Mathematical Sciences

Graduate Student Colloquium: Melissa Beerbower
2024-09-06 12:30 - 13:30 (America/Chicago) | 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.

Colloquium: Dr. Rafael S. González D'León
2024-09-06 14:00 - 15:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
On Whitney numbers of the first and second kind, or is it the other way around?
Dr. Rafael S. González D’León
Assistant Professor
Loyola University Chicago
Algebraic combinatorics is centered on the idea that a variety of algebraic objects can be understood using the tools from combinatorics. In this talk, I will illustrate this idea using a modern example of how combinatorial theories can give us a better understanding on a subject. The story I will tell involves the Whitney numbers of the first and second kind, which are a pair of invariants of partially ordered sets (posets) that are relevant in several areas of mathematics. One of their most interesting appearances is as the coefficients of a polynomial enumerating proper colorings of a graph (think of colorings of a map where neighboring countries must be distinguishable from each other). They also appear counting regions in the complement of a real hyperplane arrangement.
I will introduce and give insight into the algebraic and combinatorial objects in this story and share a very curious phenomenon: sometimes the Whitney numbers of the first and second kind of a poset are also the Whitney numbers of the second and first kind, but of a different poset. Some recent results about this phenomenon could shed light on longstanding questions in the subject.

Community of Practice Kickoff Event
2024-09-13 12:30 - 13:30 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
Do you like to talk about teaching? Do you like to socialize with other people in the department? Do you like pear bread? Do you have nothing better to do on a Friday afternoon?
Bring your thoughts:
Have you done something cool in your class that you’d love to share with others?
Do you have something you’d like to try in your class?
Would you like to learn about cool things to do in your class?
Come join us for the first event of the Math Community of Practice! We’ll have snacks!

Colloquium: Prof. Laurel Ohm
2024-09-13 14:00 - 15:30 | EMS Building, EMS E495, 3200 Cramer St, Milwaukee, WI, 53211, United States
PDE Problems in Thin Filament Hydrodynamics
Prof. Laurel Ohm
Assistant Professor of Mathematics
University of Wisconsin-Madison
Many fundamental biophysical processes, from cell division to cellular motility, involve dynamics of thin structures immersed in a very viscous fluid. Various popular models have been developed to describe this fluid-structure interaction mathematically, but much of our understanding of these models is at the level of numerics and formal asymptotics. Here we will discuss some of the interesting issues that arise in developing the PDE theory of thin filament hydrodynamics, as well as the insights this analysis perspective can shed on the underlying biophysics.

Graduate Student Colloquium: Joe Paulson
2024-09-20 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Introduction to (Partial) Z-Boundaries
Joe Paulson
PhD Graduate Student
University of Wisconsin – Milwaukee
In this talk, I’ll share an abridged story of Z-boundaries and their utility in group theory. Throughout, we’ll revisit some main characters (compactifications, homotopy groups, group actions) and introduce some new ones (group boundaries, shape invariance). As our story seemingly resolves, we’ll adapt and refocus on a theory of partial Z-boundaries (ie. Z_n-boundaries) and identify some preliminary results.

Colloquium: Prof. Bob Eisenberg
2024-09-20 14:00 - 15:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Ion Channels, the Ultimate Multiscale Device
Prof. Bob Eisenberg
Professor of Biomathematics and Physiology
Department of Physiology & Biophysics, Rush University, Chicago IL
Proteins called ion channels are the ultimate multiscale device, the ‘nanovalves of life’ controlling most biological functions the way transistors control computers. A handful of atoms control biological function on the macroscale, so analysis must link atomic distances of 10-10m with biological distances of 1 meter. Channels are of such biological and medical importance that hundreds of scientists study them every day and thousands of talks on channels are given at (for example) the annual meeting of the Biophysical Society (USA). Channels conduct ions (spherical charges like Na+ , K+ , Ca2+ , and Cl- with diameter ~ 0.2 nm) through a narrow tunnel of permanent charge (‘doping’) with diameter ~ 0.6 nm. Channels can be studied in the tradition of physical science. PNP equations familiar in physics form an adequate initial model of current voltage relations in many types of channels under many conditions, and can be extended to describe ‘chemical’ phenomena like selectivity with some success. Variational models and methods are needed to deal with the coupling of different fields. Ionic channels are studied with the powerful techniques of molecular biology. Atoms (and thus charges) can be substituted a few at a time and the location of every atom can be determined in favorable cases. Ion channels are one of the few living systems of great importance where natural biological function can be well described by a tractable set of equations. Mathematics and mathematicians are needed to apply these equations to the hundreds of channels of biological and clinical significance.

