About Me

I am a fifth-year PhD student in the Department of Mathematics at Brown University. My advisor is Bena Tshishiku. I am on the postdoc job market this fall.

I am broadly interested in low dimensional topology and geometric group theory. More specifically, I am interested in mapping class groups, and connections to low dimensional topology and algebraic geometry through the lens of bundles and monodromy.

Before coming to Brown, I was an undergraduate at the University of Virginia where I studied math and computer science. My undergraduate thesis advisor was Sara Maloni.

You can see my CV here. You can contact me at trent_lucas@brown.edu.

me!

Research

Papers and preprints

  1. Birman-Hilden theory for 3-manifolds. Submitted. (pdf, arxiv)
    Abstract

    Given a branched cover of manifolds, one can lift homeomorphisms along the cover to obtain a (virtual) homomorphism between mapping class groups. Following a question of Margalit-Winarski, we study the injectivity of this lifting map in the case of 3-manifolds. We show that in contrast to the case of surfaces, the lifting map is generally not injective for most regular branched covers of 3-manifolds. This includes the double cover of S3 branched over the unlink, which generalizes the hyperelliptic branched cover of S2. In this case, we find a finite normal generating set for the kernel of the lifting map.

  2. Veech fibrations. With Sam Freedman, submitted. (pdf, arxiv)
    Abstract

    We investigate complex surfaces that fiber over Teichmüller curves where the generic fiber is a Veech surface. When the fiber has genus one, these surfaces are elliptic fibrations; for higher genus fibers, they are typically minimal surfaces of general type. We compute the topological and complex-geometric invariants of these surfaces via the monodromy action on the mod-m homology of the fiber. We get exact values of the invariants for all known algebraically primitive Teichmüller curves.

  3. Homological representations of low genus mapping class groups. J. Algebra 647 (2024) 533-566. (pdf, arxiv)
    Abstract

    Given a finite group G acting on a surface S, the centralizer of G in the mapping class group Mod(S) has a natural representation given by its action on the homology H_1(S;Q). We consider the question of whether this representation has arithmetic image. Several authors have given positive and negative answers to this question. We give a complete answer when S has genus at most 3.

Notes and other writing

  • Hyperbolic structures on surfaces. Undergraduate thesis. (pdf)

Activities

I currently co-organize the Graduate Student Seminar at Brown with Steven Creech.

Recent and upcoming travel

    2024
  • October 2-4: Chicago Geometry & Topology/No Boundaries Seminar
  • September 17: Tufts Topology and Geometric Group Theory Seminar
  • June 10-21: Topology, Representation Theory, and Higher Structures (Sabhal Mòr Ostaig)
  • April 8-12: Young Geometric Group Theory XII (Bristol)
  • March 26: Notre Dame Topology Seminar
  • March 6-9: Spring Topology and Dynamics Conference (UNC Charlotte)
  • January 8-12: Log Cabin Workshop on Big MCGs (Young, AZ)
2023
  • November 18: GATSBY (Brown)
  • November 10-12: Texas Geometry and Topology Conference (Rice)
  • May 26-28: GTA Philadelphia (Temple)
  • May 15-19: Dynamics, Rigidity and Arithmetic in Hyperbolic Geometry (ICERM)
  • April 1: GATSBY (Yale)
2022
  • August 4-7: Mod(M^4) mini-course (Chicago)
  • July 25-29: Topology Students Workshop (Georgia Tech)
  • June 15-17: The Circle at Infinity (Harvard)
  • March 21-25: Braids in Symplectic and Algebraic Geometry (ICERM)

Teaching

Here is a list of some courses I have taught recently.

    Brown University
  • TA, CEMA 922 Set Theory (Brown Pre-College Program), Summer 2024
  • Instructor, MATH 50 Analytic Geometry and Calculus, Fall 2023
  • Instructor, MATH 180 Multivariable Calculus, Spring 2023
  • TA, MATH 180 Multivariable Calculus, Fall 2022
  • TA, MATH 90 Calculus I, Spring 2022
  • TA, MATH 90 Calculus I, Fall 2021
    • University of Virginia
  • TA, MATH 3310 Basic Real Analysis, Spring 2020
  • TA, MATH 3310 Basic Real Analysis, Fall 2019