking configuration.

Andrew Campbell

Heavy-Tailed Elliptic Random Matrices

An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$ is a random variable independent of every other entry except possibly $X_{ji}$. Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with independent entries. When the entries of an elliptic random matrix have mean zero and unit variance, the empirical spectral distribution is known to converge to the uniform distribution on the interior of an ellipse determined by the covariance of the mirrored entries.

In this talk we consider elliptic random matrices whose entries fail to have two finite moments, but are in the domain of attraction of an $\alpha$-stable random variable, for $0<\alpha<2$. We will discuss the heavy-tailed analogue of covariance for the mirrored entries which gives the empirical spectral measure converging, in probability, to a deterministic limit. This is joint work with Sean O’Rourke.

Annie Giokas

The Noetherian property of invariant rings

A Noetherian ring is a ring that has the ascending chain condition, which means that any ascending chain of ideals of the ring must stabilize after a finite number of steps. The concept of Noetherian rings came to be after the German mathematician, Emmy Noether, discovered that primary decomposition of ideals is a consequence of the ascending chain condition in 1921. It is known that for a graded ring over a field, the Noetherian property is equivalent to the ring being finitely generated over the field. Noether proved in 1926 that the ring of invariants for the action of a finite group via k-algebra automorphisms of finitely generated algebras over a field k are in fact, Noetherian. In a related direction, a Noetherian ring containing the field of rational numbers, will also have a Noetherian invariant ring under a finite group action. The proof depends on the ring containing a characteristic 0 field, because it uses an especially useful formula, the Reynold's operator, that involves the inverse of the order of the given finite group to define the map between the ring and its invariant ring that turns out to be a projection. This however is not necessarily true in other cases. We constructed a class of rings of characteristic p for each prime integer p such that each ring in the class is Noetherian with a finite group G acting on it such that the ring of invariants under this group action is not Noetherian. This class of rings is generalized from the characteristic 2 counterexample due to Nagarajan, in 1968.

Benjamin Stager

Modeling sustained transmission of Wolbachia among Anopheles: Implications for Malaria control in Haiti

Our mathematical model suggests that releasing mosquitoes infected wAlbB Wolbachia bacteria could eliminate malaria in Haiti. Preliminary studies of Anopheles (An.) mosquitoes infected with wAlbB Wolbachia bacteria show that infected mosquitoes are less capable of spreading malaria. The infection induces a cytoplasmic incompatibility that disrupts the infection cycle through population suppression, inhibiting within-vector replication of the Plasmodium falciparum parasite, and reducing vector competence. \W~ is widespread among arthropods, and there are ongoing trials for sustaining wild populations of \W-infected Aedes aegypti to prevent the spread of Dengue fever. We create and analyze a model to evaluate different approaches for maintaining wAlbB infection within An. albimanus mosquitoes and apply it to assess its potential as a malaria control strategy in Haiti.

Our nine non-linear differential equation model divides the mosquito population by sex, infection status, pregnancy status, and it includes an aquatic stage. The model involves male and female mosquitoes at different life stages, including egg, larva, and adult. It also accounts for key biological effects, such as the vertical transmission of \W, where the offspring of infected female mosquitoes are infected, and cytoplasmic incompatibility, which effectively sterilizes uninfected female mosquitoes when they mate with infected male mosquitoes. We derive and interpret relevant dimensionless numbers, including the basic reproductive number. We then show that the system presents a backward bifurcation, consisting of the disease-free, high-infection, and a threshold endemic equilibrium (corresponds to the minimal infection needs to be released). We use sensitivity analysis to rank the relative importance of the epidemiological parameters in Haiti during the rainy season.

The model simulates the combination of vector control strategies of larviciding, ultra-low volume spraying, thermal fogging, and habitat manipulation. We simulate the integrated intervention strategies, where traditional vector controls reduce the wild adult mosquitoes before releasing the \W-infected mosquitoes. Specifically, we compare the effectiveness of targeting the larvae stage (larviciding) and adult mosquitoes (thermal fogging) before releasing the infected mosquitoes. Preliminary findings indicate that the use of results in the fastest establishment of endemic wAlbB transmission.

Daniel Koizumi

Braid Monodromy Groups and Curve Complements

An important problem in low-dimensional topology is the calculation of fundamental groups of a space. For this project, we calculate the fundamental group for an algebraic curve complement. Such a computation can be done by instead looking at polynomial braid monodromy groups, which are objects of algebraic geometry. Aided with a computer visualization in Julia, we compute one such fundamental group for the curve complement defined by polynomial p(x,y)=y^2-x.