Graduate Student Colloquium: Alexander Moon
2024-09-27 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Kohnert Properties of Northeast Diagrams
Alexander Moon
Graduate Student
University of Wisconsin – Milwaukee
Kohnert polynomials and posets are combinatorial objects with deep representation theoretic meaning, generalizing both Schubert polynomials and Demazure characters, i.e., key polynomials. In this talk I will explore what Kohnert posets and polynomials are in general, then I will discuss some recent results centering on the Kohnert properties of “northeast” diagrams. I will present some conditions for the boundedness and rankedness of a “northeast” Kohnert poset and present a surprising connection between certain minimal elements and key diagrams. There will be a worksheet. This is a joint work with Aram Bingham, Beth Anne Castellano, Kimberly Hadaway, Reuven Hodges, Yichen Ma, and Kyle Salois that originated at this year’s GRWC.

Graduate Student Colloquium: Jillian Cervantes
2024-10-04 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
(t,r) Broadcast Domination of the Truncated Square Tiling Graph
Jillian Cervantes
Graduate Student
University of Wisconsin – Milwaukee
This talk will introduce graph domination theory and a generalization called (t,r) broadcast domination. We study a family of graphs that arise as a finite subgraph of the truncated square tiling, which utilizes regular squares and octagons to tile the Euclidean plane. For positive integers m and n, we let Hm,n be the graph consisting of m rows of n octagons (cycle graph on 8 vertices). For all t ≥ 2, we provide lower and upper bounds for the (t, 1) broadcast domination number for Hm,n for all m, n ≥ 1. We give exact (2, 1) broadcast domination numbers for Hm,n when (m, n) ∈ {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2)}. We also consider the infinite truncated square tiling, and we provide constructions of infinite (t, r) broadcasts for (t, r) ∈ {(2, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1)}. Using these constructions we give upper bounds on the density of these broadcasts i.e., the proportion of vertices needed to (t, r) broadcast dominate this infinite graph. We end with some directions for future study

Colloquium: Dr. Zhaosheng Feng
2024-10-04 14:00 - 15:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Parabolic System of Aggregation Formation in Bacterial Colonies
Dr. Zhaosheng Feng
Endowed Chair Professor of Mathematics
University of Texas Rio Grande Valley
The goal of this talk is to study a fourth-order nonlinear parabolic system with dispersion for describing bacterial aggregation. Analytical solution of traveling wave is found by taking into account the dispersion coefficient. Numerically, we demonstrate that the initial concentration of bacteria in the form of a random distribution over time transforms into a periodic wave, followed by a transition to a stationary solitary wave without dispersion.

Graduate Student Colloquium: Kelsey Brouwer
2024-10-11 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Combinatorial Models for Some Generalized McMullen Maps in the Case of Two Bounded Critical Orbits
Kelsey Brouwer
PhD Student
University of Wisconsin – Milwaukee
The family of generalized McMullen maps R(z)= z^n + b + a/z^n has two independent critical orbits. We consider the case in which one critical value lies in the immediate basin of an attracting cycle and the other critical value eventually lands in that immediate basin. Computer-generated images of the dynamical plane suggest the presence of both baby quadratic Julia sets and some sets which appear to be modifications of those. We present combinatorial models of the dynamics which help to explain this phenomena.

Colloquium: Dr. Daniel Stoertz
2024-10-11 14:00 - 15:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Baby Mandelbrot Sets for Maximally Generalized McMullen Maps
Dr. Daniel Stoertz
Visiting Assistant Professor of Mathematics
St. Olaf College
In Complex Dynamics, we study the iteration of holomorphic or meromorphic functions on the complex plane or the Riemann sphere. Of particular interest is the behavior of the critical orbits of function families with one or more parameters. The simplest of such families, z^2 +c, is well-known to define the famous Mandelbrot set fractal as the set of c-values for which the unique critical orbit is bounded. In this talk we will examine the function family R(z) = z^n +b +a/(z^d), and we will explore old and new results establishing the location of baby Mandelbrot sets in parameter space for increasingly general versions of this family. In the most general case, which we call maximally generalized McMullen maps, this family has multiple independent critical orbits, and the dynamics in this case are not yet well understood.

