Mathematical Sciences

Graduate Student Colloquium: Mrs. Kelsey Brouwer
2024-02-09 12:30 - 13:30 (America/Chicago) | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Julia Sets in Parameter Space for the Generalized McMullen Map
Mrs. Kelsey Brouwer
PhD Student
University of Wisconsin – Milwaukee
We will begin by introducing the field of complex dynamics and explaining the two types of pictures that dynamicists are interested in, namely Julia sets and parameter spaces. Next, we’ll introduce the generalized McMullen map and share some results about its Julia sets and parameter spaces. Finally, we will demonstrate an interesting connection between McMullen parameter space and quadratic Julia sets and discuss some potential techniques for explaining and documenting this phenomenon. A software demonstration may follow.

Graduate Student Colloquium: Mr. Joe Paulson
2024-02-16 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Introduction to Topological Combinatorics
Mr. Joe Paulson
PhD Student
University of Wisconsin – Milwaukee
With the prominence of topology and combinatorics at UWM, we’ll explore the intersection of these fields. This relatively new field originated with the solving of Kneser’s conjecture via Lovasz’s use of topological methods. We’ll explore Lovasz’s solution and other solutions to famous problems such as the Necklace problem and the Inscribed Rectangle.

Colloquium: Dr. Bogdan Nica
2024-02-16 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Elementary Matrix Operations on a Napkin
Dr. Bogdan Nica
Assistant Professor
Indiana University – Purdue University Indianapolis
Atmospheric science provides excellent opportunities for encouraging student interest in all fields of STEM (Science, Technology, Engineering, Math). A model outreach program (“Chasing the Wind: Tools for Understanding the Weather”) will be presented for use in elementary, middle and high school classrooms. Key elements of the presentation will be highlighted: sharing of personal career background; story-telling about scientists and scientific concepts; utilization of physical props and humor; messaging on the importance of fluency in mathematics, the language of STEM; and direct involvement of the audience.

Graduate Student Colloquium: Mr. Joakim Jakovleski
2024-02-23 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Representations of the Natural Numbers Using Box Arrangements
Mr. Joakim Jakovleski
PhD Student
University of Wisconsin – Milwaukee
Natural numbers can be presented as sums of numbers in boxes, where we are subject to certain rules about how we generate the boxes and the numbers in them. In this talk, we will explore this presentation and investigate the properties of boxes of different sizes. We will center on a simplified way to obtain all numbers in all boxes using a certain subset of them, and use it to ultimately produce a formula for these numbers. We then manage to provide some generalizations, and pave the field for potential future work.

Graduate Student Colloquium: Mr. John Museus
2024-03-01 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Games, Winning, and Nimbers
Mr. John Museus
PhD Student
University of Wisconsin – Milwaukee
The Nimbers are a particularly peculiar Field which arises naturally when studying the winning strategies of Impartial Games. In this talk, we will give a brief overview of the constructions leading up to the Nimbers by studying three games: Nim, Subtraction, and Turning Corners. Along the way we will see what Tweedledee and Tweedledum have to do with winning games and give an airtight proof that 2 tims 2 is actually 3.

Colloquium : Dr. Selvi Kara
2024-03-01 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Combinatorial Resolutions of Monomial Ideals
Dr. Selvi Kara
Assistant Professor of Mathematics
Bryn Mawr College
One of the central problems in commutative algebra concerns understanding the structure of an ideal in a polynomial ring. Abstractly, an ideal’s structure can be expressed through an object called its minimal resolution, but there is no explicit method to obtain a minimal resolution in general, even for the simpler and fundamental class known as monomial ideals.
In this talk, we will focus on resolutions of monomial ideals. In particular, I will introduce a new combinatorial method that provides a resolution of any monomial ideal using tools from discrete Morse theory.

Graduate Student Colloquium: Dorian Smith
2024-03-08 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Sandpile Group For Cones Over Trees
Dorian Smith
PhD Student
University of Minnesota Twin Cities
The sandpile group $K(G)$ of a graph $G$ is a finite abelian group, isomorphic to the cokernel of the reduced graph Laplacian of $G.$ We study $K(G)$ when $G = Cone(T)$. The graph $Cone(T)$ is obtained from a tree $T$ on $n$ vertices by attaching a new cone vertex attached to all other vertices. For two such families of graphs, we will describe $K(G)$ exactly: the fan graphs $Cone(P_n)$ where $P_n$ is a path, and the thagomizer graph $Cone(S_n)$ where $S_n$ is the star-shaped tree. The motivation is that these two families turn out to be extreme cases among $Cone(T)$ for all trees $T$ on $n$ vertices.

