**Topology of the moduli spaces of curves**

The moduli spaces of curves are geometric objects that move fluidly through geometry, topology, and representation theory. Tropical geometry provides for a route to understanding the properties of these moduli spaces using combinatorics. This project will explore a moduli space of rational weighted tropical curves. This is a polyhedral object whose points correspond to trees with a particular type of decoration. Fundamental topological questions about this moduli space, for instance its level of connectivity, inform the topology of a certain classical moduli space of curves of which little is known. The concrete goal will be to determine the homotopy type of the moduli spaces of rational weighted stable curves. Participants in this project will not require prior background in algebraic geometry, but will learn the context along the way. Through examples and explicit calculations, they will learn to work with tropical moduli spaces, matroids, and Bergman complexes.

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**Generators of Jacobians of random graphs**

The Jacobian of a graph is a finite abelian group associated to the graph that arises throughout combinatorics, algebraic geometry, and number theory. Although essentially every finite abelian group occurs as the Jacobian of some graph, jt is expected that the Jacobian of an Erdos-Renyi random graph is cyclic with probability about .79. This number occurs as the reciprocal of the product of values of the Riemann zeta function at odd integers greater than 1. With the understanding that the group is often cyclic, we are well placed to explore further questions about it. The aim of this project is to understand how often a particularly simple element is a generator of the Jacobian of a random graph. Participants will learn background about Jacobians and Erdos-Renyi random graphs; the only background assumed will be basic knowledge of abelian groups, linear algebra, and basic knowledge of graphs. A significant focus of the project will be computing examples, both by hand and by computer, in order to form conjectures.

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**Hidden signals in the Ulam sequence**

The Ulam sequence is defined as follows. Beginning with the numbers 1 and 2, each subsequent term is the is the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. The first few terms are 1,2,3,4,6,8,11,13,16,18,26,28,36,38,47…

This sequence exhibits some rather unexpected properties. For instance, one might expect that if we consider this sequence modulo some parameter T, the sequence is uniformly distributed. Surprisingly, this is not so! If we choose the parameter T the right way, a very clear pattern emerges. Moreover, the sequence seems to have positive density. In this project, we will explore the mysteries of the Ulam sequence. While there is a wealth of numerical data, there is as yet no known way to prove the corresponding conjectures. Students in this project will use experimentation in unison with tools like Fourier analysis, to gain a deeper understanding of the Ulam sequence, and find other sequences that exhibit similar phenomena.

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**The Sylvester-Gallai theorem in tropical geometry**

The Sylvester-Gallai theorem is a fundamental result in Euclidean geometry. It states that, given a collection of points in the plane that do not all lie on a line, there is a line that contains exactly two of these points. This result has some beautiful consequences, such as the de Brujin-Erdos theorem. The theorem is delicately poised — for instance, the analogous statement is false over the complex numbers! This project will explore the Sylvester—Gallai theorem for lines in the tropical plane. A collection of tropical lines can have exotic behaviour — distinct lines might intersect in infinitely many points. Still, these tropical lines reflect and inform both real and complex algebraic geometry. In this project, students will learn about the history of the Sylvester—Gallai theorem and its proofs, while trying to prove (or disprove) its generalization to the tropical setting.

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**Enumerating configurations of lines in the plane**

Moduli spaces are one of the cornerstones of algebraic geometry. The moduli space of arrangements of lines in the projective plane, constructed by Hacking-Keel-Tevelev and Alexeev, are a higher dimensional generalization of the celebrated moduli space of stable rational curves. They provide concrete examples of moduli spaces in higher dimensions that are easily accessible via combinatorics. One can naturally enrich the problem of classifying arrangements of lines in the plane by choosing to weight some of the lines by a number between 0 and 1. The goal of this project is to understand how to the moduli space (and the objects it parametrizes) changes as one varies the choices of weights. We will begin by working out the case of 5 lines, where this is completely understood, and will move on to 6 lines, where very little is known! This project will not assume background in algebraic geometry. Rather, the goal will be to illustrate various concepts of birational geometry through examples which have a rich combinatorial flavor.

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**Local h-vectors**

The h-polynomial of a simplicial complex is a fundamental invariant, determined by the number of faces of each dimension via inclusion-exclusion (Mobius inversion). In algebraic geometry, it is closely related to the dimensions of homology groups of toric manifolds. In commutative algebra, it appears as the Hilbert series of the quotient of a Stanley-Reisner ring by a homogeneous system of parameters. Both of these interpretations lead to easy proofs of combinatorially non-obvious facts about h-polynomials, e.g. that the h-polynomial of a triangulation of a manifold (with or without boundary) has non-negative integer coefficients.

Our project will focus on local h-polynomials of subdivisions. These may be interpreted as subtle refinements of h-polynomials; their name reflects the fact that h-polynomials are often expressible as sums of local h-polynomials. A fundamental and combinatorially non-obvious fact is that the coefficients of the local h-polynomial of any triangulation of a simplex are non-negative integers.

Our starting point is an intriguing problem posed by Richard Stanley a quarter century ago, in the landmark paper “Subdivisions and local h-vectors” in which he introduced this notion. Under what combinatorial conditions is the local h-polynomial zero? Under what combinatorial conditions is it nonzero? We will investigate this question starting in low dimensions, e.g. for triangulations of a triangle, and may also consider variants, such as relative local h-polynomials, which have appeared in subsequent work (e.g. “Local h-polynomials, invariants of subdivisions, and mixed Ehrhart theory” by Katz and Stapledon), and about which even less is known.