# Research Interests

Free probability theory; von Neumann algebras; C*-algebras; L^{2}-invariants; Poisson geometry.

# Education

1991-1993 B.S., University of Nevada, Reno (high distinction)1993-1997 Ph.D. University of California, Berkeley (thesis advisor: Prof. Dan-Virgil Voiculescu)

# Former Graduate students

Dan Shiber (Ph.D. Sept 2008)Jason Asher (Ph.D. June 2010)

Yoann Dabrowski (Ph.D. Dec 2010)

Benjamin Hayes (Ph.D. June 2014)

Paul Skoufranis (Ph.D. June 2014)

Brent Nelson (Ph.D. June 2015)

Ian Charlesworth (Ph.D. June 2017)

Joshua Keneda (Ph.D. March 2019)

David Jekel (Ph.D. June 2020)

Max Zhou (Ph.D. March 2024)

June Bahr (Ph.D. June 2024)

# List of publications

- Free Quasi-Free States, C.R. Acad. Sci. Paris, t. 322, Serie I (1996), 645-649
- Free Quasi-Free States, Pacific J. Math, 177, (1997) 329-368
- Limit Distribution of Matrices with Bosonic and Fermionic Entries, Free Probability (D.-V. Voiculescu, ed.), Fields Institute Communications, vol. 12, AMS, 1997, pp. 241-25
- R-transform of Certain Joint Distributions, ibid, 253-256
- Gaussian Random Band Matrices and Freeness with Amalgamation, International Math. Res. Notices, 20 (1996), 1013-1025
- Some Applications of Freeness with Amalgamation, J. reine angew. Math. 500 (1998), 191-212
- A-valued Semicircular Systems, J. Funct. Anal., 166 (1999), 1\4
- Free Quasi-Free states, Thesis, UC Berkeley, 1997
- Gaussian random band matrices and operator-valued free probability theory, Proceedings of the Quantum Probability Conference, Banach Center Publ., 1998
- Free Entropy with Respect to a Completely-positive Map, American J. Math., 122 (2000), 45\81
- (with A. Nica and R. Speicher), Some minimization problems for the free analogue of Fisher information, Adv. Math., 121 (1999), 282--321
- (with A. Nica and R. Speicher), Maximality of the microstates free entropy for R-diagonal elements, Pacific J. Math., 187 (1999), 333--347\
- A Microstates Approach to Relative Free Entropy, Internat. J. Math.
**13**(2002) 605--623\ - (with A. Nica and R. Speicher), R-diagonal elements and freeness with amalgamation, Canad. J. Math.
**53**(2001) 355--381 - Survey paper: von Neumann algebras and Poisson manifolds,
__http://math.berkeley.edu/~aweinst/277papers/shlyakhtenko.tex__ - Free Fisher information with respect to a completely positive map and cost of equivalence relations, Comm. Math. Phys.
**218**(2001), 133--152 - (with K. Dykema), Exactness of Cuntz-Pimsner C*-algebras, Proc. Edinburgh Math. Soc.
**44**(2001), 425-444 - Microstates Free Entropy and Cost of Equivalence Relations, Duke Math. J.
**118**(2003), 375--425 - Prime type III factors, Proc. Nat. Acad. Sci.,
**97**, 2000, 12439-12441 - (with Sorin Popa), Cartan subalgebras and bimodule decompositions of II1 factors, Math. Scand.
**92**(2003) 93--102 - (with N. Brown and K. Dykema), Topological Entropy of Free Product Automorphisms, Acta Math.
**189**(2002) 1--35 - (with Y. Ueda), Irreducible subfactors of L(F(infty)), J. Reine Angew. Math.
**548**(2002) 149--166 - (with S. Popa), Universal properties of L(F(infty)) in subfactor theory, Acta Math. 191 (2003), no. 2, 225--257.
- (with A. Nica and R. Speicher), R cyclic families of matrices in free probability, J. Funct. Anal.
