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.


Local-Global Principles for Tori over Arithmetic Curves, with Jean-Louis Colliot-Th\'el\`ene, 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
Local-Global Principles for Zero-Cycles on Homogeneous Spaces over Arithmetic Function Fields, with Jean-Louis Colliot-Th\'el\`ene, David Harbater, Julia Hartmann, R. Parimala and V. Suresh, Trans. Amer. Math. Soc., 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, Comment. Math. Helv., 89 (2014), no. 1, 215--253
Weierstrass preparation and algebraic invariants, with David Harbater and Julia Hartmann, Math. Ann., 356 (2013), no. 4, 1405--1424
Relative Brauer groups of genus 1 curves, with Mirela Ciperiani, Israel J. Math., 192 (2012), no. 2, 921--949
Appendix to: Period and index in the Brauer group of an arithmetic surface By Max Lieblich, , J. Reine Angew. Math., 659 (2011), 1--41
Patching subfields of division algebras, with David Harbater and Julia Hartmann, Trans. Amer. Math. Soc., 363 (2011), no. 6, 3335--3349
Distinguishing division algebras by finite splitting fields, with Kelly McKinnie, Manuscripta Math., 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, Trans. Amer. Math. Soc., 362 (2010), no. 9, 4781--4792
Zero cycles on homogeneous varieties, Adv. Math., 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, J. Algebraic Geom., 18 (2009), no. 3, 477--509
Index reduction for Brauer classes via stable sheaves, with Max Lieblich, Int. Math. Res. Not. IMRN, no. 8 (2008), Art. ID rnn010, 31 pp
Birational maps between generalized Severi-Brauer varieties, J. Pure Appl. Algebra, 212 (2008), no. 4, 689--703
Motives of unitary and orthogonal homogeneous varieties, J. 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, Doc. Math., 8 (2003), 527--546 (electronic)


Local-Global Principles for Curves over Semi-Global Fields, with David Harbater and Alena Pirutka
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

Other writing

Open algorithms, social justice, and predictive policing, submitted as an opinion piece to the Notices of the AMS