# Faculty Research Areas

### Mathematical Physics

#### Abdelmalek Abdesselam

Associate Professor

Professor

#### John Imbrie

Professor & Associate Chair

#### Leonid Petrov

Assistant Professor

#### Yen Do

Assistant Professor

### Group Theory

#### Mikhail Ershov

Associate Professor

Professor

#### Andrei Rapinchuk

McConnell-Bernard Professor of Mathematics

### Probability

#### Christian Gromoll

Associate Professor

#### Tai Melcher

Associate Professor

#### Yen Do

Assistant Professor

#### Leonid Petrov

Assistant Professor

Professor

### Algebraic Topology

Professor

#### Michael Hill

Associate Professor

Professor

### Statistical Mechanics

#### Leonid Petrov

Assistant Professor

### Geometric Topology

#### Thomas Mark

Associate Professor

Professor

### Operator Algebras

#### David Sherman

Associate Professor

Professor

### Representation Theory

#### Leonid Petrov

Assistant Professor

#### Brian Parshall

G. T. Whyburn Professor

Professor

#### Benjamin Webster

Assistant Professor

### Harmonic Analysis

#### Yen Do

Assistant Professor

Professor

Professor

### Algebraic Geometry

#### Andrew Obus

Assistant Professor

#### Benjamin Webster

Assistant Professor

### Analysis

Professor

Professor

#### David Sherman

Associate Professor

### Commutative Algebra

#### Craig Huneke

Marvin Rosenblum Professor & Chair

#### Abdelmalek Abdesselam

Associate Professor
Research Interests
• Mathematical Physics

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

#### Peter Abramenko

Professor
Research Interests
• Group Theory

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).

#### Greg Arone

Professor
Research Interests
• Algebraic Topology

Calculus of functors, operad formality, and rational homology of embedding spaces, Acta Mathematica 153-198 (with P. Lambrechts and I. Volic). arxiv math/0607486

Filtered spectra arising from permutative categories, Crelle's Math Journal 604 (2007), 73-136 (with Kathryn Lesh).

#### Yen Do

Assistant Professor
Research Interests
• Harmonic Analysis
• Mathematical Physics
• Probability

#### Mikhail Ershov

Associate Professor
Research Interests
• Group Theory

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.

#### Christian Gromoll

Associate Professor
Research Interests
• Probability

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.

#### Zoran Grujic

Professor
Research Interests
• Analysis
• Applied Mathematics
• Partial Differential Equations

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.

#### Ira Herbst

Professor
Research Interests
• Mathematical Physics

Analyticity estimates for the Navier-Stokes equations (with E. Skibsted), Adv. Math., 228 (2011), 1990-2033.

Smoothness and analyticity of perturbation expansions in QED (with D. Hasler), Adv. Math., 228 (2011), 3249-3299.

#### Michael Hill

Associate Professor
Research Interests
• Algebraic Topology

On the non-existence of elements of Kervaire invariant one (with M. Hopkins and D. Ravenel). http://front.math.ucdavis.edu/0908.3724

The topological Hochschild homology of \ell and ko, Amer. J. Math. 132(2) (2010), 297-330 (with V. Angeltveit and T. Lawson). Arxiv

#### Jeffrey Holt

Professor
Research Interests

The Beurling-Selberg extremal functions for a ball in Euclidean space, Duke Math. J. 83, no. 1 (1996), 203-247 (with J. D. Vaaler).

#### Craig Huneke

Marvin Rosenblum Professor & Chair
Research Interests
• Commutative Algebra

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.

#### John Imbrie

Professor & Associate Chair
Research Interests
• Mathematical Physics

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

#### Slava Krushkal

Professor
Research Interests
• Geometric Topology

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)

#### Nicholas Kuhn

Professor
Research Interests
• Algebraic Topology

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.

#### Thomas Mark

Associate Professor
Research Interests
• Geometric Topology

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).

#### Tai Melcher

Associate Professor
Research Interests
• Probability

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.

#### Andrew Obus

Assistant Professor
Research Interests
• Algebraic Geometry

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.

#### Brian Parshall

G. T. Whyburn Professor
Research Interests
• Representation Theory

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.

#### Karen Parshall

Professor
Research Interests
• History of Mathematics

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).

#### Leonid Petrov

Assistant Professor
Research Interests
• Mathematical Physics
• Probability
• Representation Theory
• Statistical Mechanics

Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes (2012), arXiv:1202.3901 [math.PR]. To appear in Probability Theory and Related Fields

#### Donald Ramirez

Professor
Research Interests
• Analysis
• Harmonic Analysis

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

#### Andrei Rapinchuk

McConnell-Bernard Professor of Mathematics
Research Interests
• Group Theory

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.

#### David Sherman

Associate Professor
Research Interests
• Analysis
• Operator Algebras

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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#### Weiqiang Wang

Professor
Research Interests
• Representation Theory

Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Inventiones Mathematicae 183 (2011), 189-224 (with Shun-Jen Cheng and Ngau Lam).

Quantum supergroups I. Foundations (with Sean Clark and David Hill), Transformation Groups 18 (2013), 1019--1053.

#### Benjamin Webster

Assistant Professor
Research Interests
• Algebraic Geometry
• Representation Theory