Colloquium: Dr. Emmanuel Asante-Asamani
2024-03-08 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
A Mechanochemical Model of Cell Migration in Confined Environments
Dr. Emmanuel Asante-Asamani
Assistant Professor of Mathematics
Clarkson University
Eukaryotic cells can move in confined environments by using pressure driven protrusions of their cell membrane, a motility mechanism known as blebbing. Blebbing has been observed to facilitate the movement of tumor cells and some cancer cells during metastasis. Many questions remain unanswered about how cells translate mechanical cues from their environment into coordinated movement during blebbing. Of particular interest is how proteins that link the cell membrane to the cortex regulate the size and frequency of blebs under different levels of environmental confinement. In this talk, I will present a multiscale model of bleb expansion that treats the cell as a viscous fluid encased by a viscoelastic boundary, whose mechanical properties are regulated by dynamic structural and motor proteins. Numerical simulation of this model supports experimental data suggesting, contrary to intuition, that weakening the adhesion of the cell membrane to the cortex produces smaller and less frequent blebs.

Graduate Student Colloquium: Dorian Smith
2024-03-15 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
On the Lucky and Displacement Statistics of Stirling Permutations
Dorian Smith
PhD Student
University of Minnesota Twin Cities
Stirling permutations are parking functions. We investigate two parking function statistics in the context of these objects: lucky cars and displacement. Among our results, we consider two extreme cases: extremely lucky Stirling permutations (those with maximally many lucky cars) and extremely unlucky Stirling permutations (those with exactly one lucky car). We show that the number of extremely lucky Stirling permutations of order n is the Catalan number, and the number of extremely unlucky Stirling permutations is (n−1)!.
This is a joint work with Laura Colmenarejo, Aleyah Dawkins, Jennifer Elder, Pamela E. Harris, Kimberly J.Harry, Selvi Kara, and Bridget Eileen Tenner.

Colloquium: Dr. Jean-Pierre Mutunguha
2024-03-15 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
The Dynamical view of Free-by-Cyclic Groups
Dr. Jean Pierre Mutanguha
Instructor
Princeton
Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss various dynamical properties of automorphisms that turn out to be group invariants of the corresponding free-by-cyclic groups (e.g. growth type). In particular, certain dynamical hierarchical decompositions of an automorphism determine canonical hierarchical decompositions of its mapping torus.

Spring Break
2024-03-17 - 2024-03-23

Graduate Student Colloquium: Daniel Quigley
2024-03-29 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
A Primer on the Mathematics of Artificial Neural Networks
Daniel Quigley
PhD Student
University of Wisconsin-Milwaukee
Artificial neural networks (ANNs, or, simply, neural networks) are ubiquitous, not least of all in the context of modern machine learning. This presentation is a primer on the mathematics that underlie the mechanics of relatively simple feedforward ANNs. A sketch of the proof for the universal approximation theorem is given, which states that a (fully connected) ANN with at least one hidden layer (of a sufficient number of neurons), together with a non-linear activation function, can approximate any continuous function on a compact set to arbitrary accuracy. This presentation contributes to the movement for providing mathematical explanations and descriptions of ANNs, favoring a functional analytical and well-founded framework at the expense of algorithmic aspects of deep learning otherwise concerned with identifying the most suitable deep ANNs for specific applications.

