Research
Exploring the fibers of fraud: Bundle Networks for generating and analyzing fraudulent transactions
A fraud detection model takes thousands of very different card transactions and gives each one a single risk score. I ask the question backwards: given a risk score, what do all the transactions behind it look like, and can we create new ones?
To answer it, I used Bundle Networks, a generative model built on fiber bundles from geometry, on real credit card data. They created more realistic fraud than the GAN models they were compared with, and adding their synthetic frauds to the training data made a standard fraud detector more accurate.
Co-Higgs bundles and Poisson structures
A new, geometric way to see stability: whether a co-Higgs bundle is stable can be read off the shape of the zeros of its Poisson structure.
Co-Higgs bundles correspond to Poisson structures, following work of Polishchuk and Matviichuk. I extended that correspondence beyond the Calabi–Yau case, classified when the induced structures are integrable over open subsets of ℂⁿ, and showed that over curves of genus g ≥ 1 every co-Higgs field is strongly integrable. For rank 2 bundles over ℙ¹, the zero locus of the Poisson structure splits into a 2:1 cover of the spectral curve and fibers over the zeros of Φ. The bundle is stable exactly when the spectral curve is irreducible and only the 2:1 cover appears.
The Riemann–Roch theorem and different ways to generalize the Weierstrass semigroup
Taking a classical invariant of points on a curve and generalizing it to vector bundles, where stability controls how it behaves.
The Weierstrass semigroup of a point on a curve of genus g records which pole orders a rational function can have there, and its gaps are always smaller than 2g. Starting from the Riemann–Roch theorem, I defined versions of it for divisors and for vector bundles. For a vector bundle F, the resulting Weierstrass set is an ideal over the classical semigroup, and when F is semistable its largest gap is less than 2g − deg(F)/rk(F).
Talks
- 2024Co-Higgs bundles and Poisson structures
Canadian Mathematical Society (CMS), Canada
- 2024Co-Higgs bundles and Poisson structures, part I
Algebraic Geometry Seminar, University of Waterloo
- 2023Schubert cycles
Intersection Theory Seminar, University of Waterloo
- 2022Sidon sets
Graduate Student Colloquium, University of Waterloo
- 2022The Weierstrass semigroup
Student Algebraic Geometry and Foliations Seminar, IMPA (virtual)
- 2022Lifting a co-Higgs field to a Poisson structure
Differential Geometry Working Seminar, University of Waterloo
- 2022Classification of Poisson surfaces
Algebraic Geometry Working Seminar, University of Waterloo
- 2021Rank-2 odd degree moduli spaces of co-Higgs bundles, part II
Algebraic Geometry Working Seminar, University of Waterloo
- 2021Rank-2 odd degree moduli spaces of co-Higgs bundles, part I
Algebraic Geometry Working Seminar, University of Waterloo
- 2020A different way to generalize the Weierstrass semigroup
Geometry and Topology Seminar, University of Waterloo
- 2020A different way to generalize the Weierstrass semigroup
IV Algebraic Geometry Summer Meeting, São Paulo, Brazil
- 2016The magic world of Sidon setsPoster
MAA MathFest, Ohio, USA