ICMA-IX
2024-10-18 - 2024-10-19
International Conference on Mathematical Modeling and Analysis of Populations in Biological Systems
The broad, general theme of the conference is the formulation, validation, analysis and simulation of mathematical models for the spatiotemporal dynamics of biological populations. A special emphasis at this ninth conference will be placed on multiple scale dynamics in ecology, epidemiology, and systems biology; many biological systems operate on multiple spatial and temporal scales and the incorporation of these scales may be essential for capturing system dynamics. Specific topics include, but are not limited to:

Immuno-epidemiological and vector-host systems
Molecular-cellular modeling in systems biology and oncology
Biological invasions and persistence
Adaption and evolutionary dynamics
Biological systems operating on multiple spacial or temporal scales

ICMA-IX aims to accomplish at least three broad tasks: scientific progress, scientific collaboration, and the training of a future generation of mathematical biologists. The conference will thus have a significant impact, both nationally and internationally, on future research in biomathematics and mathematical biology. The collaborative opportunities provided to researchers at all points of the career path from undergraduate and graduate students, to early-career faculty, to established senior researchers will establish new directions and avenues for the national and international research community.
Hosts:
Peter Hinow
Professor
EMS, E491
hinow@uwm.edu
Gabriella Pinter
Professor
EMS, E479
gapinter@uwm.edu

Graduate Student Colloquium: Gregory Mwamba
2024-10-18 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Blowup of the Nonlinear Klein-Gordon Equation in FLRW Spacetimes
Gregory Mwamba
Graduate Student
University of California – Merced
The nonlinear Klein-Gordon equations are a class of important evolution equations that describe the movement of spinless relativistic particles, which can lend understanding in many physical applications. In this talk we will demonstrate a sufficient condition for blowup of the nonlinear Klein-Gordon equation, with arbitrarily positive initial energy in Friedmann-Lemaître-Robertson-Walker spacetimes. This is accomplished using an established concavity method that has been employed for similar PDEs in Minkowski space. This proof relies on the energy inequality associated with this equation.
This talk will be online at the following zoom link: https://wisconsin-edu.zoom.us/j/94983351854 and will also be streamed in EMS E495.

Colloquium: Prof. Qingshuo Song
2024-10-18 14:00 - 15:30
Long-Time Behaviors of Stochastic Linear-Quadratic Optimal Control Problems
Prof. Qingshuo Song
Associate professor
Worcester Polytechnic Institute
In this talk, we will discuss the asymptotic behavior of the solution to a linear-quadratic stochastic optimal control problems. By establishing a connection between the ergodic cost problem and the so-called cell problem in the homogenization of Hamilton-Jacobi equations, we reveal the turnpike properties of the linear-quadratic stochastic optimal control problems from various perspectives.

Math Club Meets: Dr. Kevin A. Palencia Infante
2024-10-24 16:00 - 17:00 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Math Club Meets: Dr. Kevin A. Palencia Infante
UWM Math Club students are invited to meet with Dr. Kevin A. Palencia Infante an Assistant Professor and Calculus Coordinator at Northern Illinois University (NIU). He is part of the Creating Opportunities in Mathematics through Equity and Inclusion (COME-IN) project under Transforming Post-Secondary Education in Mathematics (TPSEMath), where he serves as a DEI consultant. Over the next two years, he will work together with our Math department with the purpose of enhancing efforts to broaden participation and inclusion. In this role, he will provide an external perspective, drawing on his experience and background to support the department through the implementation of its plan.

Graduate Students Meet: Dr. Kevin A. Palencia Infante
2024-10-25 11:30 - 12:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Graduate Students Meet: Dr. Kevin A. Palencia Infante
UWM Math Graduate students are invited to meet with Dr. Kevin A. Palencia Infante an Assistant Professor and Calculus Coordinator at Northern Illinois University (NIU). He is part of the Creating Opportunities in Mathematics through Equity and Inclusion (COME-IN) project under Transforming Post-Secondary Education in Mathematics (TPSEMath), where he serves as a DEI consultant. Over the next two years, he will work together with our Math department with the purpose of enhancing efforts to broaden participation and inclusion. In this role, he will provide an external perspective, drawing on his experience and background to support the department through the implementation of its plan.

