Alex Moon
The Homology of the 2-step Nilpotent Lie Algebra Indexed by Trees
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For a simple graph G, the associated 2-step nilpotent Lie algebra has a basis element for every vertex and edge, and the nonzero brackets are those that send a pair of vertices to the edge connecting them. These Lie algebras were introduced by Dani and Mainkar, who showed that they uniquely determine their graph. We examine the Lie algebra homology of the Lie algebras for a tree. First, we will look at the case of the star graph, using the natural symmetric group action on the leaves to determine the S_n-module structure of the homology. Then, we show more general results about a decomposition of the associated chain complex such that each part has concentrated homology, and use this to compute the Betti numbers of the chain complex of the path graph.
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Alejandro Galvan
The Combinatorial Nullstellensatz
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The Combinatorial Nullstellensatz is a discrete version of Hilbert's Nullstellensatz developed by Alon, based on his earlier work with Tarsi, Nathanson and Ruzsa. We will discuss several applications of this algebraic technique in the context of additive combinatorics and number theory, such as proofs of the Chevalley–Warning and Cauchy–Davenport theorems, as well as a generalization of the Erdős–Heilbronn conjecture.
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Allison Tsypin
Knots and immersed curves
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Exotic n-traces have been well-studied in low-dimensional topology for a long time. Recently, Hom and Wan showed that the absolute value of the concordance invariant epsilon is an invariant of the diffeomorphism class of the n-trace of a knot. Using this, they give examples of exotic n-traces using a family of pattern knots defined by Yasui and conjecture that in almost all cases, such n-traces are exotic. In this talk, we will discuss knots, knot concordance, knot traces, and exotic 4-manifolds, then compute the epsilon-invariant for a new family of satellite knots with Yasui's patterns. We use immersed curves methods for this computation, and deduce that these cases also give rise to exotic n-traces. This is joint work with Shreya Sinha.
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Aidan Hennessey
Chain complexes counting conjugacy classes
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natural question to ask is how many conjugacy classes there are in GLn(Fq). For fixed q, this is roughly exponential in n, but for fixed n, one obtains a degree n polynomial in q. We first show how to compute these polynomials, and then look for patterns. One pattern we’ll see is the vanishing of certain coefficients. It turns out that to each coefficient you can associate a chain complex whose Euler characteristic is the value of the coefficient, and in the vanishing regime there is a nice argument showing that the complex is exact.
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Mark Lovett
Influencer's games applied to a mixture-of-experts method.
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In this talk, I will introduce the influencer's game and some theory on bifurcation behavior (the main contribution of my thesis). I will then introduce mixture-of-experts (MoE) methods for a mathematics audience and how the theory behind influencer's games inspires a novel MoE method. Finally, if given time, I will introduce some of my work in AI alignment on the formalization of games and AI constitutions.
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Michaela & Patrick
Departmental Check-In
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Patrick and Michaela will be running a departmental check-in. This is a chance to discuss concerns in the department and brainstorm possible solutions. We will be bringing the things we discuss to a meeting with the GPC. Everyone is welcome! No issue is too small and if everything is going great, come and share your advice.
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Will Degroot and Rohan Kapoor
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