Schedule

Title: Julia Sets in Parameter Space for the Generalized McMullen Map

Abstract: 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.

Title: Introduction to Topological Combinatorics

Abstract: 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.

Title: Representations of the Natural Numbers using Box Arrangements

Abstract: 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 paper, we explore this presentation and investigate the properties of boxes of different sizes. We 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. 

Title: Games, Winning, and Nimbers

Abstract: 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.

Title: Sandpile Group of Cones over Trees

Abstract: 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.

Title: On the Lucky and Displacement Statistics of Stirling Permutations

Abstract: A parking function of length n is a tuple alpha=(a_1,a_2, ... ,a_n) in [n]^n whose rearrangement into weakly increasing order beta=(b_1,b_2, ... ,b_n) satisfies b_i \leq i for all i in [n]. We investigate two statistics of parking functions when alpha is a Stirling permutation: luck and displacement. A Stirling permutation of order n is a permutation of the multiset {1,1,2,2,3,3, ... ,n,n} where any value j appearing between the two instances of i satisfies j>i. A car i is said to be lucky if it parks in its preferred spot, a_i. An extremely lucky Stirling permutation is a Stirling permutation where the maximum number of cars are lucky. Another statistic for parking functions is known as its displacement. The displacement of a car is defined as the difference between where it parks and its preference. Amongst our results we show that for any positive integer n, the number of extremely lucky Stirling permutations of order n is the Catalan number C_n. We also describe a formula for the displacement of each car in an extremely lucky permutation. This is a joint work with Laura Colmenarejo, Aleyah Dawkins, Jennifer Elder, Pamela E. Harris, Kimberly J. Harry, Selvi Kara, and Bridget Eileen Tenner.

Title: A Primer on the Mathematics of Artificial Neural Networks

Abstract: 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.

Title: On Combinatorial Problems of Generalized Parking Functions

Abstract: 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.

Title: Harmonize your Fractals

Abstract: 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 some open problems. 

 Title: Counting Orbits of Defective Parking Functions

Abstract: 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.