# 2019–2020 Mathematics Seminar

**Date**: Friday, September 27 at 3:30pm

**Room**: Roosevelt 201

**Speaker**: Michael Cole, Hofstra University

**Title**: Orthogonal Polynomials in One or More Variables

**Abstract**: PDF

**Date**: Wednesday, October 16 at 11:30am

**Speaker**: Yotam Smilansky, Rutgers University

**Title**: Patterns and Partitions

**Abstract**: A colored partition of a set in **R**^{d} is its representation as a disjoint union of subsets, referred to as tiles, where each tile is also assigned a color. In the talk, we will consider sequences of colored partitions defined using multiscale substitution rules on finite collections of colored prototiles. In the substitution process, which generalizes a construction first introduced by Kakutani, tiles of maximal volume in a given partition are replaced by colorful patterns consisting of rescaled copies of colored prototiles, thus defining the next partition in the sequence. Tiles that appear in the process are modeled by a flow on a directed weighted graph, and distributional and statistical questions on sequences of partitions are reformulated as questions on the distribution of paths on graphs. Under a natural incommensurability assumption, special properties of the poles of the Laplace transforms of graph counting functions imply various explicit statistical results. In addition, computer experiments reveal the beautiful patterns in which these poles appear in the complex plane, patterns which seem to be closely related to Diophantine properties of the generating substitution rule.

**Date**: Wednesday, October 30 at 11:30am

**Speaker**: Stefan Waner, Hofstra University

**Title**: Online resources for elementary math classes

**Abstract**: Steve Costenoble and I have developed online adaptive tutorials as games—as well as numerous other online resources—to address many of the well-documented challenges facing mathematics instructors in elementary math classes. Viewing the classroom experience as a game is more than simply a convenient metaphor, but allows the designer of educational products to deploy numerous features borrowed from student experience in computer games and, more importantly, to present the entire classroom experience to the student explicitly in terms with which they are familiar from their experience outside the classroom, and to which they know how to respond effectively.

**Date**: Wednesday, November 6 at 11:30am

**Speaker**: Manon Stipulanti, Hofstra University

**Title**: A way to extend Pascal's triangle to words

**Abstract**: Pascal's triangle and the corresponding Sierpiński's triangle are well-studied objects and have connections with different areas in science. The main ingredient of this presentation is the link between them. I will first recall it and then exploit it to present a way of extending both objects to the area of combinatorics on words.

Combinatorics on words is a relatively new domain of discrete mathematics, which focuses on the study of words and formal languages. In this context, a finite word is simply a finite sequence of letters, or symbols, that belong to a finite set called the alphabet. For instance, 01101 and 01 are two finite (binary) words over the (binary) alphabet {0,1}. A language is a set of words. For instance, we let {0,1}* denote the set of all finite words over {0,1}. The binomial coefficient of two finite words *u* and *v* over some alphabet is the number of occurrences of *v* as a subsequence of *u*. For example, the binomial coefficient of 01101 and 01 is 4. This concept, which generalizes binomial coefficients of integers, has been widely studied for the last thirty years or so. Knowing the definition of the Pascal's triangle with binomial coefficients of integers, its extension to binomial coefficients of words seems somewhat natural.

**Date**: Wednesday, November 13 at 3:00pm

**Room**: Breslin 103

**Speaker**: Jim Thatcher, University of Washington-Tacoma

**Title**: Electoral Districting in more than Euclidean spaces: travel-time considerations for communities of interest

**Abstract**: With the coming 2020 census, new electoral and representational districts will be drawn across the United states. This talk discusses the historic and current role cartography, here the literal drawing of lines, plays in that process. Specifically, it discusses the different mappings that can be produced using various more-than- and non-euclidean metrics for distance and association. Travel-time across districts is presented as one means of measuring equity, access, and representation within voting and representational districts.

