# Research

My arithmetic geometry research centers around the use of p-adic methods to study global invariants and geometric structures on algebraic varieties, with an emphasis on exploiting perfectoid spaces to facilitate comparison to positive characteristic geometry. My research also includes interdisciplinary work along the boundary of mathematics and art. I have pieces across various media (from computational to textile to 3D printing) and have had my work appear as installations in galleries, as well as on the covers of books and peer reviewed journals. Here is a selection of publications and preprints:

- Twenty-Seven (2024)
*Algebraic geometry, multimedia art and sculpture*- Joint with Daniel Rostamloo
*Submitted*

- On the importance of illustration for mathematical research (2024)
*Mathematical illustration*- Joint with Rémi Coulon, Edmund Harriss, Martin Skrodzki, Katherine E. Stange, Glen Whitney
*Notices Amer. Math. Soc.*, (70 no. 1)

- Deformation Spaces and Static Animations (2023)
*Mathematical illustration, 3D printing**Proc. Sympos. Appl. Math.*, (79)

- Untilting Line Bundles on Perfectoid Spaces (2023)
*p-Adic geometry**Int. Math. Res. Not. IMRN*, (3):2058-2591 (2023)

- On Picard Groups of Perfectoid Covers of Toric Varieties (2023)
*p-Adic geometry*- Joint with Anwesh Ray and Peter Wear.
*Eur. J. Math.*, 9, 16 (2023),

- Searching for Rigidity in Algebraic Starscapes (2022)
*Algebraic number theory, computer graphics, mathematical illustration.*- Joint with Shuchang Xu
*J. Math. Arts.*, 16: 57-74 (2022).

- Projective Geometry for Perfectoid Spaces (2021)
*p-Adic geometry**Münst. J. of Math.*,14: 445-484 (2021).

- 3D Printing and Deformations (2020).
*Mathematical illustration, 3D printing**Illustrating Mathematics*Diana Davis, editor American Mathematical Society. 100-101. (2020)

- The Fabric of Spacetime (2020).
*Differential geometry, multimedia art and sculpture*- Joint with Merle Maynard
*Illustrating Mathematics*Diana Davis, editor American Mathematical Society. 16-17. (2020)

- The Combinatorics of Interval Vector Polytopes. (2013)
*Combinatorics and discrete geometry*- Joint with Matthias Beck, Jessica De Silva, Joseph Pruitt, and Amanda Ruiz.
*Electron. J. of Combin.*, 20.3 p.22 (2013)

- Here is a copy of my PhD Thesis: Projective Geometry for Perfectoid Spaces It is a longer version of the paper above with the same name, including more exposition and foundation.
**Beware**: Some errors that were corrected or removed in the published version still appear.

# Research Talks

Here is a selection of past, present, and future invited research talks.

- Casa Matemática Oaxaca // Banff International Research Station: Modern Advances in Mixed Characteristic Commutative Algebra: May 9, 2022
- Bay Area Mathematical Adventureres (a joint lectures series from SJSU and SCU): April 5, 2022
- San Jose State University Colloquium: March, 9 2022
- Art Exhibit at the Mathematical Association of America, Golden Section Meeting: Feb 26, 2022
- Santa Clara University Colloquium: January 27, 2022
- Joint Mathematics Meetings Mini Course: 3D Printing in Mathematics: January 3-4, 2022
- Saint Lawrence University Colloquium: December 16 2021
- University of California, San Diego, Number Theory Seminar: November 18 2021
- University of Utah Representation Theory and Number Theory Seminar: October 6, 2021
- Berkeley Undergraduate Number Theorey Conference: November 15, 2020
- University of California, Berkeley, RTG Seminar in Arithmetic Geometry: April 20, 2020
- Brown University Algebraic Geometry Seminar: November 22, 2019
- Cornell University Algebraic Geometry Seminar: November 6, 2019
- Dartmouth College Algebra and Number Theory Seminar: October 8, 2019
- University of Washington Algebra and Algebraic Geometry Seminar: April 16, 2019
- Boston University Number Theory Seminar: December 10, 2018
- Columbia University Algebraic Geometry Seminar: December 7, 2018
- Rice University Algebra and Number Theory Seminar: November 20, 2018
- University of Arizona Algebraic Geometry Seminar: November 14, 2018
- University of Oregon, WAGS Poster Presentation: October 6, 2018
- MIT, Arithmetic Geometry, Number Theory, and Computation Mini Talk: August 2018
- ICERM, Birational Geometry and Arithmetic, Poster Presentation: May 2018
- University of California, San Diego, Graduate Research Presentation: May 2018
- Arizona Winter School, Project Group Presentation: March 2017