Levent Alpoge The average number of integral points on elliptic curves is bounded
I'll prove the theorem stated in the title, which goes back to my senior thesis and was the clear first step in my current research path. I'll also explain that it was only thanks to Joe's persistent encouragement that I even asked for a problem for said thesis!
Yakov Berchenko-Kogan The Combinatorics of Finite Element Methods
Traditionally, finite element methods belong in applied math as a way of discretizing and numerically solving PDEs. However, recent decades have shown that bridges between the discrete and the continuous have implications well beyond their original fields of study. Focusing on combinatorics, I will discuss the link between finite element spaces and the Euler characteristic (or, more generally, the simplicial cohomology) of simplicial triangulations. Most of the talk will be expository, following the work of Arnold, Falk, and Winther, who developed a framework called finite element exterior calculus in a seminal 2006 paper that simultaneously generalized the work of both geometers and numerical analysts. In particular, they showed that the aforementioned link to cohomology is essentially the reason why some numerical methods give accurate answers and others do not. At the end of the talk, I will briefly discuss some more recent results and research directions.
Manjul Bhargava Galois groups of random polynomials
Stephen Curran A Variation of Nim Played on Boolean Matrices
We consider a version of matrix Nim played on a Boolean matrix. Each player, in turn, removes a non-zero row or column. The last player to remove a row or column wins. We investigate the Boolean matrices that represent the Ferrers diagram of an integer partition. An integer partition in which each summand is greater than the number of terms in the partition is said to be strong. The Grundy numbers of these Boolean matrices consisting of three or fewer rows are determined. This allows us to classify the P-positions and N-positions of Boolean matrices that represent the Ferrers diagram of any strong integer partition.
Samit Dasgupta Stark's Conjectures and Hilbert's 12th Problem
In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory and the special values of L-functions. The goal of explicit class field theory is to describe certain extensions of a number field via analytic means; this question lies at the core of Hilbert's 12th Problem. Meanwhile, there is an abundance of conjectures on the special values of certain important functions that arise in number theory, called L-functions. Of these, Stark's Conjecture has special relevance toward explicit class field theory.
I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture. This conjecture states the existence of certain special elements in extensions of fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units essentially generate the maximal abelian extension of a totally real field, thereby giving a solution to the question of explicit class field theory for these fields.
Colin Defant A Smorgasbord of Ungar Moves
Inspired by Ungar's solution to the famous slopes problem, I will introduce Ungar moves, which are operations that can be performed on elements of a finite lattice. I will state several results and open problems concerning manifestations of Ungar moves in algebraic combinatorics, combinatorial dynamics, combinatorial probability, and combinatorial game theory. The talk will focus on some of my solo work, an article written jointly with (2021 Duluthian) Rupert Li, and an article written jointly with (2018 Duluthian) Noah Kravitz and Nathan Williams. I will also mention results obtained in the Duluth REU by Letong (Carina) Hong (in 2021), Yunseo Choi and Nathan Sun (in 2022), and Eric Shen (in 2023).
Ellen Eischen Some congruences and consequences in number theory and beyond
I will trace a path from some problems I was thinking about when I applied to the Duluth REU to some recent developments in my research. We will start with the Bernoulli numbers, which link several seemingly unrelated problems, and end with recent progress for automorphic forms, certain functions that play key roles in number theory.
Jacob Fox Ramsey theory, information theory, and random graphs
A graph is Ramsey if its largest clique or independent set is of size logarithmic in the number of vertices. While almost all graphs are Ramsey, there is still no known explicit construction of Ramsey graphs. Alon conjectured that every finite group has a Ramsey Cayley graph. We prove that for almost all n, all abelian groups of order n satisfy Alon's conjecture. We also verify a conjecture of Alon and Orlitsky motivated by information theory that there are self-complementary Ramsey Cayley graphs. We further prove general results for clique numbers of random graph models. Along the way, we study some fundamental problems in additive combinatorics, and discover that group structure is superfluous for these problems. Based on joint work with David Conlon, Huy Pham, and Liana Yepremyan.
Maria Monks Gillespie From partition reconstruction to battery-powered tableaux: a Duluth REU-inspired journey
Partitions and Young tableaux are combinatorial objects that play an important role in the intersection of symmetric function theory, representation theory, and geometry. I will give an overview of how these fields are tied together by these simple combinatorial objects, and give a brief history of how I got interested in it, going back to my first Duluth REU project. I will also describe my most recent joint result with Sean Griffin, which gives a new formula for a collection of symmetric functions that arise in geometric representation theory, using a particular new combinatorial object that we call a battery-powered tableau.
Benjamin Gunby Antichain Codes
Let S be a subset of the Boolean cube that is both an antichain and a distance-2r+1 code. How large can S be? We discuss this question and its relation to results in anticoncentration. Joint work with Xiaoyu He, Bhargav Narayanan, and Sam Spiro.