Colloquium: Dr. Jay Pantone
2024-03-29 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Experimental Methods in Combinatorics
Dr. Jay Pantone
Assistant Professor of Mathematics
Marquette University
What number comes next in the sequence
1, 2, 4, 8, 16, 32, … ?
How about
1, 2, 3, 5, 8, 13, … ?
Or maybe
1, 3, 14, 84, 592, 4659, … ?
Many questions in combinatorics have the form “How many objects are there that have size n and that satisfy certain properties?” For example, there are n! permutations (rearrangements) of n distinct objects, there are 2^n binary strings of length n, and the number of sequences of n coin flips that never have two tails in a row is the nth Fibonacci number. The “counting sequence” of a set of objects is the sequence a_0, a_1, a_2, …, where a_n is the number of objects of size n.
As a result of theoretical advances and more powerful computers, it is becoming common to be able to compute a large number of initial terms of the counting sequence of a set of objects that you’d like to study. From these initial terms, can you guess future terms? Can you guess a formula for the nth term in the sequence? Can you guess the asymptotic behavior as n tends to infinity?
Rigorously, you can prove basically nothing from just some known initial terms. But, perhaps surprisingly, there are several empirical techniques that can use these initial terms to shed some light on the nature of a sequence.
As we talk about two such techniques — automated conjecturing of generating functions, and the method of differential approximation — we’ll exhibit their usefulness through a variety of combinatorial topics, including chord diagrams, permutation classes, and inversion sequences.

Marden Lecture: Professor Caroline J Klivans
2024-04-03 16:30 - 17:30 | Lubar Hall N140, 3202 N Maryland Ave, Milwaukee, WI, 53211, United States
The intrigue that compels us
When we witness unexpected phenomena, a mathematician finds themselves asking: why? We are compelled to understand further; what is the cause, the basic underlying principles? Mathematics is full of symmetries, patterns and visuals that we can appreciate in their own right or use to explain and illustrate concepts. And, they can also drive us to discovery. I will tell a story of chip-firing, a mathematical idea that leads to fascinatingly intricate structures. The imagery of these structures has spurred many questions to which we know surprisingly few answers.

Marden Special Topics Seminar: Prof. Caroline Klivans
2024-04-04 13:00 - 14:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
The Mathematics of Chip-Firing
Caroline Klivans
Professor of Applied Mathematics, Deputy Director of ICERM
Brown University
Chip-firing processes are discrete dynamical systems. A commodity (chips, sand, dollars) is exchanged between sites of a network according to simple local rules. Although governed by local rules, the long-term global behavior of the system reveals unexpected properties, including intricate fractal-like patterns. In this talk, I will give a broad survey of the theory of chip-firing, highlighting both classical results and recent advances.

Graduate Student Colloquium: Kimberly Hadaway
2024-04-05 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
On Combinatorial Problems of Generalized Parking Functions
Kimberly Hadaway
PhD Student
Iowa State University
In this talk, we study combinatorial problems related to generalized parking functions. Our work is motivated by two different research questions posed to us by Dr. Ken Fan and Dr. Shanise Walker. First, we reframe Dr. Fan’s probabilistic question in terms of defective parking functions, which enumerate the number of cars unable to park in the classical parking function problem, thereby providing a partial answer to his question. Second, we answer Dr. Walker’s question establishing a bijection between unit interval parking functions and the Fubini rankings, which get their name as they are enumerated by the Fubini numbers.

Colloquium: Prof. Yoichiro Mori
2024-04-12 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Mathematical Modeling of Cell Volume Control and Electrolyte Balance
Prof. Yoichiro Mori
Professor of Mathematics
University of Pennsylvania
Electrolyte and cell volume regulation is essential in physiological systems. Biophysical modeling in this area, however, has been relatively sparse. After a brief introduction to cell volume control and electrophysiology, I will discuss the classical pump-leak model of electrolyte and cell volume control. It will be shown that thermodynamic considerations lead to a new perspective of cell volume control. This classical model will then be generalized to a model with spatial extent (a system of partial differential equations) modeling cell-level electrodiffusive and osmotic phenomena. A simplified version of this model will then be applied to study osmosis-driven cell movement. I will also touch upon tissue-level models of ionic electrodiffusion and osmotic water flow which we have developed to study certain pathophysiologies of the central nervous system.

