I am currently a Maryam Mirzakhani Postdoctoral Fellow at Stanford, with mentor Ciprian Manolescu. From the fall of 2018 to the summer of 2020, I was a postdoctoral researcher at UCLA.
I got my PhD at MIT, advised by Tom Mrowka.
The main theme of my research is using tools from analysis (including differential operators, gauge theory, and operator algebras) to study geometric and topological invariants of manifolds. My research output can be divided into two main branches, low dimensional topology and non-commutative geometry.
In low dimensional topology, I primarily study knots and links using Floer theories, which are families of invariants of knots, links, and 3-manifolds built by doing an infinite-dimensional analogue of Morse theory on a space of geometric structures on the underlying knot, link, or 3-manifold.
In non-commutative geometry, I study certain properties of group C* algebras, looking, in particular, at the K-theory of such operator algebras. The study of the K theory of operator algebras associated to a group G can be used to understand the structure of differentiable manifolds with fundamental group G.
You can find my CV here.
Papers1. Non-orientable link cobordisms and torsion order in Floer homologies (with Marco Marengon), arXiv preprint, https://arxiv.org/abs/2010.06577
2. On the Kronheimer-Mrowka concordance invariant, https://arxiv.org/abs/1908.05018, accepted for publication in Journal of Topology
3. The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces (with Jianchao Wu and Guoliang Yu), https://arxiv.org/abs/1811.02086, submitted to Geometric and Functional Analysis.
4. Khovanov homology and binary dihedral representations for marked links, Journal of Geometry and Physics, Volume 140, June 2019, Pages 214-245.
5. Property RD and the Classification of Traces on Reduced Group $C^*$-algebras of Hyperbolic Groups, Journal of Topology and Analysis, (2017) Volume 9, Issue 4, Pages 707-716.
6. Finite part of operator K-theory for groups with rapid decay, Journal of Noncommutative Geometry, (2015) Volume 9, Issue 3, Pages 697-706.
7. Computational approaches to Poisson traces associated to finite subgroups of Sp(2n,C) (with Pavel Etingof, Aldo Pacchiano, Qingchun Ren, and Travis Schedler), Experimental Mathematics, (2012) Volume 21, Issue 2, Pages 141-170.
8. On Congruence Conditions for Primality (with Scott Kominers), INTEGERS: The Electronic Journal of Combinatorial Number Theory, (2010) Volume 10, Issue 3, Pages 313-317.
9. On a problem regarding coefficients of cyclotomic polynomials, Journal of Number Theory, (2009) Volume 129, Issue 12, Pages 2883-3114.
Teaching and Outreach
- UCLA Math 32a, Multivariable Calculus, Winter 2019, Winter 2020.
- UCLA Math 131a, Real analysis, Fall 2018, Spring 2019, Winter 2020.
- UCLA Math 182, Algorithms, Fall 2019
- UCLA Math 110A, Abstract Algebra, Fall 2019
- UCLA Math 31a, Single variable calculus, Fall 2018.
- TA for MIT course 18.02, multivariable calculus.
- In 2020, I ran a pre-camp program for the girls at the Math Olympiad Program (MOP) to help them get acclimated and get to know each other, running all classes on the academic side and participating in all activities.
- I was an instructor for the Puerto Rican Math olympiad teams several times throughout the past few years, and in particular in the Summer and Fall of 2020.
- I volunteered for BEAM LA, a program to help low-income students from underserved communities learn advanced mathematics and enter a career in STEM fields. I have volunteered for this program to help run events, and to teach an introductory programming class every Saturday during the Fall semester of 2019.
- I was the leader of the Puerto Rican team to the Olimpiada Iberoamericana de Matematicas, Fall 2015.
- I was the leader or deputy leader for every United States team to the European Girls' Math Olympiad from 2012-2017, and for the US team to the China Girls' Math Olympiad in 2010 and 2011.
- I taught multiple times at MOP, a training program for the United States team to the International Mathematical Olympiad (2012, 2014, 2015).
- I taught at S.T. Yau's Summer Camp for high school students at Tsinghua university, 2019.
- I was an instructor at Brazil's math olympiad training program, Summer 2012.