**Date**: Friday, November 15 at 3:30pm

**Room**: Roosevelt 201

**Speaker**: Mike Chinbayar, Hofstra University

**Title**: Pascal's Triangle and Finding the Expected Value

**Abstract**: In a recent paper, Spiegelhofer and Wallner demonstrated that the *n*th row of Pascal's Triangle, when put under the 2-adic valuation, typically follows a normal distribution. In addition, they discussed the possibility of generalizing their result for some prime number *p*. We show a possible conjecture on what that general result may be.

**Date**: Wednesday, November 20 at 11:30am

**Speaker**: Kawkab Abid, Hofstra University

**Title**: Digit patterns of the Collatz function in base 3

**Abstract**: We study the patterns of the Collatz function in base 3. When the Collatz function is evaluated, we observe an array consisting of 0, 1 and 2. We prove several theorems describing the patterns that form.

**Date**: Wednesday, January 29 at 11:30am

**Speaker**: Célia Cisternino, University of Liège

**Title**: Properties of the alternate base representations

**Abstract**: Depending on the considered base, numbers can be represented differently. Émilie Charlier and I introduced the new theory of the alternate bases representations. In this generalization of the beta representations, we use a *p*-tuple *β*_{1}, *β*_{2}, ..., *β*_{p} as a base to represent any real number in [0,1]. During this presentation I will first recall the theory, already widely studied, of the beta representations. Then, some properties of the alternate base representations will be stated as well as the generalization of Parry's Theorem.

**Date**: Friday, February 7 at 3:30pm

**Speaker**: Peter Winkler, Dartmouth / MoMath

**Title**: When Can You Avoid Backward Steps?

**Abstract**: Suppose the whole computer system at Hofstra requires upgrades, but all changes must be done without reducing service below a certain level. If this can be done, can it be done without at some point *down*-grading some component? Inspired by a still-open problem in metric topology, we develop a model for answering *some* questions of this form. The consequence is a set of general conditions under which optimal scheduling can be done without backward steps. Among the applications are observations about searching for a lost child in a forest, and a fast algorithm for scheduling multiple processes without overusing a resource. This is joint work in part with Graham Brightwell (LSE) and in part with Lizz Moseman (NSA).

**Date**: Wednesday, February 26 at 11:30am

**Speaker**: Marina Jacobo, Hofstra University

**Title**: Extremal Graphs With Large Rank Numbers

**Abstract**: A *k*-ranking of a graph *G* is a function *f* : *V*(*G*) → {1, 2, ..., *k*} such that if *f*(*u*) = *f*(*v*) then every *uv* path contains a vertex *w* such that *f*(*w*) > *f*(*u*). The rank number of *G*, denoted *χ*_{r}(*G*), is the minimum *k* such that a *k*-ranking exists for *G*. The rank number is a variant of graph colorings. It is known that given a graph *G* and a positive integer *t* the question of whether *χ*_{r}(*G*) ≤ *t* is NP-complete. The characteristics of any *n*-vertex graph whose rank number is equal to *n*−1 or *n*−2 is known; in this talk we extend this question to *n*−3. Also, we examine the extremal graphs such that their rank number is equal to *n*, *n*−1, *n*−2 and *n*−3.

**Date**: Wednesday, March 4 at 11:30am

**Speaker**: Lara Pudwell, Valparaiso University

**Title**: What's in your wallet?!

**Abstract**: You may associate this title with credit card commercials, but it is also an invitation to some interesting probabilistic mathematics. A Markov chain is a model for analyzing sequences of events where each event only depends on the result of the previous event. In this talk, we will explore Markov chains in general, and use them to answer the particular question "what is the most likely distribution of coins to have in your wallet?".

**Date**: Friday, March 13 at 3:30pm [This talk has been canceled.]

**Speaker**: Emily Gunawan, University of Connecticut

**Title**: Cluster algebras and binary subwords

**Abstract**: We establish a connection between binary subwords and perfect matchings of a snake graph, an important tool in the theory of cluster algebras. Every binary expansion *w* can be associated to a piecewise-linear poset *P* and a snake graph *G*. We will construct a tree structure called the antichain trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and Stipulanti. We then present bijections from the subwords of *w* to the antichains of *P* and to the perfect matchings of *G*. This is joint work with Rachel Bailey (https://arxiv.org/abs/1910.07611).