Graduate Student Colloquium: Matt McClinton
2024-04-19 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Harmonize your Fractals
Matt McClinton
Graduate Student
University of Wisconsin-Milwaukee
The Sierpinski Gasket (SG) is a known fractal object. A simple observation shows that SG is path connected. Unfortunately, the infinitely jagged structure of the Gasket prevents these paths from being differentiable. If only there existed a means of smoothing out SG into an object where continuous and differentiable paths existed between pairs of points. As a matter of fact there is!
I will demonstrate the technique known as “minimizing the graph energy” as described in the late Robert Strichartz’s book “Differential Equations on Fractals”. This technique involves finding the solution to a system of equations where the solution produces a graph that has differentiable paths, and even better satisfies the Laplacian. Using a homeomorphic mapping defined by Jun Kigami in 1989, by finding the graph energy minimizing values on level sets of SG, we produce a fractal object known as the Harmonic Sierpinski Gasket (HSG).
This talk is intended for those that are interested in analysis, algebra, combinatorics, topology, fractal geometry, and/or graph theory. Any necessary background information will be provided during the talk, and I will end with some open problems.

Colloquium: Prof. Roger Howe
2024-04-23 13:00 - 14:15 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Mathematics around the Heisenberg Group
Prof. Roger Howe
Professor Emeritus
Yale University
In the mid 1920s, Werner Heisenberg formulated the CCR – canonical commutation relations – describing the relationship between the operations of measuring position and of measuring momentum of a particle in quantum mechanics. These have been fundamental to the later, dramatically successful development of subatomic physics. Shortly after Heisenberg’s work, Hermann Weyl pointed out that the CCR defined the relations of a Lie algebra, whose associated group is a two-step nilpotent group with one dimensional center. Today, this group (and its increasingly large set of cousins) is known as the “Heisenberg group”. Over the remainder of the 20th century, appreciation grew of the fundamental role of the Heisenberg group in disparate mathematical topics, including harmonic analysis, partial differential equations, invariant theory and representation theory, in both finite and infinite dimensions. This talk will review and attempt to summarize some of the manifold connections between these topics that are mediated by the Heisenberg group

PhD Dissertation Defense: Mr. William Braubach
2024-04-26 10:00 - 12:00 | EMS Building, Room E425, E425; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Coarse Homotopy Extension Property and its Applications
Mr. William Braubach
University of Wisconsin-Milwaukee
A pair (X, A) has the homotopy extension property if any homotopy of A can be extended to a homotopy of X. The main goal of this dissertation is to define a coarse analog of the homotopy extension property for coarse homotopies and prove coarse versions of results from algebraic topology involving this property.
First, we define a notion of a coarse adjunction metric for constructing coarse adjunction spaces. We use this to redefine coarse CW complexes and to construct a coarse version of the mapping cylinder. We then prove various pairs of spaces have the coarse homotopy extension property. In particular, pairs of coarse CW complexes. We then prove results involving the coarse homotopy extension property, leading to the result that a coarse map f from X into Y is a coarse homotopy equivalence if and only if the coarse mapping cylinder coarse deformation retracts onto its copy of X. We use this to prove our main result, a coarse version of Whitehead’s Theorem: If a cellular coarse map f between coarse CW complexes induces isomorphisms between coarse homotopy groups, then f is a coarse homotopy equivalence.
Advisor: Prof. Boris Okun
Committee Members:
Profs. Boris Okun, Craig Guilbault, Jeb Willenbring, Jonah Gaster, and Chris Hruska

Graduate Student Colloquium: Alex Moon
2024-04-26 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Counting Orbits of Defective Parking Functions
Alex Moon
PhD Student
University of Wisconsin-Milwaukee
Parking functions are well-studied objects in combinatorics and representation theory which constitute tuples of preferred parking spots for cars under a linear parking scheme. This talk will generalize to defective parking functions. I will enumerate the orbits of defective parking functions under the action of the symmetric group by characterizing them as nondecreasing tuples and sketching a bijection to standard nondecreasing parking functions. I will also introduce the concept of the conjugate of a nondecreasing parking function in order to simplify the case where the number of cars and spots differ.
This is a joint with Pamela E. Harris, Aaron Ortiz, Lauren J. Quesada, Cynthia Marie Rivera Sánchez, and Dwight A. Williams II.

