Organizers: Padi Fuster, Robyn Brooks, Swati Patel

This conference is funded by a grant through the Institute of Advanced Study, School of Mathematics through the Women and Mathematics program and Lisa Simonyi, as well as by the Tulane University Math Department, The School of Science and Engineering, The Tulane University Graduate Studies Student Association, The Tulane University Center for Engaged Learning and Teaching, the Tulane Honors Program, the Tulane SIAM Chapter and the Tulane AMS Chapter.

The Conference will take place at Tulane University: Dinwiddie Hall 102&103 and Mussafer Hall. Check the schedule and the map for locations.

Abbe Herzig: Graduate education in the mathematical sciences: Who, what, where, when, why?

Kristi VanDusen: The Mystery of the Primes

Keisha Cook: Stochastic modeling in cellular internalization

Elliot Hill: Getting to the bottom of it: a friendly introduction to mathematical optimization

Evangelos Nastas: On the Distribution Estimation of Prime Numbers

Selvi Kara: Weighted Stanley-Reisner Correspondence

Michael Joyce: Parameterizing Paths Between Matchings

Erin Griffin: Gradient Bach Flow on 4-dimensional Homogeneous Manifolds

Aram Bingam: Ternary arithmetic, factorization, and the class number one problem

Riley Juenemann: A First-Pass Statistical Dashboard for Categorizing Diverse Particle Movement Patterns

Leah Kaisler: Dynamics of a two-strain cholera model with environmental component

Susan Rogowski: Modeling the effects of Taxol and Carboplatin on Breast Cancer Cells

Rainey J Lyons: Biological Modeling with Measures

Victor Bankston: No Three in a Line

Vaishavi Sharma: The Valuation of Polynomial Sequences

Yusuf Afolabi: Basic Results Related to Sequential Caputo Fractional Differential Operator

Benjamin Cooper Boniece: On optimal wavelet-based thresholding for integrated variance estimation

Yusuf Afolabi

Basic Results Related to Sequential Caputo Fractional Differential Operator.

The aim of this talk is to compare the sequential and the "traditional" Caputo fractional differential operators, and to highlight some salient features of the former.

Victor Bankston

No Three in a Line

We will discuss a variant of the no-three-in-a-line problem over the fields Z_p. What is the maximum number of points in Z_p x Z_p with no three in a line?

Aram Bingam

Ternary arithmetic, factorization, and the class number one problem

Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of ‘3-primality’ – primality with respect to ternary multiplication – emerges, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields Q(√-n), n>0, with odd discriminant and whose ring of integers admits unique factorization. We also present algorithms for determining representations of numbers as ternary products, as well as related algorithms for usual primality testing and integer factorization.

Benjamin Cooper Boniece

On optimal wavelet-based thresholding for integrated variance estimation

The problem of disentangling jumps from otherwise continuous path behavior of a stochastic process is an important problem in financial econometrics, made mathematically interesting by the fact that in all practical applications, only a discretely observed path is available.

Wavelets have long been recognized as powerful tools for the analysis of functions over multiple scales, and have been used in several contexts to detect jumps by exploiting shrinkage effects and thresholding of wavelet coefficients. A celebrated “universal threshold” of Donoho and Johnstone has become a gold standard following their seminal works on wavelet shrinkage methods in the 90s, and in recent years this threshold has been used to estimate jumps and important quantity underpinning many semimartingale models called the integrated variance.

In this talk, we present a new wavelet-based method and optimality characterization of the universal threshold for integrated variance estimation for a class of finite jump activity semimartingales, and discuss some extensions to an infinite activity case. This is joint work with José E. Figueroa-López (Washington University in St. Louis).