Patricia Hersh Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology
Given a polytope P and a nonzero vector c, the question of which point in P has largest inner product with c is the main goal of linear programming in operations research. Key to efficiency questions regarding linear programming is the directed graph G(P,c) on the 1-skeleton of P obtained by directing each edge e(u,v) from u to v for c(u) < c(v) and in particular the diameter of G(P,c). We will explore the question of finding sufficient conditions on P and c to guarantee that no directed path ever revisits any polytope face that it has left; this is enough to ensure that linear programming is efficient under all possible choices of pivot rule. It turns out that poset-theoretic techniques and poset topology can help shed light on this question. In fact, a high school student named Dominik Preuss recently proved a conjecture of ours that the monotone Hirsch conjecture holds for any simple polytope P and cost vector c such that G(P,c) is the Hasse diagram of a lattice. We will provide history and background along the way in telling this story.
Adam Hesterberg Conflict-free graph coloring
A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. I'll discuss the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). In particular, I'll discuss a proof (with collaborators) of the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.
Wade Hindes Counting points of bounded height in semigroup orbits
We improve known estimates for the number of points of bounded height in semigroup orbits of polarized dynamical systems. In particular, we give exact asymptotics for generic semigroups acting on the projective line. The main new ingredient is the Wiener-Ikehara Tauberian theorem, which we use to count functions in semigroups of bounded degree.
Nathan Kaplan Numerical Semigroups and the Duluth REU
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is genus of the semigroup and the largest element of the complement is its Frobenius number. How many numerical semigroups have genus g? How many numerical semigroups have Frobenius number F? What does a random numerical semigroup of genus g ‘look like’? In this talk, I will give an overview of some excellent work done on these kinds of questions by students at the Duluth REU.
Kiran Kedlaya Tetrahedra with rational dihedral angles
In joint work with Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein, we describe the classification of similarity classes of tetrahedra with the property that the dihedral angles along all six edges have rational measures (in degrees). This confirms a conjecture made by Poonen and Rubinstein in 1995 on the basis of numerical computations; our proof reduces the problem to this computation plus some further algebraic computations in some cyclotomic fields. In the process, we also classify configurations of lines through the origin in R^3 (of arbitrary size) with the property that the angle between any two of the lines have rational measures.
Harun Kir The refined Humbert invariant with automorphism group of a genus 2 curve
Ernst Kani recently showed that the automorphism group Aut(C) of a genus 2 curve C can be determined from its associated refined Humbert invariant q_C , which is a positive definite quadratic form. In this talk, for a given automor- phism group of a genus 2 curve C, where J_C is a product of two isogenous CM elliptic curves E_1 and E_2, we will list all possible quadratic forms q_C in the form of Eisenstein-reduced integral ternary quadratic form. For this purpose, we will classify ternary integral quadratic forms according to their automorphism groups. As an application of our list, we will mention new results on the intersection of Humbert surfaces. We will also illustrate how these results reprove some interesting known results in the literature, and what they say for the extension of these results.
Sandor Kiss Generalized Sidon sets of perfect powers
For any fixed integer $k \ge 2$ and an infinite set of positive integers $A$, let $R_{A,k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ distinct terms from $A$. Given positive integers $g \ge 1$, $h \ge 2$, we say a set of positive integers $A$ is a $B_h[g]$ set if every postive integer can be written as the sum of $h$ not necessarily distinct terms from $A$ at most $g$ different ways. A set $A$ of positive integers is called a basis of order $k$ if every positive integer can be written as the sum of $k$ terms from $A$. We say a basis $A$ of order $k$ is thin if the number of representations of $n$ as the sum of $k$ terms from $A$ is positive but small for every positive integer $n$. A few years ago, Van Ha Vu proved the existence of a thin basis of order $k$ formed by perfect powers. In my talk I would like to speak about $B_{h}[g]$ sets formed by perfect powers. I prove the existence of a set $A$ formed by perfect powers with almost possible maximal density such that $R_{A,h}(n)$ is bounded. The proof is based on the probabilistic method. This is a joint work with Csaba S\'andor.
Noah Kravitz An update on the Lonely Runner Conjecture
Dirichlet's Theorem says that for any real number t, there is some v in {1,2,...,n} such that tv lies within 1/(n+1) of an integer. The Lonely Runner Conjecture of Wills and Cusick asserts that the constant 1/(n+1) in this theorem cannot be improved by replacing {1,2,...,n} with a different set of n nonzero real numbers. The conjecture, although now more than 50 years old, remains wide open for n larger than 6. In this talk I will describe the "Lonely Runner spectra" that arise when one considers the "inverse problem" for the Lonely Runner Conjecture, and I will explain the (a priori surprising) "hierarchical" relations among these spectra. Based on joint work with Vikram Giri.
Nikola Kuzmanovski Macaulay Posets and Rings
Macaulay posets are posets in which an analog of the Kruskal-Katona Theorem holds. Macaulay rings (often called Macaulay-Lex rings in the literature) are rings in which an analog of Macaulay’s Theorem for lex ideal holds. The study of both of these objects started with Macaulay almost a century ago. The talk will present an equivalence between these two objects. Another proof of the Mermin-Murai theorem for colored square free rings will be given, and the theorem will be extend to rings which are not square free.