**188**(2002) 227-271 - Free Fisher Information for Non-Tracial States, Pacific J. Math
**211**(2003) 375--390 - (with G. Pisier) Grothendieck's Theorem for operator spaces, Invent. Math.
**150**(2002) 185--217 - (with A. Nica and R. Speicher) Operator-valued distributions: I. Characterizations of freeness, Int. Math. Res. Notices
**29**(2002) 1509-1438 - On the classification of Full Factors of Type III, Trans. AMS
**356**(2004) 4143--4159 - On multiplicity and free absorption for free Araki-Woods factors,
__http://xxx.lanl.gov/abs/math.OA/0302217__ - Some estimates for non-microstates free entropy dimension, with applications to
*q*-semicircular families, Int. Math. Res. Notices**51**(2004) 2757--2772 - (with A. Connes) L2-homology for von Neumann algebras, J. Reine Angew. Math. 586 (2005), 125--168.
- (with I. Mineyev) Non-microstates free entropy dimension for groups,
*g*Geom. Funct. Anal. 15 (2005), no. 2, 476--490. - (with O. Radko) Picard groups of topologically stable Poisson structures, Pacific J. Math. 224 (2006), no. 1, 151--183
- (with K. Dykema and K. Jung) The Microstates Free Entropy Dimension of any DT-operator is 2, Doc. Math. 10 (2005), 247--261 (electronic)
- Remarks on free entropy dimension, Operator Algebras: The Abel Symposium 2004, 249--257, Abel Symp., 1, Springer, Berlin, 2006.
- Notes on free probability theory,
__http://xxx.lanl.gov/abs/math.OA/0504063__ - A free analogue of Shannon's problem on monotonicity of entropy, Adv. Math. 208 (2007), no. 2, 824--833.
- (with K. Jung), Any generating set of an arbitrary property T von Neumann algebra has free entropy dimension $\le1$. J. Noncommut. Geom. 1 (2007), no. 2, 271--279.
- (with A. Guionnet), On classical analogues of free entropy dimension, J. Func. Anal. 251 No. 2 (2007), 738--771
- (with A. Guionnet), Free diffusions and matrix models with strictly convex interaction, Geom. Funct. Anal. 18 (2009), 1875-1916
- (with an appendix by H. Schultz), Shannon's monotonicity problem for free and classical entropy, Proc. Nat. Acad. Sci. 104 No. 39 15254-15258 (2007)
- Lower estimates on microstates free entropy dimension, Analysis & PDE, Vol. 2 (2009), No. 2, 119-146
- (with A. Guionnet and V.F.R. Jones), Random matrices, free probability, planar algebras and subfactors, in “Quanta of maths”, pp. 201–239, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010
- (with V.F.R. Jones and K. Walker), An orthogonal approach to the subfactor of a planar algebra, Pacific J. Math., Vol. 246 (2010), No. 1, 187-197
- (with S. T. Belinschi) Free probability of type B: analytic interpretation and applications, American Journal of Mathematics Vol. 134 (2012), 193-234.
- Free probability, Planar algebras, Subfactors and Random Matrices, Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), Vol. III, Hindustan Book Agency, 2010, pp. 1603--1623
- (with C. Houdayer) Strongly solid II
_{1}factors with an exotic MASA, International Mathematics Research Notices, Vol. 2011, No. 6, 1352–1380 - (with A. Guionnet and V.F.R. Jones), A semi-finite algebra associated to a planar algebra, J. Func. Anal., 261 (2011), 1345-1360
- On operator-valued free convolution powers, Indiana Univ. Math. J. 62 (2013), 91-97
- (with A. Guionnet; V.F.R. Jones and P. Zinn-Justin) Loop models, random matrices and planar algebras. Comm. Math. Phys. 316 (2012), no. 1, 45–97.
- On the Symmetric Enveloping Algebra of Planar Algebra Subfactors (with S. Curran and V.F.R. Jones), Trans. AMS 366 (2014) no. 1, 113–133