Community of Practice Meeting
2024-10-25 12:30 - 13:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Community of Practice: Broaden Participation and Inclusion via the TPSE COME-IN Project
Kevin A. Palencia Infante is an Assistant Professor and Calculus Coordinator at Northern Illinois University (NIU). He is part of the Creating Opportunities in Mathematics through Equity and Inclusion (COME-IN) project under Transforming Post-Secondary Education in Mathematics (TPSEMath), where he serves as a DEI consultant. Over the next two years, he will work together with the University of Wisconsin-Milwaukee’s math department with the purpose of enhancing efforts to broaden participation and inclusion. This event is envisioned partly as a listening session for Dr Palencia Infante to learn about the successes and struggles of our teaching faculty and partly as a discussion and conversation where he can share best practices as an experienced Faculty Teaching Mentor. During his visit he will meet all constituents of our department: undergraduate and graduate students, teaching faculty and faculty. This session provides an opportunity for our teaching faculty to share their experiences with him.

Math Department Town Hall
2024-10-25 14:00 - 15:00 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Town Hall

Our department is engaged in a two-year project “Creating Opportunities in Mathematics through Equity & Inclusion (COME-IN)” through the professional organization Transforming Post Secondary Education in Mathematics (TPSE). Our consultant, Dr. Kevin Palencia Infante, is visiting our department from Northern Illinois University and he will work together with our math department with the purpose of enhancing efforts to broaden participation and inclusion. In this role, he will provide an external perspective, drawing on his experience and background to support the department in shaping and implementing its plan. In this Town Hall we will provide a short “State of our Department” and introduce the COME-IN project and our initial plans.
Speakers will include Craig Guilbault, Suzanne Boyd, Jeb Willenbring and Gabriella Pinter. We aim for a lively discussion with our audience through a question/answer session.

Graduate Student Colloquium: Kim Harry
2024-11-01 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
A q-analog of Kostant’s Weight Multiplicity Formula and a Product of Fibonacci Numbers
Kim Harry
PhD Graduate Student
University of Wisconsin-Milwaukee
Using Kostant’s weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root µ in the adjoint representation of sl_{r+1}(C), which we denote L(˜α), where ˜α is the highest root of sl_{r+1}(C). We prove that the number of terms contributing a nonzero value to the multiplicity of the positive root µ = α_i + α_i+1 + · · · + α_j with 1 ≤ i ≤ j ≤ r in L(˜α) is given by the product F_i · F_(r−j+1), where F_n is the nth Fibonacci number. Using this result, we show that the q-multiplicity of the positive root µ = α_i + α_i+1 + · · · + α_j with 1 ≤ i ≤ j ≤ r in the representation L(˜α) is precisely q^{r−h(µ)}, where h(µ) = j − i + 1 is the height of the positive root µ. Setting q = 1 recovers the known result that the multiplicity of a positive root in the adjoint representation of sl_{r+1}(C).

Colloquium: aBa Mbirika & Morgan Fiebig
2024-11-01 14:00 - 15:30
A graphical approach to the Fibonacci sequence (Fn) n≥0 modulo m extended to the Lucas sequences (Un(p,q))n≥0 and (Vn(P,q))n≥0
aBa Mbirika & Morgan Fiebig
University of Wisconsin – Eau Claire
The goal of this talk is twofold: (1) extend theory on statistics in the Fibonacci and Lucas sequences modulo m to the Lucas sequences U :=(Un(p,q))n≥0 and V :=(Vn(p,q)n 0, and (2) apply this theory to a novel graphical approach of U and V modulo m. The statistics we explore are the period π(m), entry point e(m), and order ω(m) := pi(m)/e(m). We generalize a wealth of known Fibonacci and Lucas statistical results to the U and V setting. Based on ω(m), we describe behaviors shared by infinite families of nondegenerate U and V sequences with parameters q = ± 1. In our graphical approach we place the cycle of repeating terms of the periods of U and V in a circle, and patterns which would otherwise be overlooked emerge. In particular, we exhibit some tantalizing examples in the following three sequence pairs: Fibonacci and Lucas, Pell and associated Pell, and, balancing and Lucas-balancing. Our proofs utilize results from primary sources ranging from the ground-breaking papers of Lucas in 1878 and Carmichael in 1913, to the seminal works of Wall in 1960 and Vinson in 1963, amongst others.

