September 6: Melissa Beerbower (Loyola University Chicago)

Title: On the Lucky Sets of Fubini Rankings

Abstract: One special subset of parking functions is the set of Fubini rankings, which encode the outcomes of n competitors in a race where ties are allowed. The number of lucky cars in a Fubini ranking is equivalent to the number of distinct ranks, k. We enumerate Fubini rankings and some subsets recursively through fixed sets of lucky competitors. Our enumerations explain twin coefficients for minimum powers in the lucky polynomials of l -interval Fubini rankings.

September 20: Joe Paulson (UW-Milwaukee)

Title:  Introduction to (Partial) Z-Boundaries

Abstract: 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. 

September 27: Alex Moon (UW-Milwaukee)

Title: Kohnert Properties of Northeast Diagrams

Abstract: 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.

October 4: Jillian Cervantes (UW-Milwaukee)

Title: (t,r) Broadcast Domination of the Truncated Square Tiling Graph

Abstract: 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.

October 11: Kelsey Brouwer (UW-Milwaukee)

Title: Combinatorial models for some generalized McMullen maps in the case of two bounded critical orbits

Abstract: 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.

October 18: Gregory Mwamba (University of California Merced)

Title: Blowup of the Nonlinear Klein-Gordon Equation in FLRW Spacetimes

Abstract: 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.

November 1: Kim Harry (UW-Milwaukee)

Title: A q-analog of Kostant's Weight Multiplicity Formula and a Product of Fibonacci Numbers

Abstract: 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 slr+1(C), which we denote L(˜α), where ˜α is the highest root of slr+1(C). We prove that the number of terms contributing a nonzero value to the multiplicity of the positive root µ = αi + αi+1 + · · · + αj with 1 ≤ i ≤ j ≤ r in L(˜α) is given by the product Fi · Fr−j+1, where Fn is the nth Fibonacci number. Using this result, we show that the q-multiplicity of the positive root µ = αi + αi+1 + · · · + αj with 1 ≤ i ≤ j ≤ r in the representation L(˜α) is precisely qr−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 slr+1(C).

November 15: Eric Redmon (Marquette University)

Title: Finite State Machines and Bounded Permutations

Abstract: We define a k-bounded permutation π of length n to be a permutation such that for each pair of adjacent entries π and π(i + 1) for i = 1, 2, 3, . . . , n − 1 we have |π(i) − π(i + 1)| ≤ 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.

December 6: 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 quesitons you might have.