Daniel Qin

On Mutually Unbiased Bases and Hadamard Matrices

The problem of finding maximal sets of mutually unbiased bases in arbitrary dimensions is fairly young (61 or so years old), despite being related to a wide range of ``older'' objects from finite projective planes to unitary operators. Of these equivalent objects, our discussion will focus on the standard constructions of mutually unbiased bases by unitary operators and complex Hadamard matrices. We aim to understand why these existing constructions work in prime power dimensions (and fail in non-prime dimensions) and fail in non-prime power dimensions--in particular, in the first composite dimension, six. We will wrap up by working a few explicit computations and suggesting some other methods to explore to progress along the existence problem.

Ghanshyam Bhatt

Construction of Mercedes-Benz frames in Rm

A set of lines through the origin in an Euclidean space is equiangular when any pair of lines from this set intersects with each other at a common angle. This set forms an equiangular tight frame if this angle attains its value given by the Welch bound. The Mercedes-Benz frames in $\Bbb R^m,$ which are special cases of equiangular tight frames, are known to exist. They are presented here by a direct constructive method unlike previously known induction methods or the use of signature matrices. Being full spark frames, they are simplices in $\R^m$ too. The constructive method presented thus constructs simplices in any dimension.

Govinda Pageni

Study of Three Systems of linear and non-linear Caputo Fractional Differential Equations with initial conditions and Applications

In this talk, We shall provide an analytical method to solve three linear coupled systems of Caputo fractional differential equations with fractional initial conditions. Since the Mittag-Leffler function doesn't satisfy all the properties of the exponential function, we cannot use the integer order methods. Here we have used an efficient and convenient method, called the Laplace transform method, to solve the three systems of linear Caputo fractional differential equations with fractional initial conditions when the order of the fractional derivative is q and 0 < q <1. In addition, the Laplace-Adomian Decomposition Method allows us to obtain an approximation of the non-linear SIR epidemic model of fractional order q. The method yields integer results as a special case. Our method also works for scalar linear sequential Caputo fractional differential equations of order $nq$, since it can be reduced to n systems of q^th order linear Caputo fractional differential equations with initial conditions.

José São João

A representation of finite semidistributive lattices

Lattices appear all over mathematics: from logic to collections of substructures to geometry. An important result in lattice theory is Birkhoff’s representation of finite distributive lattices via posets from the 1930’s. Priestley gave representations of arbitrary distributive lattices which generalizes Birkhoff’s representation, using topology. Some recent research has been concerned with representing general finite and infinite lattices. It has been found that there is a one-to-one correspondence between finite lattices and a special class of digraphs called TiRS graphs. In this talk I will present on some results we found on characterizing the TiRS graphs of semidistributive lattices. Semidistributive lattices appear in the study of free lattices, convex geometries and Coxeter groups. This is joint work with Andrew Craig (University of Johannesburg, South Africa) and Miroslav Haviar (Matej Bel University, Slovakia).

Kriti Goel

Affine semigroups of maximal projective dimension

A submonoid of N^d is of maximal projective dimension (MPD) if the associated affine semigroup ring has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of MPD-semigroups in N^d. We prove that under suitable conditions these semigroups satisfy the generalized Wilf's conjecture.

Liza Jacoby

Non-edge-to-edge Tilings, Branched Covers, and Star Polyhedra

A complete classification of edge-to-edge tilings of the 2-sphere by regular polygons of three or more sides is given by the Platonic solids, the Archimedean solids, the prisms and antiprisms, and twenty-five of the ninety-two Johnson solids, beginning in 350 BCE and culminating in 1966. We complete the classification of non-edge-to-edge tilings of the 2-sphere using regular polygons of three or more sides, introducing 33 new families of non-edge-to-edge tilings. We then extend these results to branched coverings of the 2-sphere and determine tilings by regular polygons of three or more sides of the double, triple, n-tuple branched cover of the 2-sphere with two branch points. We introduce new infinite families of non-uniform star polyhedra to prove the existence of certain tilings.

Mallory Dolorfino, Daniel Qin, Lucas Rizzolo

On Minimal Generating of Invariants for Finite Abelian Groups

Given some arbitrary polynomial ring, an invariant polynomial is a polynomial that is unchanged by the action of a group $G$. We investigate the ring of invariant polynomials under the action of some abelian groups with the goal of finding generators for this ring. When considering an abelian group, we can always find a basis such that the action is diagonal, so there exists monomial generators $m_i$ for the invariant ring. By Noether's degree bound, the minimal set of generating monomials $\langle m_1,\ldots,m_k\rangle$ is finite and the degree of each generating monomial $m_i$ is less than $|G|$. Motivated by the previous work of Gandini and Derksen, we present a new approach to find invariants for $\mathbb{Z}_p \oplus \mathbb{Z}_p$ and show that this approach can fail for $\mathbb{Z}_n \oplus \mathbb{Z}_n$ when $n$ is not a prime.

