Mathematical Sciences

MS Thesis Defense: Ms. Helen Kafka
2024-04-30 13:00 - 14:00 (America/Chicago) | 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.

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.