Graduate Student Colloquium: Eric Redmon
2024-11-15 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Finite State Machines and Bounded Permutations
Eric Redmon
Graduate Student
Marquette University
We define a k-bounded permutation π of length n to be a permutation such that for each pair of adjacent entries $pi$ and $pi(i + 1)$ for $i = 1, 2, 3, . . . , n − 1$ we have $|pi(i) − pi(i + 1)| leq k$. Previous work has shown that the generating function for this family of permutations is rational, and has computed generating functions for small values of $k$. In this talk, we will discuss the nature of finite state machines and how we can leverage the insertion encoding devised by Albert, Linton, and Ruškuc to build a finite state machine that we can use to find generating functions for larger values of $k$.

Colloquium: Dr. Lei Hua
2024-11-15 14:00 - 15:30
Unified Tail Dependence Measures and Its Applications in High-Frequency Financial Data
Dr. Lei Hua
Associate Professor, Director of Statistical Consulting Service
Northern Illinois University
In this presentation, I will first motivate the necessity for a unified tail dependence measure, followed by an examination of the theoretical framework for developing such measures utilizing random variables characterized by regularly varying tails. Specific instances that result in unified tail dependence measures applicable in practical scenarios will be demonstrated. Ultimately, I will explore the application of the unified tail dependence measure in the analysis of high-frequency financial market data and discuss novel empirical insights from financial markets.

Colloquium: Nick Mayers
2024-11-22 12:30 - 13:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Well-Behaved Kohnert Posets
Dr. Nicholas Mayers
Postdoctoral Research Scholar
North Carolina State University
Kohnert polynomials form a family of polynomials indexed by diagrams that consist of unit cells arranged in the first quadrant. Many families of well-known polynomials have been shown to be examples of Kohnert polynomials, including key, Schur, and Schubert polynomials. Given a diagram D, the monomials occurring in the corresponding Kohnert polynomial encode diagrams formed from D by applying sequences of certain moves, called “Kohnert moves,” each of which alters the position of at most one cell. In this talk, we focus on the underlying sets of diagrams which generate the monomials of Kohnert polynomials. With each such collection of diagrams, one can associate a poset structure which is known to not, in general, be well-behaved. In particular, the corresponding “Kohnert posets” generally do not have a unique minimal element, are not ranked, and are not lattices. Here, we will focus on recent attempts to find conditions under which Kohnert posets are well-behaved in the sense that they have a unique minimal element, are ranked, or are EL-Shellable. No background knowledge concerning posets is assumed.

Graduate Student Colloquium: Math Graduate Student Panel
2024-12-06 12:30 - 13:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Graduate Student Colloquium: Math Graduate Student Panel
This is our last Math Graduate Student Colloquium of the semester. We will have a panel of senior graduate students happy to discuss our experiences here in UWM’s math department and answer any questions you might have. This can range from department life, teaching, research, and more. Please come and join us in the conversation and bring any topics you would like to talk about or questions you might have.

Colloquium: Dr. Sarah Sword
2025-01-24 14:00 - 15:00 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Studying Successful Doctoral Students from Underrepresented Groups
Dr. Sarah Sword
Principal Research Scientist
Education Development Center, Inc.
In this talk, we will share early findings from an NSF study of 75 doctoral students and recent PhDs in mathematics from underrepresented groups. The project has conducted in-depth, semi-structured interviews with four cohorts: newly accepted students, early graduate students (pre-qualifying exams), advanced graduate students (dissertation level), and recent PhDs (0-5 years since graduation). The research questions in the study are: (1) What are lived experiences of successful underrepresented students who are pursuing mathematics PhDs? and (2) What formal and informal structures are perceived by doctoral students as effective supports? Themes drawn from the interviews can inform department policies and practices to create environments that support students from underrepresented groups as they earn PhDs in mathematics.

Graduate Student Colloquium: Soft Open
2025-01-31 12:30 - 13:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Graduate Student Colloquium: Soft Open
This is our first Math Graduate Student Colloquium. We will be hanging out, eating snacks, and getting back into the swing of semester life. Please, join us and get a feel of what to expect in this semester’s colloquium. If you want to bring any snacks, feel free to show off any baking, cooking, or generosity skills.

