Ethan Pesikoff

I mostly think about 4-manifolds. I'm particularly interested in mapping class groups, Lefschetz fibrations, and algebraic surfaces. As an undergrad, I worked on more discrete and combinatorial type problems, mostly stemming from REUs.

Papers:

• Homological Nielsen realization for the manifolds \( \#_n \mathbb{CP}^2 \). arXiv pdf
  Abstract

  Given a smooth, oriented, simply-connected \(4\)-manifold \(M\), the homological Nielsen realization problem asks: when does a finite group of isometries \(G \leq O(H_2(M;\mathbb{Z}))\) preserving the intersection form lift isomorphically to a finite group of orientation-preserving diffeomorphisms? We study this question for the smooth, positive-definite \(4\)-manifolds \(M_n := \#_n \mathbb{CP}^2\). Even though every isometry of \(H_2(M_n;\mathbb{Z})\) is induced by some orientation-preserving diffeomorphism, not necessarily of finite order, we show that Nielsen realization is sparse: as \(n \to \infty\), a random subgroup of \(O(H_2(M_n;\mathbb{Z}))\) is asymptotically almost never realizable in \(\mathrm{Diff}^+(M_n)\); the same is true for random odd order elements of \(O(H_2(M_n;\mathbb{Z})\). We present both positive realization results in certain cases and a range of obstructions to realization in other cases. The proofs combine equivariant connected-sum constructions, fixed-point theory for group actions on \(4\)-manifolds, finite group actions on surfaces, analytic combinatorics, and previous work of Hambleton--Tanase.

• Irreducibility over the Max-Min Semiring, with B. Baily, J. Dell, H. L. Fleischmann, F. Jackson, S. J. Miller, and L. Reifenberg. Journal of Integer Sequences (2025). arXiv pdf
  Abstract

  For subsets \(A,B \subset \mathbb{N}\), write their sumset as \(A+B=\{a+b:a\in A,\ b\in B\}\). A set is irreducible if it cannot be expressed as such a sum with both summands having at least two elements. This question is equivalent to a factorization problem for Boolean polynomials. We establish irreducibility results for polynomials and power series over the max-min semiring, which generalizes the Boolean setting. Combinatorial and probabilistic arguments show that almost all max-min polynomials are irreducible, resolving a conjecture of Applegate, Le Brun, and Sloane. Measure-theoretic methods and Borel's theorem on normal numbers further show that almost all max-min power series are asymptotically irreducible.

• Optimal Point Sets Determining Few Distinct Angles, with H. L. Fleischmann, S. J. Miller, E. A. Palsson, and C. Wolf. Australasian Journal of Combinatorics (2023). arXiv pdf
  Abstract

  We characterize the largest planar point sets that determine at most one, two, or three distinct angles. If \(P(k)\) denotes the greatest possible size of a point set determining no more than \(k\) angles, then \(P(2)=5\) and \(P(3)=5\). More generally, we prove \(k+2 \leq P(k) \leq 6k\), with a possible improvement to the upper bound depending on progress on the Weak Dirac Conjecture. Thus \(P(k)=\Theta(k)\), in marked contrast with the corresponding distinct-distance problem, where the best upper bound is quadratic and the lower-bound behavior remains poorly understood.

• The maximum hook length of d-distinct simultaneous core partitions, with B. Przybocki and J. Xia. Electronic Journal of Combinatorics (2023). arXiv pdf
  Abstract

  Whenever only finitely many \((s,t)\)-core partitions with \(d\)-distinct parts exist, we determine exactly the largest hook length that such a partition can have. We also give an explicit algorithm for constructing a \(d\)-distinct \((s,t)\)-core partition attaining this maximum.

• Distinct Angles in General Position, with H. L. Fleischmann, S. V. Konyagin, S. J. Miller, E. A. Palsson, and C. Wolf. Discrete Mathematics (2023). arXiv pdf
  Abstract

  Erdős's distinct-angle problem asks for the minimum number of different angles determined by \(n\) non-collinear planar points. For point sets in general position, we improve the known upper bound from \(O(n^{\log_2(7)})\) to \(O(n^2)\). The construction uses the geometry of a logarithmic spiral directly, avoiding earlier arguments based on generic projections from higher-dimensional space. The same configuration improves the bound for the largest guaranteed subset whose angles are all distinct, from \(O(n^{\log_2(7)/3})\) to \(O(n^{1/2})\).

• Distinct Angle Problems and Variants, with H. L. Fleischmann, H. B. Hu, F. Jackson, S. J. Miller, E. A. Palsson, and C. Wolf. Discrete and Computational Geometry (2023). arXiv pdf
  Abstract

  We study a broad collection of variants of Erdős's distinct-angle problem and prove both upper and lower bounds. In particular, \(n\) points in general position can determine \(O(n^{\log_2(7)})\) distinct angles, the first nontrivial estimate of this kind. We construct a new family of asymptotically optimal configurations with no four points on a circle, analyze how such configurations behave under perturbation, and compare them with higher-dimensional variants of Lenz's construction. We also bound the smallest possible size of a maximal general-position subset with all angles distinct between \(\Omega(n^{1/5})\) and \(O(n^{\log_2(7)/3})\), and obtain estimates for partite forms of the problem.

• The Bergman Game, with B. Baily, J. Dell, I. Durmić, H. L. Fleischmann, F. Jackson, I. Mijares, S. J. Miller, L. Reifenberg, A. Smith Reina, and Y. Yang. Fibonacci Quarterly (2022). Article pdf
  Abstract

  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.

• The Generalized Bergman Game, with B. Baily, J. Dell, I. Durmić, H. L. Fleischmann, F. Jackson, I. Mijares, S. J. Miller, L. Reifenberg, A. Smith Reina, and Y. Yang. arXiv pdf
  Abstract

  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.

• The Complexity of the Zeckendorf Graph Game, with B. Baily, J. Dell, H. L. Fleischmann, F. Jackson, S. J. Miller, and L. Reifenberg. In preparation.
• Large Sets are Sumsets, with B. Baily, J. Dell, S. Dever, A. Dionne, H. L. Fleischmann, F. Jackson, S. J. Miller, L. Goldmakher, G. Gross, H. Pham, L. Reifenberg, and V. Venkatesh. In preparation.