Ethan Pesikoff

An upper level coruse in Field and Galois Theory. Primary instructor Akhil Mathew.

An introduction to the theory of commutative rings and modules, with special attention to their prime spectra. Primary instructor Oron Propp.

An introduction to the representation theory of finite groups. Primary instructor Benson Farb.

An introductory course in linear algebra.

An introductory course in complex analysis.

We construct an "exotic" copy of the rational elliptic surface \(E(1)\) using Lefschetz fibrations, i.e. a manifold M which is homeomorphic but not diffeomorphic to \(E(1)\). Lots of tools come together in surprising ways to tell this story, including general 4-manifold theory, Lefschetz fibrations, 2- and 4-manifold mapping class groups, reflections groups, and the theory of unimodular lattices.  So we won't prove everything from scratch, but we will review all the 4-manifold theory and Lefschetz fibrations background we need.  The plan is:  review some 4-manifolds, then introduce \(E(1)\).  Then we review Lefschetz fibrations and use them to construct our exotic M.  The homeomorphism is proven homologically; the key ingredient to prove \(E(1)\) and \(M\) are not diffeomorphic is the idea of minimal genus, along with input from reflection groups and 4-dimensional Dehn twists.  

You all live in a democracy (although not all of you are allowed to vote here...). You may also wonder whether our elections are fair. Depending on what you mean by 'fair', you might conclude yes or no. Personally, I think no. But say you get to design your own election system -- could you come up with something really, truly fair? Spoiler: No. We discuss different voting systems and fairness axioms, and then we'll prove some impossibility theorems. Tldr: For many reasonable fairness-axiom sets, there is no consistent voting rule which satisfies them all.

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erdős’ distinct angle problem, the problem of finding the minimum number of distinct angles between \(n\) non-collinear points in the plane. We provide new upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by \(n\) points in general position is \(\Omega(n)\) and \(O(n^2)\) using a logarithmic spiral construction. We also introduce several new asymptotically optimal configurations in the unrestricted setting and analyze their sensitivity to perturbation. In higher dimensions we show that a variant of Lenz’s construction admits fewer distinct angles than the previously known optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of \(n\) points in general position admitting only unique angles is \(\Omega(n^{1/5})\) and \(O(n^{1/2})\). The lower bound arises from a probabilistic argument.

Recent work by Fomin-Pylyavskyy-Shustin-Thurston elucidates a novel and remarkable connection between the combinatorial theory of quiver mutations and the topology of complex plane curve singularities. Specifically, the authors describe a method to extract a quiver from a real morsification of a plane curve and prove that if a quiver is obtained from a real morsification of some singularity of a complex plane curve, then that quiver uniquely determines the complex topological type of the singularity. Furthermore, the authors postulate the even stronger statement that two singularities have the same complex topological type if and only if the quivers associated with the two morsifications are mutation equivalent. In light of this recent work, we investigate the mutation equivalence classes of quivers arising from algebraic divides. Specifically, we provide a classification of which such quivers are of finite mutation type. Pending a proof of the above conjecture, this work then provides as a corollary a classification of which underlying plane curve singularities have finite topological equivalence class. Our work then provides a deeper topological understanding of complex plane curves via this novel combinatorial approach.

A partition is a weakly decreasing tuple of positive integers \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_n)\), and partition theory has been studied intensively for centuries both as pure number theory and for its applications to group representation theory. A partition can be visualized by its Young diagram, which is a left-justified array of cells where row \(i\) contains \(\lambda_i\) cells for all \(i\in [n]\). For each cell, we define its hook length to be the number of cells to its right, below it, and itself. A notion of interest in representation theory is that of an \(s\)-core partition, a partition whose Young diagram contains no cells with hook length \(s\). Recent years have seen growing interest in \(s,s+k\)-core partitions, which contain no hook lengths of either \(s\) or \(s+k\), as well as in \(d\)-distinct partitions, in which each pair of parts differs by at least \(d\). We combine these restrictions and ask what the largest hook length \(H_d\) is over all \(s,s+k\)-core partitions with \(d\)-distinct parts. We completely answer this general question with a closed-form solution for \(H_d\). For \(s\) and \(k\) coprime, we convert the question of largest hook length into one about principal order ideals of a certain poset, and we subsequently extend our result to all \(s\) and \(k\) without the coprimality restriction.

Erdős’ distinct distance problem is perhaps the most famous problem in discrete geometry. Less well known is Erdős’ later distinct angle problem, the problem of characterizing non-collinear point sets admitting few distinct angles. We introduce and provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by \(n\) points in general position is \(O(n^{\log_2(7)})\), the first non-trivial bound for such restricted sets. We introduce a new classes of asymptotically optimal point configurations and study their sensitivity to perturbation. In higher dimensions we show that a variant of Lenz’s construction admits fewer distinct angles than the optimal configurations in two dimensions. Furthermore, we demonstrate that the minimum size of a maximal subset of \(n\) points in general position which does not admit repeated angles is \(\Omega(n^{1/5})\) and \(O(n^{\log_2(7)})\). Finally, we examine partite variants of the standard distinct angle problem and provide asymptotically tight bounds.

The Zeckendorf Game (2018) is a two-player combinatorial game which can be analyzed number-theoretically. Players alternate making moves arising from the Fibonacci recurrence, and the last player to move wins. Regardless of play, the final state is unique and encodes the number \(n\) of starting chips as a sum of non-adjacent Fibonacci numbers. The second player is known to have a winning strategy when \(n>2\). However, the proof is non-constructive, and no explicit winning strategy is known. We study what happens when an extra dimension is added to the game, extending the gameboard from a tape to a planar directed graph. We show that the resulting game, the Zeckendorf Graph Game, is PSPACE-hard, and we describe a family of graphs on which it is PSPACE-complete. Additionally, we generalize this game to a broad class of games based on positive linear recurrences, and we show that the same complexity results hold in this setting. Our results both shed light on a broad and novel class of PSPACE-complete games and provide evidence that the original Zeckendorf Game is intractable to solve.

Every positive integer has a unique base-\(\beta\) decomposition when \(\beta\) is the dominant root of a non-increasing positive linear recurrence. Motivated by the two-player game that produces Zeckendorf decompositions, we define the Generalized Bergman Game on infinite tuples of nonnegative integers, with play transforming an integer into its base-\(\beta\) expansion. Starting from a state \(S\) with \(n\) summands, the longest possible game has \(\Theta(n^2)\) moves, while the shortest possible game has length between \(\Omega(n)\) and \(O(n^2)\). We also prove a linear upper bound for the largest tuple length encountered during play.

Every positive integer may be written uniquely as a base-\(\varphi\) decomposition–that is a legal sum of powers of \(\varphi\), the golden mean. Guided by earlier work on a two-player game which produces the Zeckendorf Decomposition of an integer (see [1]), we define a related game played on an infinite tuple of non-negative integers which decomposes a positive integer into its base-\(\varphi\) expansion. We call this game the Bergman Game. We prove that the longest possible Bergman game on an initial state \(S\) with \(n\) summands terminates in \(\Theta(n^2)\) time, and we also prove that the shortest possible Bergman game on an initial state terminates in \(\Theta(n)\) time. We also show a linear bound on the maximum length of the tuple used throughout the game.