Danny Krashen (research)

Using shapes to understand the arithmetic of algebraic structures.


Field patching is a technique for working with algebraic structures over fields of transcendence degree 1 over complete discretely valued fields, such as Qp(t). Over the past few years, I have been working with David Harbater and Julia Hartmann on developing this theory in order 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 and its relation to field arithmetic has been major theme in my recent research. Algebraic structures such as central simple algebras and quadratic forms give rise to a number of 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 fundmental problems in field arithmetic involve understanding these invariants and their relationship to each other, and there are many more invariants defined via algebra and algebraic geometry which promise to hold very rich and subtle information about field arithmetic.

Preprints



Conformal Blocks on Smoothings via Mode Transition Algebras, with Chiara Damiolini and Angela Gibney
Brauertsch Fields, with Max Lieblich and Minseon Shin
Factorization Presentations, with Chiara Damiolini and Angela Gibney
Transcendental splitting fields of division algebras, with Max Lieblich
A Tannakian approach to patching, with Bastian Haase and Max Lieblich
Schubert cycles and subvarieties of generalized Severi-Brauer varieties, with Caroline Junkins and Nicole Lemire
The Clifford algebra of a finite morphism, with Max Lieblich

Research Publications



Local-Global Principles for Constant Reductive Groups over Semi-Global Fields, with J.-L. Colliot-Thelene, D. Harbater, J. Hartmann, R. Parimala and V. Suresh, Michigan Mathematical Journal, vol. 72 (August 2022), 77-144
Bounding cohomology classes over semiglobal fields, with David Harbater and Julia Hartmann, To appear in Israel Journal of Mathematics
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
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
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, in the Journal of Mathematical Cryptology, Vol. 14, no. 1 (2020)
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
Period-index bounds for arithmetic threefolds, with Benjamin Antieau, Asher Auel, Colin Ingalls and Max Lieblich, Inventiones Mathematicae, 216 (2019), no. 2, 301-335
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
Derived categories for torsors for Abelian schemes, with Benjamin Antieau and Matthew Ward, Advances in Mathematics, 306 (2017), 1-23
Period and index, symbol lengths, and generic splittings in Galois cohomology, Bulletin of the London Mathematical Society, 48 (2016), no. 6, 985-1000
Local-global principles for torsors over arithmetic curves, with David Harbater and Julia Hartmann, American Journal of Mathematics, 137 (2015), no. 6, 1559--1612
Diophantine and cohomological dimensions, with Eliyahu Matzri, Proceedings of the AMS, 143 (2015), no. 7, 2779--2788
Refinements to patching and applications to field invariants, with David Harbater and Julia Hartmann, International Math. Research Notices, doi: 10.1093/imrn/rnu278 (2015)
Local-global principles for Galois cohomology, with David Harbater and Julia Hartmann, Commentarii Mathematici Helvetici, 89 (2014), no. 1, 215--253
Weierstrass preparation and algebraic invariants, with David Harbater and Julia Hartmann, Mathematische Annalen, 356 (2013), no. 4, 1405--1424
Relative Brauer groups of genus 1 curves, with Mirela Ciperiani, Israel Journal of Mathematics, 192 (2012), no. 2, 921--949
Appendix to: Period and index in the Brauer group of an arithmetic surface, by Max Lieblich, Journal für die reine und angewandte Mathematik, 659 (2011), 1--41
Patching subfields of division algebras, with David Harbater and Julia Hartmann, Transactions of the American Mathematical Society, 363 (2011), no. 6, 3335--3349
Distinguishing division algebras by finite splitting fields, with Kelly McKinnie, Manuscripta Mathematica, 134 (2011), no. 1-2, 171--182
Field patching, factorization, and local-global principles, Quadratic forms, linear algebraic groups, and cohomology, 57--82, Dev. Math., 18, Springer, New York, 2010
Corestrictions of algebras and splitting fields, Transactions of the American Mathematical Society, 362 (2010), no. 9, 4781--4792
Zero cycles on homogeneous varieties, Advances in Mathematics, 223 (2010), no. 6, 2022--2048
Applications of patching to quadratic forms and central simple algebras, with David Harbater and Julia Hartmann, Inventiones Mathematicae, 178 (2010), no. 2, 231--263
Pointed trees of projective spaces., with Linda Chen and Angela Gibney, Journal of Algebraic Geometry, 18 (2009), no. 3, 477--509
Index reduction for Brauer classes via stable sheaves, with Max Lieblich, International Mathematics Research Notices, no. 8 (2008), Art. ID rnn010, 31 pp
Birational maps between generalized Severi-Brauer varieties, Journal of Pure Applied Algebra, 212 (2008), no. 4, 689--703
Motives of unitary and orthogonal homogeneous varieties, Journal of Algebra, 318 (2007), no. 1, 135--139
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
Severi-Brauer varieties of semidirect product algebras, Documenta Mathematica, 8 (2003), 527--546 (electronic)

Other Writing



Organize a Laid-Back Conference in the Rocky Mountains that Participants Want to Return to Year after Year, with Kelly McKinnie, Notices of the American Mathematical Society, Early Career, Volume 67, No. 11, December 2020
Response to ``Letter to AMS Notices: Boycott collaboration with police", Notices of the American Mathematical Society, Letters to the Editor, Volume 67, No. 9, October 2020
Open Algorithms, Social Justice, and Predictive Policing, Notices of the American Mathematical Society, Opinion, Volume 67, No. 9, October 2020