Math

(low-dimensional geometric topology)

Interests

I enjoy building smooth 4-manifolds. In particular, I'm interested in using horizontal handles [LP22] to construct fillings of 3-manifolds (which give rise to interesting embeddings, H-slice knots, etc.)

Preprints

Rationally slice knots that are not biprojectively H-slice, 2026. Preprint, arXiv:2610.41996.

We show that rationally slice knots need not be biprojectively H-slice (BPH-slice), answering a question in the study of the rational knot concordance group. Our main tool is a lattice embedding obstruction to H-sliceness in #nℂℙ2 for knots with vanishing signature, extending work of Kjuchukova–Miller–Ray–Sakallı. We apply the obstruction to an infinite family of rationally slice knots, beginning with 818. As additional applications, we obstruct the BPH-sliceness of a family of 2-bridge knots which is not obstructed by the instanton knot concordance invariant r0, we obstruct a non-alternating family of Montesinos knots, and we give prime alternating knots that are topologically but not smoothly biprojectively H-slice. We discuss why these knots remain plausible candidates for detecting exotic smooth structures on #nℂℙ2 and suggest natural sources of further candidates.

In addition to the acknowledgments already listed in the paper, thank you to Brendan Owens, Oğuz Şavk, and Abhishek Shivkumar for feedback on earlier drafts.

Notes

Spring 2026: Notes for my short presentation on the (purely) cosmetic surgery conjecture for Maggie Miller's topics course on open problems in low-dimensional topology.

Spring 2025: Illustrated the course notes written by Abhishek Shivkumar for Lisa Piccirillo's topics course on closed smooth 4-manifolds.

Research & Expository Talks

Upcoming

February 9th, 2027: Stanford Topology Seminar.

Past

April 25th, 2026: Graduate Student Topology and Geometry Conference (at Rutgers), "Dehn Twister: Using Heegaard diagrams to construct 4-manifolds" (slides).

April 22nd, 2026: UT Austin Junior Topology Seminar, "Using Heegaard diagrams to construct 4-manifolds".

January 22, 2026: UT Austin, "Progress towards proving the (purely) cosmetic surgery conjecture" (notes).

November 18, 2025: UT Austin, "Horizontal handle decompositions and applications to embeddings" candidacy talk (slides).

April 2, 2025: UT Austin Junior Topology Seminar, "Horizontal Handle Decompositions and Fillings for S3p/q(K) ".

September 9, 2022: UT Austin Sophex Seminar, "Let's talk about SET" (slides).

August 9, 2019: YMC (at Ohio State), "Chebyshev Linearizations of Matrix Polynomials: Are They Well-Conditioned?" (poster on the same topic).

Math Outreach Talks

Upcoming

TBA

Past

March 5, 2026: Temple College, "Drawing shapes for a living (while getting a PhD in mathematics)" invited outreach talk for a community college audience (slides).

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