Colloquium: Fredric Ancel
2024-04-26 14:00 - 15:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States

MS Thesis Defense: Ms. Helen Kafka
2024-04-30 13:00 - 14:00 | EMS Building, E408, 3200 N Cramer St, Milwaukee, WI, 53211, United States
Markov Chain Model of Three-Dimensional Daphnia Magna Movement
Ms. Helen Kafka
University of Wisconsin-Milwaukee
Daphnia magna make turns through an antennae-whipping action. This action occurs every few seconds, hence, during the intervening time, the animal either remains in place or continues movement roughly along its current course. We view their movement in three dimensions. We divide the movement in the three dimensions into the movement on a two-dimensional lattice and the movement between the different planes. For the movement on the lattice, we construct a second-order Markov chain model to make predictions about which region of the lattice the animal moves to based on where it was at the last two time points. The movement between the different planes is simulated by a first-order Markov chain.
Advisor: Prof. David Spade
Committee Members:
Profs. David Spade, Jeb Willenbring, and Chudamani Poudyal

PhD Dissertation Defense: Mr. Russell Latterman
2024-05-02 13:00 - 15:00 | EMS Building, Room E424A, E424A; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Bayesian Change Point Detection In Segmented Multi-Group Autoregressive Moving-Average Data For The Study Of COVID-19 In Wisconsin
Mr. Russell Latterman
University of Wisconsin-Milwaukee
Changepoint detection involves the discovery of abrupt fluctuations in population dynamics over time. We take a Bayesian approach to estimating points in time at which the parameters of an autoregressive moving average (ARMA) change, applying a Markov Chain Monte Carlo method. We specifically assume that data may originate from one of two groups. We provide estimates of all multi-group parameters of a model of this form for both simulated and real-world data sets. We include a provision to resolve the problem of confounding ARMA parameter estimates and variance of segment data. We apply our model to identify events that may have contributed to the 2020 and 2021 outbreaks of COVID-19 in Waukesha County, Wisconsin.
Advisor: Prof. David Spade
Committee Members:
Profs. Richard Stockbridge, Istvan Lauko, Chao Zhu, and Vytaras Brazauskas

MS Thesis Defense: Mr. Lucas Fellmeth
2024-05-02 16:00 - 17:00 | EMS Building, E408, 3200 N Cramer St, Milwaukee, WI, 53211, United States
Utilizing ARMA Models for Non-Independent Replications of Point Processes
Mr. Lucas Fellmeth
University of Wisconsin-Milwaukee
The use of a functional principal component analysis (FPCA) approach for estimating intensity functions from prior work allows us to obtain component scores of replicated point processes under the assumption of independent replications. We show these component scores can be modeled using classical autoregressive moving average (ARMA) models, thus allowing us to also apply the FPCA model to non-independent replications. The Divvy bike-sharing system in the city of Chicago is showcased as an application.
Advisor: Prof. Daniel Gervini
Committee Members:
Profs. Daniel Gervini, David Spade, and Chudamani Poudyal

MS Thesis Defense: Mr. Sven Bergmann
2024-05-03 08:30 - 09:00 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Adding a Third Normal to CLUBB
Mr. Sven Bergmann
University of Wisconsin-Milwaukee
The Cloud Layers Unified By Binormals (CLUBB) model uses the sum of two normal probability density function (pdf) components to represent subgrid variability within a single grid layer of an atmospheric model. This binormal approach, while computationally efficient, restricts the model’s ability to capture the full spectrum of potential shapes encountered in real-world atmospheric data. This thesis proposes to introduce a third normal pdf component strategically positioned between the existing two, significantly enhancing the model’s representational flexibility. This trinormal representation allows for a wider range of grid-layer shapes while permitting analytic solutions for certain higher order moments. The core of this work lies in deriving the necessary mathematical transformations for incorporating the third normal pdf seamlessly into the CLUBB framework. This thesis lists all formulas, inputs, and outputs associated with the extended model as well as gives an outline
on how to check those equations. Additionally, it describes certain asymptotic behavior of the trinormal pdf under various parameter settings.
Advisor: Prof. Vince Larson
Committee Members:
Profs. Vince Larson, Peter Hinow, and David Spade