Gaku Liu Unimodular triangulations of sufficiently large dilations
An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope is a triangulation in which all simplices are integral with minimum volume. A classic result of Knudsen, Mumford, and Waterman states that for every integral polytope, there exists a positive integer c such that cP has a unimodular triangulation. We strengthen this result by showing that for every integral polytope P, cP has a unimodular triangulation for all but finitely many positive integers c.
Joel Louwsma Arithmetical structures and their critical groups
An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group, which may be regarded as a generalization of the sandpile group of a graph. We present results about the groups that occur as critical groups of arithmetical structures on several families of graphs, including star graphs, complete graphs, and trees. These results are joint work with subsets of Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis Garcia Puente, and Darren Glass.
Yelena Mandelshtam The combinatorics of m=1 Grasstopes
The amplituhedron is an object introduced by physicists in 2013 arising from their study of scattering amplitudes which has garnered much recent attention from physicists and mathematicians alike. Mathematically, it is a linear projection of a nonnegative Grassmannian to a smaller Grassmannian, via a map induced by a totally positive matrix. A Grassmann polytope, or Grasstope, is a generalization of the amplituhedron, defined to be such a projection by any matrix, removing the positivity condition. In this talk, I will discuss joint work with Dmitrii Pavlov and Lizzie Pratt in which we study these objects, with hope that we may gain new insights by broadening our horizons and studying all Grasstopes.
Alison Beth Miller Seifert Surfaces and Binary Quadratic Forms
A recent paper of Hayden-Kim-Miller-Park-Sundberg answered a 40-year-old question of Livingston by constructing a pair of Seifert surfaces for the same knot that do not become isotopic when pushed into the 4-ball. Remarkably, the surfaces can be distinguished by a classical invariant, whose values are binary quadratic forms. We study algebraically the natural generalization of this construction to pairs of Seifert surfaces bounded by a separating curve in a genus 2 surface. We show that the underlying algebra is closely connected to ideal classes of quadratic rings and Gauss composition of binary quadratic forms. Using this, we characterize exactly which pairs of binary quadratic forms can be attained by this construction. This talk is based on joint work in progress with Menny Aka, Peter Feller, and Andreas Wieser.
Dave Witte Morris Hamiltonian checkerboards
Place a checker on some square of an m x n rectangular checkerboard. Asking whether the checker can tour the board, visiting all of the squares without repeats, is the same as asking whether a certain graph has a hamiltonian path. The question becomes more interesting if we allow the checker to step off the edge of the board. This topic has connections with ideas from elementary topology and group theory, and has been studied by Duluth REU students and other mathematicians for over 40 years. No advanced mathematical training will be needed to understand most of this talk.
Rishi Nath Core Partitions and the Duluth REU
The Duluth REU, under the direction of Joe Gallian, has played an important role in the development of the theory of core partitions over the last decade. In this talk we discuss how this came to be, and highlight some of the main results that undergraduate researchers at Duluth produced. We will also discuss the influence that these results have had on the current directions in the field.
Kelly O'Connor The Principal Chebotarev Density Theorem
Let K/k be a finite Galois extension. We define a principal version of the Chebotarev density theorem which represents the density of prime ideals of k that factor into a product of principal prime ideals in K. We find explicit equations to express the principal density in terms of the invariants of K/k and give an effective bound which can be used to verify the non-splitting of the Hilbert exact sequence.
Andrew O'Desky Polynomials with abelian Galois group
How many monic integer polynomials are there with bounded height and given abelian Galois group? This can be translated into a Diophantine problem about toric varieties. We will discuss a proof for an asymptotic formula of the form c H^a log^(b-1)(H)(1+o(1)).
Shariefuddin Pirzada On Laplacian eigenvalues of graphs
We talk about the Laplacian eigenvalues of graphs. In particular we discuss the sum of k-largest Laplacian eigenvalues of graphs and some recent developments on Brouwer's conjecture.
Bjorn Poonen Integral points on curves
I will give a survey of approaches to determining the set of integral points on a curve and explain recent work with Aaron Landesman showing that one of these approaches will not succeed in general.
James Sellers An m-ary partition generalization of a past Putnam problem
Given Joe Gallian's interest in, and significant involvement with, the Putnam Exam over the past few decades, the focus of this talk will be the following problem which appeared as Problem B-2 on the Putnam Exam in 1983: ``For positive integers $n$, let $C(n)$ be the number of representations of n as a sum of nonincreasing powers of 2, where no power can be used more than three times ... Prove or disprove that there is a polynomial $P(x)$ such that $C(n) = \lfloor P(n) \rfloor$ for all positive integers $n$.'' We use generating functions to generalize this problem to the enumeration of a two-parameter family of $m$-ary integer partitions, which we denote by $b_{m,j}^*(n)$. In addition, we use generating functions and a bijection to give an identity between $b_{m,j}^*(n)$ and a related family of $m$-ary partitions.
Lauren Williams Combinatorics of hopping particles and positivity in Markov chains
The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. I will give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics and special functions. I will also mention some open problems and observations about positivity in Markov chains.
Yufei Zhao Graph theory and additive combinatorics
I will give a reflection of my work at the Duluth REU and how it led me to a quest to better understand the connections between graph theory and additive combinatorics.