| Presenter | Coauthors | Title & Abstract |
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Thomas Brazelton University of Pennsylvania |
Joshua Harrington Cedar Crest College Matthew Litman University of California Davis Wing Hong Tony Wong Kutztown University of Pennsylvania |
Residue Sums of Dickson Polynomials Over Finite Fields Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over these residues can sometimes be computed independently of the choice of prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this talk we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over all primes, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree. |
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Lindsay Dever Bryn Mawr College |
Prime geodesics on compact hyperbolic 3-manifolds The study of hyperbolic 3-manifolds draws deep connections between number theory, geometry, and topology. An important geometric invariant of a hyperbolic 3-manifold is the set of its closed geodesics, which are parametrized by their length and holonomy. It turns out that for geodesics of increasing lengths, holonomy is equidistributed throughout the circle; it is equally likely to land in any interval of a given size. In this talk, I will introduce compact hyperbolic 3-manifolds and present results on the distribution of holonomy, including equidistribution in shrinking intervals and bias in the finer distribution of holonomy. |
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Ronald J. Gould Emory University |
On the Saturation Spectrum of Odd Cycles Given a graph $H$, a graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but the addition of any missing edge to $G$ creates at least one copy of $H$ in $G$. We then say $G$ is $H$-saturated. The extremal number for $H$ is the maximum number of edges in an $H$-saturated graph of order $n$, while the saturation number is the minimum number of edges in an $H$-saturated graph of order $n$. The saturation spectrum is the collection of all pairs $(n,m)$ for which there exists an $H$-saturated graph of order $n$ with $m$ edges. In this talk we will explore the saturation spectrum for any odd cycle. We will also demonstrate the complete spectrum for $C_5$. Since the exact saturation number for any larger odd cycle is not know exactly, we cannot give the complete spectrum for these cycles, but we will provide a large range of values that includes all but at most $O(kn)$ possible sizes for the odd cycles $C_{2k+1}$ for $k \geq 3$. |
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Helen G. Grundman Bryn Mawr College & Brown University |
The Many Flavors of Happy Functions The happy function maps a positive integer (expressed in base 10) to the sum of the squares of its digits. A happy number is a positive integer that, under repeated applications of the happy function, maps to 1. The iterative behavior of the happy function and its many variations has been the subject of study for over 75 years. In this talk, I will discuss a range of results and open questions concerning these functions and numbers, focusing on the ideas behind the methods, rather than the details of the proofs. |
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Alyssa Haskins Southern Connecticut State University |
Abby Allen Southern Connecticut State University |
Counting the Number of Proper Skolem Labelings with the Skolem Number Building on existing results for counting Skolem arrays, we will explore how to count proper Skolem Labelings of Ladder Graphs. When relaxing constraints on the number of times a label can appear on a graph, the number of Skolem Labelings increases significantly. In this presentation, we will look deeper into counting the Skolem Labelings of Ladder Graphs as it relates to Skolem arrays. We will discuss how a Skolem Labeling works for both sequences and arrays. We will provide examples and a case analysis for labeling a ladder graph as we begin to explore the idea of counting the number of distinct Skolem Labelings. We will discuss ways to count and determine to the total number of distinct labelings using multiple strategies. On top of this, we will show an analysis of a specific size ladder graph and display our results to this point. |
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Max Lind Princeton University |
Eugene Fiorini DIMACS (Rutgers University) Griffin Johnston Villanova University Andrew Woldar Villanova University Tony Wong Kutztown University |
Cycles in Assignment Graphs A pebble assignment $(S_G)$ on a finite simple graph $G$ is a distribution of a finite number of pebbles on the vertices of $G$. A standard pebbling move removes two pebbles from a vertex $x$ of $G$ while adding one pebble to a vertex adjacent to $x$. An assignment graph $[S_G]$ is a single-rooted Hasse diagram depicting all possible states resulting from a given pebble assignment $S_G$. We construct assignment graphs of every possible (even) girth and give necessary and sufficient conditions for $[S_G]$ to have girth 4. We extend the notion of an assignment graph to that of a multiassignment graph (a multirooted Hasse diagram formed by amalgamating two or more assignment graphs on $G$) and resolve the question: When can a multiassignment graph be embedded in an assignment graph? Resolution of this question is critical to our main result: Every possible cycle type of girth at most $2n$ can be simultaneously realized in a suitable assignment graph. |
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Matthew Litman University of California Davis |
Cameron Bjorklund University of California Davis |
Error Approximation for Backwards and Simple Continued Fractions In this talk, we provide a new framework for studying continued fractions utilizing the backwards continued fraction (BCF). We show an approximation theory for BCFs, the correspondence between continued fractions and their backwards continued fractions counterpart, and illustrate a rich approximation theory for continued fractions (CFs) utilizing the methods of the approximation theory for the backwards continued fractions. In particular, we construct explicit functions that bound the BCF or CF error over any BCF or CF cylinder set, and along the way work out the details to pass seamlessly between the BCF and CF expansion of any real number. This is joint work with Cameron Bjorklund. |
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Jessica McDonald Auburn University |
3-Flows with Large Support We'll discuss flows in graphs, and in particular 3-flows in 3-edge-connected graphs. Not all such graphs have nowhere-zero 3-flows, but with DeVos, Pivotto, Rollova, and Samal we proved that they all have 3-flows with support size at least 5/6 of the edges. This result is tight, via $K_4$ and via an infinite family. |
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Michael Mossinghoff Center for Communications Research |
Greg Martin University of British Columbia Timothy Trudgian UNSW Canberra at ADFA |
Fake Mu's The sum of the Möbius function $\mu(n)$ over positive integers $n < x$ appears unbiased in sign: suitably normalized, it appears to (roughly) oscillate about the $x$-axis. The same process using the similarly defined Liouville function $\lambda(n)$ however has a negative bias: the values of its normalized summatory function appear to roughly oscillate around a real number near $-0.7$. In fact, a well-known problem posed by Pólya more than a century ago (and since resolved) asked if the latter sum ever achieved positive values for large $x$. We consider a natural family of functions, dubbed fake $\mu$'s which includes both the Möbius and Liouville functions. We determine when their summatory functions exhibit a similar bias, exhibit extremal examples, and discuss some connections to other problems, including the Riemann hypothesis. |
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Daniel White Gettysberg College |
Extreme values of Hecke $L$-functions to angular characters We prove the frequency-aspect analogue of Soundararajan's result on extreme values of the Riemann zeta function for Hecke $L$-functions to angular characters over imaginary quadratic number fields. This result relies on the resonance method, which is applied for the first time to this family of $L$-functions, where the classification and extraction of diagonal terms depends on the geometry of the associated field's complex embedding. |
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Wing Hong Tony Wong Kutztown University of Pennsylvania |
Carrie Finch-Smith Washington and Lee University Joshua Harrington Cedar Crest College |
Sums of distinct polynomial residues Let $p\geq 5$ be a prime. In 1801, Gauss proved that the sum of distinct quadratic residues modulo $p$ is congruent to $0$ modulo $p$. A study by Stetson in 1904 showed that the sum of distinct triangular residues modulo $p$ is congruent to $-1/16$ modulo $p$. Both of these results were extended in 2017 by Gross, Harrington, and Minott, who studied the sum of distinct quadratic polynomial residues modulo $p$. In this article, we determine the sum of distinct cubic polynomial residues modulo $p$ and prove a conjecture of Gross, Harrington, and Minott. We further consider the sum of distinct residues modulo $p$ for polynomials of higher degree. |