- Computational and combinatorial aspects of algebraic geometry
- Fundamental algorithms for polynomial system solving and their implementation
- Applications of the above (eg. minimal problems in computer vision)

- Numerical equality tests for rational maps and signatures of curves

(With Michael Ruddy. )

Submitted.

CODE - Trifocal Relative Pose from Lines at Points and its Efficient Solution

(With Ricardo Fabbri , Hongyi Fan , Margaret Regan, David da Costa de Pinho, Elias Tsigaridas, Charles Wampler, Jonathan Hauenstein, Benjamin Kimia, Anton Leykin, Tomas Pajdla.)

stay tuned. - PLMP - Point-Line Minimal Problems in Complete Multi-View Visibility

(With Kathlén Kohn, Anton Leykin, Tomas Pajdla.)

Oral and poster presentation at ICCV 2019

Won best student paper!

CODE - Certification for polynomial systems via square subsystems

(With Nickolas Hein and Frank Sottile.)

Long version submitted. Extended abstract presented at*MEGA 2019,*arXiv1812.02851. - Monodromy solver: sequential and parallel

(With Nathan Bliss, Anton Leykin, Jeff Sommars.)

*ISSAC 2018*, arXiv 1805.12212

CODE -
Solving polynomial systems via homotopy continuation and monodromy

(With Cvetelina Hill, Anders Jensen, Kisun Lee, Anton Leykin, Jeff Sommars.)

*IMA Journal of Numerical Analysis*, 2018, arXiv1609.08722

CODE -
Polynomial automata: Zeroness and applications

(With Micheal Benedikt, Aditya Sharad, James Worrell.)

*LICS (ACM/IEEE Symposium on Logic in Computer Science), 2017.* -
Robust graph ideals

(With Adam Boocher, Bryan Brown, Laura Lyman, Takumi Murayama, Amy Nesky, Karl Schaefer.)

*Annals of Combinatorics*, 2015, arXiv1309.7630.

- "In a sea of mathematics, the Brunn-Minkowski inequality appears like an octopus, tentacles reaching far and wide, its shape and color changing as it roams from one area to the next." -- R.J. Gardner (on inequalities)
- "My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously." -- Emmy Noether (on anonymity)
- "We justify this sudden introduction of differentials by saying that this is 'just another way of rerewriting the differential equation,' or some equally atrocious lie." -- Gian Carlo Rota (on teaching)
- "If you can solve it, it is an exercise; otherwise itâ€™s a research problem." -- Richard Bellman (on research)
- "Never get so attached to a poem that you forget truth that lacks lyricism. Never draw so close to the heat that you forget that you must eat" -- Joanna Newsom (on truth)
- "There is no clear-cut distinction between example and theory." -- Michael Atiyah (on examples)
- "Differential topology inspires elegant proofs. And elegance is a joy for its beauty. But elegance also is indicative of the right point of view. And indeed, on a cliff at the edge of the land of differential topology we can stand above many scattered results in algebraic geometry." -- Steven Kleiman (on point of view)
- "Most of my heroes don't appear on no stamps" -- Chuck D (on heroism)
- "Brute force has no hope. But clever, inspired brute force is the future." -- Doron Zeilberger (on the future)
- "For various classes of objects (functions, mappings, differential equations) one can often single out those in general position, which, first, constitute the overwhelming majority, and, second, behave in a much simpler way than an arbitrary object." -- Askold Khovanskii (on generality)
- "The beauty of mathematics only shows itself to more patient followers." -- Maryam Mirzakhani (on mathematics)
- "I don't know you, and I like that... because I don't like you, and I know that!" -- unknown (on knowledge)
- "Let's not try to define knowledge, but try to define zero-knowledge." -- Shafi Goldwasser (on limitations)