I am broadly interested in geometric topology and algebraic topology.
So far, my research output falls under the following headings: maps of non-zero degree between low-dimensional manifolds,
mapping class groups of finite and infinite type surfaces, and the Goldman Lie algebra.
Recently, I have started working on the obstruction to the existence of almost complex structures.
We show that every orientable infinite-type surface without boundary is properly rigid as a consequence of a more general result.
Namely, we prove that if a homotopy equivalence $f\colon\ \Sigma'\to \Sigma$ between any two non-compact orientable surfaces without boundary is a proper map,
then $f$ is properly homotopic to a homeomorphism, provided $\Sigma$ is homeomorphic to neither the plane nor the punctured plane.
Thus, all non-compact orientable surfaces without boundary, except the plane and the punctured plane, are topologically rigid in a strong sense.
We show that an oriented surface $\Sigma$ without boundary is of finite-type if and only if every proper self-map $f\colon\ \Sigma\to \Sigma$ of degree $\pm 1$ is homotopic to a homeomorphism.
Thus, an oriented surface without boundary is Hopfian if and only if it is of finite-type.
[Joint with Siddhartha Gadgil and Ajay Kumar Nair] The Goldman bracket characterizes homeomorphisms between non-compact surfaces.
Algebraic & Geometric Topology, Volume 25-6 (2025), p. 3585-3601.
arXiv:2307.02769v2.
We show that a homotopy equivalence $f\colon\ \Sigma'\to \Sigma$ ($f$ may not be proper) between two non-compact oriented surfaces without boundary is homotopic to
a homeomorphism if and only if $\iota\left(\widehat{\alpha},\widehat{\beta}\right)=\iota\left(\widehat{f\circ \alpha}, \widehat{f\circ \beta}\right)$ for any two closed curves $\alpha,\beta\colon\ \Bbb S^1\to \Sigma'$, and
an orientation preserving homeomorphism if and only if $\left[\widehat{\alpha},\widehat{\beta}\right]=\left[\widehat{f\circ \alpha}, \widehat{f\circ \beta}\right]$ for any two closed curves $\alpha,\beta\colon\ \Bbb S^1\to \Sigma'$;
provided $\Sigma$ is homeomorphic to neither the plane nor the punctured plane. Here, $\iota(-,-)$ and $[-,-]$ denote the geometric intersection number and the Goldman Lie bracket, respectively, and $\widehat\quad$ denotes the free homotopy class of a closed curve.
We classify all $\pi_1$-injective proper maps between non-compact surfaces up to proper homotopy.
Pre-print:
Geometric kernels of proper maps between non-compact surfaces. arXiv:2508.21057v1.
A map between connected $2$-manifolds has a geometric kernel if it sends a non-contractible simple loop to a null-homotopic loop.
While every non-$\pi_1$-injective map between compact surfaces admits a geometric kernel, this generally fails for compact bordered or non-compact surfaces.
In this paper, we use Brown's proper fundamental group to give a sufficient condition under which a degree-one map between non-compact surfaces admits a geometric kernel.
Furthermore, we characterize conjugacy classes in the proper fundamental group and use this characterization to establish sufficient conditions for the existence of geometric kernels.
[Joint with Rekha Santhanam] Brown's asymptotic limit functor and proper homology. arXiv:2609.14063.
In 1974, E. M. Brown introduced the $\wp$-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped.
In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem.
Bypassing abstract pro-categorical machinery, we show that $\wp$ has major advantages over the classical limits $\varprojlim$ and $\varprojlim^1$:
it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence.
As an application, we prove that the Brown-Grossman proper fundamental group of any open contractible 3-manifold other than $\mathbb{R}^3$ is uncountable and perfect.
PhD Thesis:
Here is my PhD thesis, titled Maps Between Non-compact Surfaces.