- (with A. Guionnet), Free Monotone Transport, Inv. Math. 197 (2014), 613-661
- (with P. Skoufranis), Freely independent random variables with non-atomic distributions, Trans. AMS 367 (2015), 6267-6291
- (with I. Charlesworth), Free entropy dimension and regularity of non-commutative polynomials. J. Funct. Anal. 271 (2016), no. 8, 2274–2292
- (with S. Curran and Y. Dabrowski), Free analysis and planar algebras. Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones' 60th birthday, 115–142, Proc. Centre Math. Appl. Austral. Nat. Univ., 46, Austral. Nat. Univ., Canberra, 2017.
- A (co)homology theory for subfactors and planar algebras. Notices Amer. Math. Soc. 65 (2018), no. 4, 408–409
- (with S. Popa and S. Vaes) Cohomology and L2-Betti Numbers for Subfactors and Quasi-Regular Inclusions. Int. Math. Res. Not. IMRN 2018, no. 8, 22412331
- Free probability of type B and asymptotics of finite-rank perturbations of random matrices, Indiana Univ. Math. J. 67 (2018), 971-991.
- Random matrices and free probability. Random matrices, 389--459, IAS/Park City Math. Ser., 26, Amer. Math. Soc., Providence, RI, 2019.
- (with C. Houdayer and S. Vaes) Classification of a family of non-almost-periodic free Araki-Woods factors.J. Eur. Math. Soc. (JEMS) 21 (2019), no. 10, 3113--3142.
- (with S. Popa) Representing interpolated free group factors as group factors. Groups Geom. Dyn. 14 (2020), no. 3, 837--855.
- (with S. Popa and S. Vaes) Classification of regular subalgebras of the hyperfinite II1 factor. J. Math. Pures Appl. (9) 140 (2020), 280--308.
- Von Neumann algebras of sofic groups with β(2)1=0 are strongly 1-bounded.J. Operator Theory 85 (2021), no. 1, 217--228.
- (with Y. Dabrowski, Yoann and A. Guionnet) Free transport for convex potentials. New Zealand J. Math. 52 (2021 [2021--2022]), 259--359.
- (with D. Jekel and W. Li) Tracial smooth functions of non-commuting variables and the free Wasserstein manifold. Dissertationes Math. 580 (2022), 150 pp.
- (with Wilfrid Gangbo, David Jekel, Kyeongsik Nam) Duality for optimal couplings in free probability. Comm. Math. Phys. 396 (2022) , 903-981
- An inequality for non-microstates free entropy dimension for crossed products by finite abelian groups. L’Enseignement Mathématique, 69 (2023), 381-397.

# Awards and honors

1993 University of Nevada, Reno Gold Medal1994 US Department of Education National Needs fellowship

1994 NSF Graduate Research Fellowship

1996 Charles B. Morrey Award (UC Berkeley)

1998 NSF Postdoctoral Fellowship

1999 MSRI Postdoctoral Fellowship

2001 Sloan Foundation Fellowship

2002 Clay Mathematics Institute Special Project award

2004 R. Sorgenfrey Distinguished Teaching Award

2004 Bourbaki Seminar ”Free quasi-free states and factors of type III” by S. Vaes on my work.

2010 Invited talk, Int. Congress Mathematicians, Hyderabad, India

2012 Plenary talk, German Mathematical Society Annual Meeting

2018 Invited talk, AMS Western Sectional Meeting

# Editorial Boards

Transactions AMS (2007-10)Memoirs AMS (2007-10)

Groups, Geometry and Dynamics

Pacific Journal of Mathematics

# Positions Held

Hedrick Assistant Professor (1998-99), UCLA;Assistant Professor (1999-01), Associate Professor (2001-03), Professor (2003-present), UCLA;

I served as the Graduate Vice-Chair in 2008-10, and as the Department Chair in 2012-15.

Director, Institute for Pure and Applied Mathematics (2017-).