My research is in algebraic geometry.
I study singularities, birational geometry, the minimal model program, K-stability, and moduli spaces of varieties.
My research is supported in part by NSF grant DMS-2302163.
(1) The K-moduli space of a family of conic bundle threefolds, with L. Ji, P. Kennedy-Hunt, M. Quek.
We provide a complete description of the K-moduli space generically parametrizing elements of Fano family 2.18. We find an irrational value corresponding to a K-moduli wall crossing.
(2) Rational unicuspidal plane curves of low degree, with N. Singh.
We provide a complete list of all rational unicuspidal plane curves of degree at most 30. We use topological constraints to generate a complete potential list and prove the existence of each curve. This is joint work with a current UMass undergraduate.
(3) Moduli of boundary polarized Calabi-Yau pairs, with K. Ascher, D. Bejleri, H. Blum, G. Inchiostro, Y. Liu, and X. Wang, submitted.
We develop the moduli theory of boundary polarized slc log Calabi-Yau pairs (X,D), proving the existence of an S-complete and Theta-reductive moduli stack, and in some cases, the existence of an (asymptotically) good moduli space.
(4) Smooth limits of plane curves of prime degree and Markov numbers, with D. Stapleton, submitted.
We study the existence of smooth limits of families of plane curves of prime degree, prove the existence of a non-planar limit of families of degree equal to a Markov number, and prove that any smooth limit of degree 7 plane curves is again planar.
(5) K-stability and birational models of moduli of quartic K3 surfaces, with K. Ascher and Y. Liu, Inventiones Mathematicae.
We use explicit wall crossings of K-moduli spaces to completely interpolate between the GIT moduli space of quartic K3 surfaces and the Baily-Borel moduli space.
(6) K-moduli of curves on a quadric surface, with K. Ascher and Y. Liu, Journal of the Institute of Mathematics of Jussieu.
We study (d,d) curves on the quadric surface and, for d = 4, use K-moduli to interpolate between GIT and the associated Baily-Borel moduli space of hyperelliptic quartic K3 surfaces.
(7) Wall crossing for K-moduli spaces of plane curves, with K. Ascher and Y. Liu, submitted.
We prove a foundational wall crossing phenomenon for K-moduli spaces of log Fano pairs. We apply this to analyze K-moduli spaces of degree at most 6 plane curves.
(8) Maximal Chow constant and cohomologically constant fibrations, with D. Stapleton, Communications in Contemporary Mathematics.
We generalize the notion of rationally connected fibration to fibrations by maximally Chow-trivial and cohomologically trivial subvarieties.
(9) Moduli of surfaces in P3, Compositio Mathematica.
We extend Hacking's work on a 'minimal' moduli space of plane curve pairs of general type to the case of hypersurfaces in Fano varieties. We provide a description of several boundary components of the moduli space of degree d surfaces in three-dimensional projective space.
(10) Hyperbolicity and uniformity of varieties of log general type, with K. Ascher and A. Turchet, International Math Research Notices.
We define a notion of almost ampleness for the logarithmic cotangent sheaf and prove that, if in addition it is globally generated, varieties with almost ample log cotangent contain finitely many integral points.
(11) An introduction to the minimal model program, slides from a general audience talk at the Fields Institute Symposium in honor of Caucher Birkar. This talk was for the general public, and the slides were intended to be accessible to anyone who knows what a polynomial is. The talk recording will be posted to the Fields website.
(12) Advice for the virtual job market, Early Career section of the AMS Notices September 2021 Issue
This is a reflection of my experience on the job market over Zoom, with broadly applicable advice for early career mathematicians on the market.
(13) Reflections on the Covid job market, with R. Palak Bakshi, Mathematical Association of America Math Values Blog
This is an honest summary of my thoughts and experiences on the job market in 2020-2021.
(14) How to have Lunch in the Time of Covid, with A. Kobin, Early Career section of the AMS Notices January 2021 Issue
This is an account of our regular 'Lunch in the Time of Covid' meetings, panels that highlighted issues relevant to early career mathematicians in the wake of the pandemic and beyond.
(15) Actionable advice for early career mathematicians on the academic job market, Early Career section of the AMS Notices November 2020 Issue
This is an article aimed at early career mathematicians trying to find balance between their present job stage and responsibilities, future career prospects and success on the job market, and experiencing the weight of a global pandemic.
(16) Moduli of higher dimensional varieties, with exercises, with D. Bejleri, in preparation.
This is a book for graduate students to learn the K-moduli theory of Fano varieties and the KSB-moduli theory of canonically polarized varieties, with the necessary tools to do research in this area. It contains many exercises and is intended to be accessible to students with a background in algebraic geometry at the level of Hartshorne. The complete draft is available on request, but the K-moduli chapter can be found on the next line.
(17) An introduction to K-moduli, with exercises.
These notes on K-moduli grew out of notes written for the 2022 AGNES Summer School on moduli of varieties and are meant to be accessible to graduate students with at least a first course in algebraic geometry. They contain many exercises!
(18) Older notes on K-stability.
(19) Some examples of blow ups.
2023 - 2024:
Stacks Project Workshop, August
Boston Algebraic Geometry Day, September
Clay Research Conference on K-stability and Birational Geometry, September
Fields Symposium in honor of Caucher Birkar, October
AGNES Penn, October
KIAS Conference on Fano varieties, their Geometry and Moduli, October
Higher Dimensional Algebraic Geometry: celebrating James McKernan, January
Moduli and Invariants in Santiago, March
AGNES Boston College, March
Isaac Newton Institute: New equivariant methods in algebraic and differential geometry, May
Geometrie Algebrique en Liberte, GAeL, June
Oberwolfach Workshop: Algebraic Geometry: Wall Crossing and Moduli Spaces, Varieties and Derived Categories, July
AGGITatE: an LMS School on Moduli Theory in Algebraic Geometry, July
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Last updated January 2024.