Keisha Cook

Stochastic modeling in cellular internalization

Live cell imaging and single particle tracking techniques have become increasingly popular amongst the mathematical biology community. We study endocytosis, the cellular internalization and transport of bioparticles. We are specifically interested in titanium dioxide nanoparticles (TiO$_2$) in human lung cells (A549), observed locally in enlarged lysosomes. Using fluorescence microscopy, we track, classify, and analyze the movement in the cells. Single particle tracking techniques allow us to collect data in order to develop statistical methods for analyzing the movement in the cells. We classify the movement as active, diffusive, sub-diffusive, or stuck. The question becomes, how does the change in the size of the lysosomes alter transport type? The larger the lysosomes, the more obstacles present themselves inside the cell. We want to ensure that the classification of active movement is really active, given short path trajectories. We employ stochastic analysis techniques, including Bayesian inference methods, to analyze and determine the best method to classify active transport.

Erin Griffin

Gradient Bach Flow on 4-dimensional Homogeneous Manifolds

Geometric flow is a very important idea in modern differential geometry. In this talk I will begin by motivating my research, discussing the basic definitions and why I chose to pursue classifying this specific type of gradient Bach soliton. After defining the idea of a manifold ‘splitting’ into components with distinct metrics, I will present results by Ho about solitons that are ‘split’ into two 2-dimensional components and my own result about manifolds of this form which cannot be gradient Bach solitons. In closing, we will look briefly at other types of splittings and possible research directions.

Abbe Herzig

Graduate education in the mathematical sciences: Who, what, where, when, why?

The American Mathematical Society (AMS) engages some 30,000 individuals and 570 institutions worldwide in research, education, advocacy, community, and passion about mathematics and its relationship to other disciplines. Key to our mission is training new generations of mathematicians by providing quality and equitable educational opportunities for all students with the interest and commitment. A variety of career opportunities await mathematicians with bachelor’s, master’s, and doctoral degrees, with each subsequent level of education opening more varied and interesting doors. What does graduate study entail? How do you decide whether and where to go to graduation school? How do you prepare yourself? In this talk, I will provide an overview of graduate education, the careers it can lead to, and how to identify the graduate programs that are the best fit for you.

Elliot Hill

Getting to the bottom of it: a friendly introduction to mathematical optimization

Humans have a natural tendency to optimize. Businesses try to maximize their profits, researchers attempt to minimize the error in their models, and you and I try to minimize the time we spend driving to work. Once we have identified an objective we want to optimize and formulated it mathematically, hundreds of optimization algorithms are available for solving these types of problems. But, not all methods work equally well on all problems, so how do we choose an appropriate method for a particular problem? In this talk, we will explore the considerations that go into selecting an optimization algorithm. Then, we will compare and contrast two of the most popular classes of nonlinear optimization algorithms and analyze their strengths and weaknesses on some example problems.

Michael Joyce

Parameterizing Paths Between Matchings

A matching of 2n vertices is an assignment of n edges so that every vertex is incident to exactly one edge. We describe certain local moves that allow you to move from one matching to another, and then we describe all the paths that go from a certain starting matching to a certain final matching. Our calculation was motivated by a desire to understand the geometric structure of a certain highly symmetric high dimensional space. The results we describe are joint with Mahir Can and Ben Wyser.

Riley Juenemann

A First-Pass Statistical Dashboard for Categorizing Diverse Particle Movement Patterns

In contrast to in vitro particle tracking experiments, wherein there are great controls on particle and environmental homogeneity, live cell (in vivo) tracking features tremendous diversity in particle movement. In this work we have developed a first-pass statistical dashboard to categorize disparate types of particle trajectories. The tools we developed for the categorization process include the correlation between consecutive increments and effective diffusivity from a maximum likelihood estimation. The standard deviation for the major and minor axis and the creation of a parameterized path to represent a fictional moving anchor employed principal components analysis. This anchor estimation allowed the computation of the average distance the particle deviated from the anchor. Based on these data measures, K-means clustering was utilized to distinguish between free diffusion, stuck diffusion, directed transport, tracker error, subdiffusion, and skating diffusion. This automated categorization process proved to be successful on data simulated using stochastic differential equations and provided interesting results on live cell data.

