Overview
My main research work involves dynamical systems/ergodic theory and geometric structures, especially associated to (higher-rank) Lie groups. I am particularly interested in problems surrounding the dynamics of flows and group actions in geometric contexts. Below are some more specific keywords:
- Higher Teichmüller-Thurston theory
- Geodesic flows in negative curvature; geodesic currents
- Anosov representations; discrete subgroups of Lie groups
- Thermodynamic formalism; Hölder regularity
- Hyperbolic groups; Gromov hyperbolic spaces
In the past, my research projects also dabbled with analytic number theory (sparse equidistribution), convex analysis, differential topology, TQFTs, and more.
My advisor is Jeff Danciger. I have been supported by the Groups & Dynamics RTG.
Research Papers and Articles
Note that all of the papers below may be found on my arXiv page as well.
| Title | Coauthors | Journal |
|---|---|---|
| Weakly Mixing Systems with Dense Prime Orbits | None | Colloquium Mathematicum 169 (2022) |
| Extremizers of the $J$ functional with respect to the $d_1$ metric | Sam Bachhuber Benjamin Christophel Tamás Darvas |
Analysis Mathematica 48 (2022) |
| Certifying Maximal Stretch Ratios for Fuchsian Free Group Representations | Joseph Hummel Kaius Noble Prajith Velicheti |
TBD |
Talks and Presentations
Please be warned that notes linked below are very rough in some cases; they're really just designed to help me prepare for the corresponding talk. Slides (or other files), on the other hand, will generally be much more polished.
The Green-Tao Theorem
UT M 392C — Dynamical Systems
The Congruence Subgroup Conjecture
UT M 392C — Open Problems in Topology
Convex Real Projective Structures on Surfaces
UT Junior Geometry Seminar
The Nielsen Realization Theorem
UT Junior Topology Seminar
Dynamics of Geodesic Flow
UT M 392C — Riemannian Geometry
The Eight Geometries and Thurston's Geometrization
UT Junior Topology Seminar
The Pressure Metric for Hitchin Components
UT Groups & Dynamics Seminar (candidacy talk)
Geodesic Flows and Ergodicity in Negative Curvature
UT Junior Geometry Seminar
Ratner's Theorem on Unipotent Flows
UT M 390C — Arithmetic Groups
Symbolic Dynamics for Hyperbolic Groups
UT M 391C — Analytic Group Theory
Geometry from Ancient Greece to the Modern Day
UT Sunday Morning Math Group
Teichmüller Space via Hyperbolic Geometry
UT Junior Topology Seminar
Hyperbolic Groups and Gromov's Geodesic Flow
UT Junior Geometry Seminar
Measures on Homogeneous Spaces
UT M 392C — Geometry of Locally Compact Groups
Geodesic Currents, Laminations, and Teichmüller Space
UT Junior Topology Seminar
Symmetric Spaces
UT Lie Groups Seminar
Group Actions, Dynamics, and Geometry
UT Math Club
Mysterious Patterns in the Primes
UT Sunday Morning Math Group
Deforming Convex Real Projective Structures
UT M 392C — Geometric Structures on Manifolds
Sarnak's Conjecture
UT Junior Groups & Dynamics Seminar
An Introduction to Dynamics and Ergodic Theory
UT Math Club
Free Groups
Lewis Bowen's summer reading group
Topological Entropy
Lewis Bowen's summer reading group
An Introduction to Ergodic Theory and Prime Orbits
UT Sophex Seminar
An Introduction to Prime Number Theory
UT Sophex Seminar
Frobenius' Theorem on Integrable Distributions
UMD Math 742 — Geometric Analysis
Clifford Algebras and Spinor Groups
UMD Math 744 — Lie Groups
Embedded Cobordisms
UVa Topology REU
Exponential Mixing for Skew Products
UMD reading course under Dr. Giovanni Forni
Weakly Mixing Systems with Dense Prime Orbits
Penn State Dynamics Workshop
A Topological Proof of the Fundamental Theorem of Algebra
UMD DRP Presentation
A Proof of the Prime Number Theorem
UMD Math 299N — Topics in Number Theory
A Proof of the Prime Number Theorem
MBHS — Complex Analysis
The Historical Development of Non-Euclidean Geometries
MBHS — Logic Math
Conferences and Workshops
UW Madison
Mini-school in Analytic Group Theory
UT Austin
Big Ideas in Dynamics and Geometry
UIC · attended virtually
Workshop on Groups and Dynamics
Texas A&M
Ergodic Theory and its Connections
Rice University · in honor of Michael Boshernitzan
Penn State