Community of Practice: Introduction to Transparency in Learning and Teaching
2025-02-07 12:30 - 13:30 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
Community of Practice: Introduction to Transparency in Learning and Teaching
What is the core purpose of the assignments, tasks, and learning opportunities in our courses? Better yet, what do our students think is the primary purpose? In this workshop we will discuss Transparency in Learning and Teaching (TILT) and how it can inform our teaching. We’ll develop a list of best practices and brainstorm how to implement in our courses. To get the most out of the session, bring an assignment such as a homework set, an out-of-class task, or a project from a class you’re teaching – ideally something coming up this semester or early next semester—and bring along a colleague, too! We’ll actively workshop together, and you’ll walk away with a new and improved transparent assignment (or at least the tools to build one)!

For a sneak peek, check out tilthighered.com

Facilitated by Hayley Nathan and Suzanne Boyd.

Colloquium: Ning Wei
2025-02-07 14:00 - 15:00 | EMS Building, E495, 3200 N Cramer St, Milwaukee, WI, United States
The Impact of Ephaptic Coupling and Ionic Electrodiffusion on Arrhythmogenesis in the Heart
Ning Wei
Assistant Professor
Purdue University
Cardiac myocytes synchronize through electrical signaling to contract heart muscles, facilitated by gap junctions (GJs) in the intercalated disc (ID). GJs provide low-resistance pathways for electrical impulse propagation between myocytes, serving as the primary mechanism for electrical communication in the heart. However, research indicates that conduction can persist without GJs. For instance, GJ knockout mice still exhibit slow, discontinuous electrical propagation, suggesting alternative communication mechanisms. Ephaptic coupling (EpC) serves as an alternative way for cell communication, relying on electrical fields within narrow clefts between neighboring myocytes. Studies show that EpC can enhance conduction velocity (CV) and reduce conduction block (CB), especially when GJs are compromised. Reduced GJs and significant electrochemical gradients are prevalent in various heart diseases. However, existing models often fail to capture their combined influence on cardiac conduction, which limits our understanding of both the physiological and pathological aspects of the heart. Our study aims to address this gap by developing a two-dimensional (2D) multidomain electrodiffusion model that incorporates EpC. This is the first model to capture the dynamics of all ions across multiple domains, enabling us to reveal the impact of EpC in the heart. In particular, we investigated the interplay between ionic electrodiffusion and EpC on action potential propagation, morphology, electrochemical properties and arrhythmogenesis in both healthy and ischemic hearts. Our findings indicate that ionic electrodiffusion enhances CV and reduces CB under strong EpC. Specifically, the electrodiffusion of Ca2+ and K+ intensifies the effects of EpC on action potential morphology, whereas Na+ diffusion mitigates these effects. Ionic electrodiffusion also facilitates action potential propagation into ischemic regions when EpC is substantial. Moreover, strong EpC can effectively terminate reentry, prevent its initiation, and lower the maximum dominant frequency (max DF), irrespective of GJ functionality. However, weak EpC may help counteract proarrhythmic effects when GJ coupling is slightly to moderately reduced, contributing to the stabilization of conduction patterns. Additionally, strong EpC notably alters ionic concentrations in the cleft, significantly increasing [K+] and nearly depleting [Ca2+], while causing moderate changes in [Na+]. This multidomain electrodiffusion model sheds light on the mechanisms of EpC in the heart.

Graduate Student Colloquium: Liam Jemison
2025-02-14 12:30 - 13:30 | EMS Building, Room E495, E495; 3200 N Cramer St., Milwaukee, WI, 53211, United States
Finite Elements for Mathematicians
Liam Jemison
PhD Graduate Student
University of Wisconsin-Milwaukee
We will discuss the finite element method, a powerful approach for numerically solving differential equations. We will introduce the weak formulation of a differential equation from the functional analysis viewpoint with a simple application of the galerkin method, and then discuss generalizations, some error estimates, and software implementations.

Colloquium: Dr. Alexander Wilson
2025-02-14 14:00 - 15:30
Symmetries and Diagram Algebras
Dr. Alexander Wilson
Visiting Assistant Professor of Mathematics
Oberlin College
In this talk I will introduce you to the world of symmetric group representations through diagram algebras, which trace their origin to the Temperley-Lieb algebra with applications in integrable models, knot theory, and quantum groups. For representation theory, these algebras offer a sneaky path toward solving difficult problems by understanding the ways that graph-theoretic diagrams combine. The only background I will assume is some familiarity with linear algebra, so if you like (or at least tolerate) playing around with pretty combinatorial objects, I hope you’ll attend!