Leah Kaisler

Dynamics of a two-strain cholera model with environmental component

Cholera, unlike many diseases, can survive, proliferate, and compete in the aquatic environment, a necessary factor to consider when modelling its long-term dynamics. Previous SIRP models have examined a single cholera population while considering the pathogen environmental concentration, which gives insight into the dynamics of the disease. It is necessary, however, to consider strain diversity and analyse multi-strain models to better understand transmission dynamics and long-term behavior (e.g. coexistence versus competitive exclusion). These models may consider strain differences in environmental and host fitness parameters or can consider distinct serotypes, strains which confer only partial cross-immunity after host infection. In this talk, I will extend the single-strain model to a two-strain SIRP cholera model and present the analysis, results, and simulations carried out when modeling these scenarios.

Selvi Kara

Weighted Stanley-Reisner Correspondence

In this talk, we will learn about a weighted version of the well-known Stanley-Reisner correspondence and discuss the relations between algebraic invariants of weighted and unweighted Stanley-Reisner ideals.

Rainey J Lyons

Biological Modeling with Measures

In recent years, many mathematical biologist have been arguing for the necessity of population models set in the space of finite Radon measures. One of the many benefits of the measure setting is the unification of discrete and continuous structures as Dirac measures (or point masses) can be used to represent a cohort of individuals. In this introductory talk, we will review the relevant definitions needed to discuss this topic, present some examples of biological models where the space of measures is convenient, and add a discussion of numerical techniques one can use for approximating solutions to differential equations in this setting.

Evangelos Nastas

On the Distribution Estimation of Prime Numbers

It is known that the zeros are symmetrical with respect to Re(s) = 1/2 line to the ζ(s). Some of the most optimal approximations of π(x) are O(xexp(−cln(x)^(1/2)). It can be shown that a sufficient approximation of O(x^{1/2+ε}) for π(x) implies the Riemann Hypothesis (RH). First, the reciprocal function of the integral is computed, allowing to approximate the prime numbers by a formula. Second, it is shown that a close approximation of π(x) implies RH. Third, via other means, it is shown that an analytic extension implies RH. A formula for li^{-1} is given and the π(x) function and RH are considered.

Susan Rogowski

Modeling the effects of Taxol and Carboplatin on Breast Cancer Cells

We would like to create a compartmental model in which proliferating cells go into cell cycle arrest as a result of a combination of specific drug treatment. Taxol is known to be an antimitotic cancer drug which stops cells from completing mitosis. Carboplatin is a DNA-targeting drug which leads to cell-cycle arrest. To begin, we will create models for Taxol and Carboplatin treatment separately. Then, we would like to combine the models to create a set of equations which represents a combination of both Taxol and Carboplatin treatment. We will use this model to analyze how these drugs interact and predict which parameter conditions may cause synergy or antagonism. In this talk, we will also discuss several parameter identifiability and estimation issues based off of the current literature and discuss potential solutions.

Vaishavi Sharma

The Valuation of Polynomial Sequences

Given a prime p and any positive integer n, the p-adic valuation of n, denoted by \nu_p(n), is the highest power of p that divides n. This notion is extended to rationals by \nu_p(a\b)=\nu_p(a)-\nu_p(b) and by setting \nu_p(0) = infinity. For any sequence {a_n} and a fixed prime p, the sequence of valuations { \nu_p(a_n)} often presents interesting challenges. One of the goals is to obtain a closed-form for these valuations. In this talk I will discuss p-adic valuations of polynomials, sequences, focusing on polynomials of low degree (2, 3). It will be shown that the sequence of 2-adic valuation of polynomials can be represented as a binary tree. Sometimes this is a finite tree. Examples of trees with infinite branches will be presented. The number of these branches is shown to be connected to the number of roots of x^n - l in the ring of p-adic integers. This is joint work with Diego Villamizar.

Kristi VanDusen

The Mystery of the Primes

It's easy to know whether a number is prime or not. But what else do we know about primes? I'll discuss some properties of prime numbers and some conjectures about primes which remain unproven.