A complete renormalization group trajectory between two fixed points, Comm. Math. Phys. 276(3) (2007), 727-772. preprint version arXiv:math-ph/0610018
Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture, Adv. Math. 208(2) (2007), 491-520 (with J. Chipalkatti). preprint version arXiv:math.AG/0411110
Twin Buildings and Applications to S-Arithmetic Groups, Lecture Notes in Mathematics 1641, Springer (1996).
Buildings: Theory and Applications, Graduate Texts in Mathematics 248, Springer (2008) (with Ken Brown).
Property (T) for noncommutative universal lattices (with Andrei Jaikin-Zapirain), Inventiones Mathematicae 179(2) (2010), 303-347.
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Kazhdan quotients of Golod-Shafarevich groups (with appendices by Andrei Jaikin-Zapirain), Proceedings of the London Mathematical Society 102(4) (2011), 599-636.
The fluid limit of a heavily loaded processor sharing queue, Annals of Applied Probability 12 (2002), 797-859 (with A. L. Puha and R. J. Williams).
Diffusion approximation of a processor sharing queue in heavy traffic, Annals of Applied Probability 14 (2004), 555-611.
Coherent vortex structures and 3D enstrophy cascade (with R. Dascaliuc), Comm. Math. Phys. 317 (2013), 547-561.
A geometric measure-type regularity criterion for solutions to the 3D Navier-Stokes equations, Nonlinearity 26 (2013), 289-296.
Decay of eigenfunctions of elliptic PDEs I (with E. Skibsted), Adv. Math. 270, (2015), 138--180.
Decay of eigenfunctions of elliptic PDEs II (with E. Skibsted), arXiv 1504.07128.
The Beurling-Selberg extremal functions for a ball in Euclidean space, Duke Math. J. 83, no. 1 (1996), 203-247 (with J. D. Vaaler).
Commutative algebra was born out of three classical fields: number theory, algebraic geometry, and invariant theory, but now is used to study many other topics in mathematics. My own focus is the study of solutions of polynomial or power series equations in many variables. I am especially interested in a method called "reduction to characteristic p." Here are a few of my papers:
The structure of linkage (with B. Ulrich), Annals of Math. 126 (1987), 277-334.
Tight closure, invariant theory, and the Briançon-Skoda theorem (with M. Hochster), J. Amer. Math. Soc. 3 (1990), 31-116.
Infinite integral extensions and big Cohen-Macaulay algebras (with M. Hochster), Annals of Math 135 (1992), 53-89.
Direct methods for primary decomposition (with D. Eisenbud and W. Vasconcelos), Inventiones Math. 110 (1992), 207-236.
Uniform bounds in noetherian rings, Inventiones Math. 107 (1992), 203-223.
Comparison of symbolic and ordinary powers of ideals (with M. Hochster), Invent. Math. 147 (2002), 349-369.
The regularity of Tor and graded Betti numbers (with D. Eisenbud and B. Ulrich), Amer. J. Math. 128 (2006), 573-605.
Branched polymers and dimensional reduction, Annals of Mathematics, 158 (2003), 1019-1039 (with D. Brydges). arXiv:math-ph/0107005
On Many-Body Localization for Quantum Spin Chains, http://arxiv.org/pdf/1403.7837.pdf
Embedability between right-angled Artin groups. Joint with Sang-hyun Kim. (Geom. Topol. 17 (2013), no. 1, 493-530)
Anti-trees and right-angled Artin subgroups of planar braid groups. Joint with Sang-hyun Kim. (Geom. Topol. 19 (2015), no. 6, 3289-3306)
Quotients of surface groups and homology of finite covers via quantum representations. Joint with Ramanujan Santharoubane. (Invent. Math. (2016))
A counterexample to the strong version of Freedman's conjecture, Annals of Mathematics 168 (2008), 675-693. Preprint version: http://arxiv.org/PS_cache/math/pdf/0610/0610865v2.pdf (.pdf)
Tutte chromatic identities from the Temperley-Lieb algebra, Geometry and Topology 13 (2009), 709-741 (with Paul Fendley). Preprint version: http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.0016v3.pdf (.pdf)
Localization of Andre-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces, Advances in Math. 201 (2006), 318-378.
Primitives and central detection numbers in group cohomology, Advances in Math. 216 (2007), 387-442.
Product formulae for Ozsvath-Szabo 4-manifold invariants, Geom. Topol. 12 (2008), 1557-1651 (with S. Jabuka).
On the Heegaard Floer homology of a surface times a circle, Adv. Math. 218 (2008), 728-761 (with S. Jabuka).
Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups (with F. Baudoin and M. Gordina), Transactions of the AMS, 365, 4313-4350, 2013.
Heat kernel analysis on semi-infinite Lie groups, J. Functional Analysis 257(11) (2009), 3552-3592.
Cyclic extensions and the local lifting problem (with S. Wewers), Ann. of Math. 180, No. 1 (2014), 233--284.
Fields of moduli of three-point G-covers with cyclic p-Sylow, I, Algebra Number Theory 6, No. 5 (2012), 833--883.
A new approach to the Koszul property in representation theory (with L. Scott), J. Inst. Math. Jussieu 12 (2013), 153-197.
Cohomology of quantum groups via the geometry of the nullcone (with C. Bendel, D. Nakano, and C. Pillen), Memoirs of the Amer. Math. Soc. 229 (2014), 93 pages.
James Joseph Sylvester: Jewish Mathematician in a Victorian World, Baltimore: Johns Hopkins University Press, 2006.
Episodes in the History of Modern Algebra 1800-1950, HMATH, vol. 32, Providence: American Mathematical Society and London: London Mathematical Society, 2008 (co-edited with Jeremy J. Gray).
Anomalies in the Foundations of Ridge Regression, International Statistical Review, 76, 89-105 (2008) (with D. Jensen). http://arxiv.org/abs/math.ST/0703551
Anomalies in the Analysis of Calibrated Data, Journal of Statistical Computation and Simulation 79(3)(2009), 299-314 (with D. Jensen). http://arxiv.org/abs/math.ST/0703550
Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, J. Amer. Math. Soc. 15 (2002), 929-978 (with Y. Segev and G. Seitz).
Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. IHES 109(2009), 113-184.
Unitary orbits of normal operators in von Neumann algebras, Journal für die Reine und Angewandte Mathematik 605 (2007), 95-132.
Model theory of operator algebras I: Stability, Bull. London Math. Soc., to appear (with Ilijas Farah and Bradd Hart).
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Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Inventiones Mathematicae 183 (2011), 189-224 (with Shun-Jen Cheng and Ngau Lam).
Brundan-Kazhdan-Lusztig conjecture for general linear Lie superalgebras (with Shun-Jen Cheng and Ngau Lam), Duke J. Math. 164 (2015), 617-695.