Mehsin Jabel Atteya

Elements Projection of *-Rings

In 1914 in Fraenkel's paper which under the title ""On zero-divisors and the decomposition of ring"" gives the modern definition of the abstract ring appeared while a systematic study of non-commutative rings started in the 20th century while commutative rings have appeared though in a covered way much before, and as many other theories, come back to Fermat's Last Theorem. Ring theory is a showpiece of mathematical unification, bringing together several branches of the subject and creating a powerful machine for the study of problems of considerable historical and mathematical importance. Rings with derivations are not the kind of subject that undergoes tremendous revolutions. However, this has been studied by many algebraists in the last years, especially the relationships between derivations and the structure of rings. A map d: R → R is called a derivation if d (x + y) = d(x) + d(y) and d(xy) = d(x)y + xd(y) for all x, y ∈ R. An additive mapping *: R → R is called an involution on R if (xy)* = y*x* and (x*) * = x for all x, y ∈ R. A ring R equipped with an involution * is called a ring with an involution * or a *-ring. An element x in a ring with involution is said to be Hermitian if x* = x and skew-Hermitian if x* = -x. An element e of a *-ring is a projection if e = e2 and e = e*.

In this paper, we focus on the action of the element projection on the *-ring. Precisely, we prove the commutativity with other cases of a *-ring that satisfied certain conditions. These results are in the sprite of the well-known theorem of the ring with derivations satisfying certain polynomial constraints.

Mirjeta Pasha

Computational and learning methods for large-scale inverse problems.

Inverse problems are ubiquitous in many fields of science such as engineering, biology, medical imaging, atmospheric science, and geophysics. Three main challenges on obtaining meaningful solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints.

This talk utilizes a combination of tools from applied linear algebra, optimization, parameter estimation, and statistics to overcome computational challenges that arise in data-intensive inverse problems.

In particular, we present a general framework to learn optimal lp and lq norms for Lp-Lq regularization and learn optimal parameters for regularization matrices defined by covariance kernels. Further, we describe some efficient methods for computing solutions with preserved edges to dynamic inverse problems, where both the quantities of interest and the forward operator change at different time instances.

Numerical examples such as tomographic reconstruction, and image deblurring illustrate the performance of the discussed approaches in terms of both accuracy and efficiency.

Vincent Martinez

Parameter Estimation for Dynamical Systems

In this talk, we will discuss a recent algorithm for inferring unknown parameters of dynamical systems based on partial observations that is amenable to robust computational studies, as well as rigorous mathematical analysis, making its study accessible across many levels.

Achouak Bekkai

The Nonexistence of global solutions for a time fractional Schrödinger system with nonlinear memory

We study a time fractional nonlinear Schrödinger system with nonlinear memory, we prove that the problem admit no global weak solution by using the test function method.

Annie Giokas and Daniel Koizumi

Taylor Resolution

Study of (minimal) free resolutions is an important area in commutative algebra because free resolutions provide a method for describing the structure of modules. By introducing simplicial complexes in the context of free resolutions of monomial ideals, we get a combinatorial perspective on the topic, and we manage to construct important simplicial resolutions that resolve monomial ideals. One such resolution is the Taylor resolution, which we will construct. Additionally, we will talk about some corollaries related to Betti numbers and regularity of monomial ideals.

Cormac LaPrete

Household SIR Model: Transmission Rate Heterogeneity

Mathematically modeling SARS-CoV-2 transmission, taking into consideration the different rates of transmission locally (within a household) and globally (between households), to be used to study heterogeneity in transmission, reproductive numbers, and other metrics of disease dynamics in a population. This can also be applied to improve testing, outbreak mitigation, and vaccination strategies for for various population structures.

Daniela Florez

Modeling the transmission of Wolbachia among Anopheles albimanus mosquito populations in Haiti

"Daniela A. Florez (1), Kerlly Bernabe (2) , Benjamin P. Stager (1), Alyssa J. Young (2), Zhuolin Qu (3), James M. Hyman (1).