PhD Dissertation Defense: Mr. Dan Noelck
2024-05-03 13:30 - 14:00 | EMS Building, Room E423, E423; 3200 N Cramer St, Milwaukee, WI, 53211, United States
Contraction Rates For McKean-Vlassov Stochastic Differential Equations
Mr. Dan Noelck
University of Wisconsin-Milwaukee
This work focuses on the contraction rates for McKean-Vlasov stochastic differential equations (SDEs), McKean-Vlasov Stochastic differential delay equations (SDDEs), and path dependent McKean-Vlasov stochastic differential equations.
Under suitable conditions on the coefficients of the SDE, this dissertation derives explicit quantitative contraction rates for the convergence in Wasserstein distances of McKean-Vlasov SDEs using the coupling method. The contraction results are then used to prove a propagation of chaos uniformly in time, which
provides quantitative bounds on convergence rate of interacting particle systems, and establishes exponential ergodicity for McKean-Vlasov SDEs. The dissertation further develops suitable conditions on the coefficients of the McKean-Vlasov SDDE to obtain a contraction in Wasserstein distance using the coupling method again. These results are used to establish exponential ergodicity for McKean-Vlasov SDDEs. Last the dissertation obtains suitable conditions on the coefficients of the path dependent McKean-Vlasov SDE for a contraction in Wasserstein distance.
Advisor: Prof. Chao Zhu
Committee Members:
Profs. Lijing Sun, Jeb Willenbring, Richard Stockbridge, and Peter Hinow

Colloquium: Prof. Genevieve Walsh
2024-05-03 14:00 - 15:30 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Hyperbolic groups, their boundaries and drilling
Prof. Genevieve Walsh
Professor of Mathematics
Tufts University
We will define and describe groups with a particular geometry, hyperbolic groups. We will define the boundary of a hyperbolic group and give many examples. If time permits, we will define a drilling of a hyperbolic group and explore how this operation changes the boundary. Any new work is joint with Groves, Haissinsky, Manning, Osajda and Sisto.

MS Thesis Defense: Mr. Silas Winnemoeller
2024-05-06 13:00 - 14:00 | EMS Building, Room E416, 3200 N Cramer St, Milwaukee, WI, 53211, United States
A Finite Element Block Modified Backward Euler Method For Solving A One-Dimensional Poisson-Nernst-Planck Ion Channel Model
Mr. Silas Winnemoeller
University of Wisconsin-Milwaukee
In this thesis, a finite element block modified backward Euler method is introduced to solve a one-dimensional Poisson-Nernst-Planck ion channel (1D PNPic) model. This model is defined as a system of time-dependent nonlinear partial differential equations, called Poisson-Nernst equations and Poisson equation, describing the transport of charged ionic species across a cell membrane via an ion channel pore. For an electrolyte with n ionic species, its numerical solution gives a prediction to n ionic concentration functions and an electrostatic potential function. However, solving the 1DPNPic model numerically is challenging due to the model’s strong nonlinearity and numerical stability issues. To address the numerical stability issues, the traditional backward Euler implicit time scheme is often selected to solve the 1DPNPic model but it may be too costly to be practical in application since it has to solve a system of n + 1 strongly nonlinear partial differential equations at each time step. Hence, its modification becomes necessary to reduce its computing cost while retaining its numerical stability properly. In this thesis, the new method is constructed by semi-discretization and finite element techniques such that its each time iteration only involves calculation within two blocks with each block only containing two linear differential equations. Consequently, the new method can reduce the
computing cost of the Euler scheme sharply. In this thesis, the new method is implemented as a software package in Python based on the finite element library from the FEniCS project. Numerical tests are then done for an electrolyte with two ionic species, demonstrating the convergence and high performance of the new method.
Advisor: Prof. Dexuan Xie
Committee Members:
Profs. Lei Wang, Vincent Larson, and Dexuan Xie

Graduate Student Colloquium: Melissa Beerbower
2024-09-06 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
On the Lucky Sets of Fubini Rankings
Melissa Beerbower
Loyola University Chicago
Recall that Fubini rankings of length n are rankings of n competitors allowing for ties. We can say that the number of rankings, k, with k less than or equal to n, is equal to the number of lucky competitors. A lucky competitor is the first to attain its rank. We enumerate Fubini rankings recursively through fixed sets of lucky competitors. Also recall that unit Fubini rankings are Fubini rankings with at most two competitors for each rank. We enumerate unit Fubini rankings through fixed lucky sets. Our enumerations explain twin coefficients for minimum powers in the lucky polynomial of ell-interval Fubini rankings.
