mathematics

algebra and arithmetic geometry

field patching

Field patching is a technique for working with algebraic structures over fields of transcendence degree 1 over complete discretely valued fields, such as Q_p(t). With David Harbater and Julia Hartmann, I have been developing this theory to answer questions about division algebras, quadratic forms, Galois cohomology and torsors for linear algebraic groups over such fields. These techniques also give rise to a number of new local-global principles for algebraic structures over these fields.

the complexity of algebraic structures

Algebraic structures such as central simple algebras and quadratic forms give rise to important arithmetic invariants of a field, such as its Brauer dimension and its u-invariant. These invariants are related to each other as well as to the field's Diophantine dimension, cohomological dimension, and symbol length bounds for Galois cohomology. Many fundamental problems in field arithmetic involve understanding these invariants and their relationships, and there are many more invariants defined via algebra and algebraic geometry which promise to hold rich and subtle information about field arithmetic.

research statement all preprints on arXiv MathSciNet

preprints

2025

On rational connectedness and parametrization of finite Galois extensions

with Danny Neftin

arXiv
2025

Finiteness of formal pushforwards

with David Harbater and Julia Hartmann

arXiv
2024

Morita equivalences for Zhu's algebra

with Chiara Damiolini and Angela Gibney

arXiv
2023

Brauertsch fields

with Max Lieblich and Minseon Shin

arXiv
2020

A Tannakian approach to patching

with Bastian Haase and Max Lieblich

arXiv
2017

Schubert cycles and subvarieties of generalized Severi-Brauer varieties

with Caroline Junkins and Nicole Lemire

arXiv
2015

The Clifford algebra of a finite morphism

with Max Lieblich

arXiv

publications

2026

A Looming of phantoms

with Kimoi Kemboi, Tianle Liu, Yeqin Liu, Eoin Mackall, Svetlana Makarova, Alexander Perry, Antonios-Alexandros Robotis, and Sridhar Venkatesh · Advances in Mathematics, 499 (2026), Paper No. 111046

arXiv DOI
2026

Classes in H^(n+1)_(p^m)(F) of lower exponent

with Adam Chapman and Kelly McKinnie · Annals of K-Theory, 11 (2026), no. 1, 37–46

arXiv DOI
2025

Conformal blocks on smoothings via mode transition algebras

with Chiara Damiolini and Angela Gibney · Communications in Mathematical Physics, 406 (2025), no. 6, Paper No. 131

arXiv DOI MathSciNet
2025

Factorization Presentations

with Chiara Damiolini and Angela Gibney · Higher dimensional algebraic geometry – a volume in honor of V. V. Shokurov. London Math. Soc. Lecture Note Ser., Vol. 489 (2025), 163-191

arXiv DOI MathSciNet pdf
2024

Transcendental splitting fields of division algebras

with Max Lieblich · Contemporary Mathematics, Vol. 800 (2024)

arXiv DOI MathSciNet
2023

Bounding cohomology classes over semiglobal fields

with David Harbater and Julia Hartmann · Israel Journal of Mathematics, Vol. 257 (2023), 353-387

arXiv DOI MathSciNet
2022

Local-Global Principles for Constant Reductive Groups over Semi-Global Fields

with Jean-Louis Colliot-Thélène, David Harbater, Julia Hartmann, R. Parimala, and V. Suresh · Michigan Math. Journal., Vol. 72 (2022), 77-144

arXiv DOI MathSciNet
2021

Division algebras with common subfields

with Eliyahu Matzri, Andre Rapinchuk, Louis Rowen, and David J. Saltman · Manuscripta Math., Vol. 169 (2022), no.1-2, 209–249.

DOI MathSciNet
2020

Local-Global Principles for Curves over Semi-Global Fields

with David Harbater and Alena Pirutka · Bulletin of the London Mathematical Society, 53 (1), Feb. 2021, 177–193

arXiv DOI MathSciNet
2020

Local-Global Principles for Tori over Arithmetic Curves

with Jean-Louis Colliot-Thélène, David Harbater, Julia Hartmann, R. Parimala, and V. Suresh · Algebraic Geometry, 7 (5) (2020) 607–633

arXiv DOI MathSciNet
2020

Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves

with Dan Boneh, Darren Glass, Kristin Lauter, Shahed Sharif, Alice Silverberg, Mehdi Tibouchi, and Mark Zhandry · Journal of Mathematical Cryptology, Vol. 14, no. 1 (2020)

arXiv DOI MathSciNet
2019

Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields

with Jean-Louis Colliot-Thélène, David Harbater, Julia Hartmann, R. Parimala, and V. Suresh · Transactions of the American Mathematical Society, 372 (2019), no. 8, 5263–5286