**Date**: Friday, April 3 at 3:30pm

**Speaker**: Benjamin Gaines, Iona College

**Title**: Playing to Win: Winning Strategies in the Game of Cycles

**Abstract**: A combinatorial game is a two player game that has a well-defined ruleset and no element of chance. This means that if both players play optimally, the winner can be determined before the game even begins. The Game of Cycles is a new combinatorial game played on any simple connected planar graph, introduced by Su (2020). In this talk I will introduce the basics of combinatorial game theory, the rules for the Game of Cycles in particular, and discuss results we have found about which player has a winning strategy on various classes of gameboard. This is joint work with Ryan Alvarado, Maia Averett, Christopher Jackson, Mary Leah Karker, Malgorzata Aneta Marciniak, Francis Edward Su, and Shanise Walker.

**Date**: Wednesday, April 15 at 11:30am

**Speaker**: Michael Cole, Hofstra University

**Title**: Cubic and Quartic Analogues of the Trigonometric Functions

**Abstract**: PDF

**Date**: Wednesday, April 22 at 11:30am

**Speaker**: Brandon Crofts, Hofstra University

**Title**: An Exploration and Generalization of the KRC Sequence

**Abstract**: For the sequence defined as KRC(*n*) = "the amount of integer triples (*a*,*b*,*c*) which satisfy *a*^{2} + 2*bc* = 0 where *a*,*b*,*c* are bounded by *n*," previous algorithms were simplistic and costly. These inefficiencies were due to their recursive nature. An algorithm not relying on recursive principles was theorized, developed, and partially optimized. Upper and lower bounds of this function were considered, as well as attempting to write this function algebraically.

From this point, the sequence defined as KRC3(*n*) = "the amount of integer triples (*a*,*b*,*c*) which satisfy *a*^{2} + 2*bc* = 0 where *a*,*b*,*c* are bounded by *n*," was brought to the forefront of the project. Our previous algorithm was adapted and updated to fit. Further optimizations were discovered, and a general algorithm was written for KRCP(*n*, *p*) = "the amount of integer triples (*a*,*b*,*c*) which satisfy *a*^{2} + *pbc* = 0 where *a*,*b*,*c* are bounded by *n*, and *p* is prime."

**Date**: Wednesday, April 29 at 11:30am

**Speaker**: Briana Schmidt, Hofstra University

**Title**: Defining Long Island's Sea Breeze Event Using Hofstra Data and D-Basis Algorithm

**Abstract**: Weather prediction is important. From daily weather forecasts to hurricane paths, we rely on it on a daily basis. Studying one localized weather event, like the Long Island sea breeze event, can improve small-scale weather predictions and pave the way to more accurate predictions of large-scale storms. Hear about the process of defining the sea breeze event using data collected at Hofstra. Future goals include a new application of the D-basis algorithm, developed at Hofstra Math Department, to predict the sea breeze event. My supervisor was Dr. Adaricheva and we collaborated with Dr. Bernhardt.

**Date**: Wednesday, May 6 at 11:30am

**Speaker**: Daniel Dimijian, Hofstra University

**Title**: Searching the Apéry numbers efficiently

**Abstract**: The Apéry numbers constitute a sequence with many interesting number-theoretic properties. However, terms in the sequence quickly become very expensive to compute, so it can be difficult to study experimentally. In part of an attempt to determine a nice formula for the *p*-adic valuation of arbitrary terms in the sequence with regard to certain primes *p*, we searched the sequence for values that satisfied a particular rare property. In order to make the search computationally feasible, we employed several optimization strategies and attempted to split the sequence into chunks to be searched independently and in parallel.

Our Many Cheerful Facts seminar took place over the summer.