1 Department of Mathematics, Tulane University, New Orleans, LA, USA

2 School of Public Health and Tropical Medicine, Tulane University, New Orleans, LA, USA

3 Department of Mathematics, University of Texas San Antonio, San Antonio, TX, USA

Anopheles albimanus mosquitoes infected with the wAlB strain of Wolbachia bacterium are less capable of spreading malaria. We develop and analyze an ordinary differential equation model to evaluate the effectiveness of different vector control strategies in establishing a sustained Wolbachia infection among wild Anopheles mosquitoes in Haiti. The model involves both male and female mosquitos at different life stages, including egg, larva, and adult. It also accounts for critical biological effects, such as the vertical transmission of Wolbachia, where the offspring of infected female mosquitoes are infected, and cytoplasmic incompatibility, which effectively sterilizes uninfected female mosquitoes when they mate with infected male mosquitos. We derive and interpret relevant dimensionless numbers, including the basic reproductive number. We then show that the system presents backward bifurcation, consisting of the disease-free, complete-infection, and a threshold endemic equilibrium (corresponds to the minimal infection that needs to be released). We use sensitivity analysis to rank the relative importance of the epidemiological parameters in Haiti during the rainy season. Lastly, we simulate the integrated intervention strategies, where traditional vector controls are implemented to reduce the wild adult mosquitoes before releasing the Wolbachia-infected mosquitoes. Specifically, we compare two vector controls targeting the larvae stage (larviciding) and adult mosquitoes (thermal fogging), respectively, and we assess the efficacy of the controls by measuring the speed of establishment of Wolbachia among the mosquito population.

Emma Coates

Modeling Viral Interactions Between Seasonal Endemic Viruses and Invader Viruses

During the 2009 pandemic of an invading Influenza A virus, data indicated that the spread of the virus may have been interrupted by the seasonal, endemic rhinovirus. In this study, we investigate viral interference and interactions between an endemic virus and invader virus using a modified SIR model. The model simulates the spread of two active viruses with temporary immunity and cross-immunity through a population and assumes nine states based on the current infection and resistance status. The structure of the model assumes that viruses can simultaneously co-infect a host. We replicate the seasonality in transmission by periodically forcing the system’s baseline transmission parameter. A single virus SIR model exhibits damped oscillations to steady state, but periodic forcing produces sustained oscillations that emulate viral seasonality similar to the patterns of rhinoviruses. After periodically forcing the model, we find that increased cross immunity from the endemic virus for the invader virus minimizes the initial pandemic peak of the invader virus. We additionally find an increased severity of the viruses each seasonal cycle with large amplitudes of the periodic forcing terms.

Hunter Shepard

Characteristics of Orthogonal Projection to GPS

The purpose of this research was to find the orthogonal projection in relation to Global Positioning System (GPS). An orthographic projection is a process of prediction in which an object is characterized or surface mapped using parallel lines to project its shape onto a plane. In this research an orthographic projection method was used to project three-dimensional plane in two measurements. Orthographic projection is a type of equal projection, where all the projection lines are symmetrical to the projection plane, bringing about each plane of the scene showing up in relative change on the review surface. The orthographic projection method was used to arrange the framework and the information on a level surface. Numerical figures were utilized to change over to arrange the framework to utilize the GPS. These projections help to navigate the coordinates of the GPS.

Logan Richard

On the Embedding Problem for Polygons, Polyhedra, & Beyond!

Given a polytope P, the problem of characterizing for which smaller polytopes P' contained in P do there exist continuous paths of decreasing polytopes from P to P' is known as the Embedding Problem. The problem has been studied extensively in the literature for the case of triangles, but in motivation of a more general solution for all polytopes, we extend the methods from the literature to outline a strategy for solving the general case, including addressing novel parts of the problem which are not as explicit in the `smaller' cases as they are in the `larger'. Research ongoing.

Naufil Sakran

Real Nash Geometry

The objective of this poster is to give a glimpse of the Nash geometry and its formulation in modern Mathematics. We will present analogous definitions of various Mathematical objects in the Nash geometry and give some examples. Finally, we will state some open questions in the field.

Victor Bankston

Combinatorial structure in the Pauli Group

The Pauli Group is a fundamental tool in the study of quantum error correcting codes, and can be interpreted as collections of measurements. We will show that these measurements have the structure of an association scheme, and so we may use ideas from the combinatorial theory of designs in investigating the structure of Pauli measurements.

Zachary Ramsey

Iterative cycles and the determinant of a certain sparse matrix class

Let f be an integer-valued function and let the matrix, F_n, be n times n. All entries of F_n are either x, 1, or 0 with the matrix being overwhelmingly sparse. All diagonal entries are 1. Entry f_{ij} = x if f(i)=j. All other off-diagonal entries are 0. Derived from the standard definition of determinant, an amended formula for the determinant of F_n is established. This new formula explicitly reveals the relationship between iterative cycles in f and det(F_n). As an illustrative example, a Collatz-like iterative function is examined. It is shown that if and only if det(F) = det(F_272) for all n > 272, then no undiscovered 3x-1 cycles exist.

PLENARY SPEAKERS ABSTRACTS