arXiv DOI MathSciNet
2019

Local-global Galois theory of arithmetic function fields

with David Harbater, Julia Hartmann, R. Parimala, and V. Suresh · Israel Journal of Mathematics, vol. 232, no. 2 (2019), 849-882

arXiv DOI MathSciNet
2019

Period-index bounds for arithmetic threefolds

with Benjamin Antieau, Asher Auel, Colin Ingalls, and Max Lieblich · Inventiones Mathematicae, 216 (2019), no. 2, 301-335

arXiv DOI MathSciNet
2017

Derived categories for torsors for Abelian schemes

with Benjamin Antieau and Matthew Ward · Advances in Mathematics, 306 (2017), 1-23

arXiv DOI MathSciNet
2016

Period and index, symbol lengths, and generic splittings in Galois cohomology

Bulletin of the London Mathematical Society, 48 (2016), no. 6, 985-1000

arXiv DOI MathSciNet
2015

Local-global principles for torsors over arithmetic curves

with David Harbater and Julia Hartmann · American Journal of Mathematics, 137 (2015), no. 6, 1559–1612

arXiv DOI MathSciNet
2015

Diophantine and cohomological dimensions

with Eliyahu Matzri · Proceedings of the AMS, 143 (2015), no. 7, 2779–2788

arXiv DOI MathSciNet
2015

Refinements to patching and applications to field invariants

with David Harbater and Julia Hartmann · International Math. Research Notices, doi: 10.1093/imrn/rnu278 (2015)

arXiv DOI MathSciNet
2014

Local-global principles for Galois cohomology

with David Harbater and Julia Hartmann · Commentarii Mathematici Helvetici, 89 (2014), no. 1, 215–253

arXiv DOI MathSciNet
2012

Weierstrass preparation and algebraic invariants

with David Harbater and Julia Hartmann · Mathematische Annalen, 356 (2013), no. 4, 1405–1424

arXiv DOI MathSciNet
2012

Relative Brauer groups of genus 1 curves

with Mirela Ciperiani · Israel Journal of Mathematics, 192 (2012), no. 2, 921–949

arXiv DOI MathSciNet
2011

Appendix to: Period and index in the Brauer group of an arithmetic surface

Journal für die reine und angewandte Mathematik, 659 (2011), 1–41

arXiv DOI MathSciNet
2010

Patching subfields of division algebras

with David Harbater and Julia Hartmann · Transactions of the American Mathematical Society, 363 (2011), no. 6, 3335–3349

arXiv DOI MathSciNet
2010

Distinguishing division algebras by finite splitting fields

with Kelly McKinnie · Manuscripta Mathematica, 134 (2011), no. 1-2, 171–182

arXiv DOI MathSciNet
2010

Corestrictions of algebras and splitting fields

Transactions of the American Mathematical Society, 362 (2010), no. 9, 4781–4792

arXiv DOI MathSciNet
2010

Zero cycles on homogeneous varieties

Advances in Mathematics, 223 (2010), no. 6, 2022–2048

arXiv DOI MathSciNet
2010

Field patching, factorization, and local-global principles

Quadratic forms, linear algebraic groups, and cohomology, 57–82, Dev. Math., 18, Springer, New York, 2010

arXiv DOI MathSciNet
2009

Applications of patching to quadratic forms and central simple algebras

with David Harbater and Julia Hartmann · Inventiones Mathematicae, 178 (2009), no. 2, 231–263

arXiv DOI MathSciNet
2008

Pointed trees of projective spaces.

with Linda Chen and Angela Gibney · Journal of Algebraic Geometry, 18 (2009), no. 3, 477–509

arXiv DOI MathSciNet
2008

Birational maps between generalized Severi-Brauer varieties

Journal of Pure Applied Algebra, 212 (2008), no. 4, 689–703

arXiv DOI MathSciNet
2008

Index reduction for Brauer classes via stable sheaves

with Max Lieblich · International Mathematics Research Notices, no. 8 (2008), Art. ID rnn010, 31 pp

arXiv DOI MathSciNet
2007

Motives of unitary and orthogonal homogeneous varieties

Journal of Algebra, 318 (2007), no. 1, 135–139

arXiv DOI MathSciNet
2004

Severi-Brauer varieties and symmetric powers

with David J. Saltman · Algebraic transformation groups and algebraic varieties, 59–70, Encyclopaedia Math. Sci., 132, Springer, Berlin, 2004

DOI MathSciNet
2003

Severi-Brauer varieties of semidirect product algebras

Documenta Mathematica, 8 (2003), 527–546 (electronic)

arXiv DOI MathSciNet