## Professor Viorel Nitica

### Basic Information

### Book

Rigidity in Higher Rank Abelian Group Actions: Volume 1, Introduction and Cocycle Problem

By: Anatole Katok and Viorel Nitica

Cambridge University Press, June 2011. ISBN: 9780521879095

Book Excerpt**Review / Order:**

**Summary**

This self-contained monograph presents rigidity theory for a large class of dynamical systems, differentiable higher rank hyperbolic and partially hyperbolic actions. This first volume describes the subject in detail and develops the principal methods presently used in various aspects of the rigidity theory. Part I serves as an exposition and preparation, including a large collection of examples that are difficult to find in the existing literature. Part II focuses on cocycle rigidity, which serves as a model for rigidity phenomena as well as a useful tool for studying them. The book is an ideal reference for applied mathematicians and scientists working in dynamical systems and a useful introduction for graduate students interested in entering the field. Its wealth of examples also makes it excellent supplementary reading for any introductory course in dynamical systems.

**Table of Contents**

- Introduction: an overview
- Part I. Preliminaries from Dynamics and Analysis:
- 1. Definitions and properties of abelian group actions
- 2. Principal classes of algebraic actions
- 3. Preparatory results from analysis
- Part II. Cocycles, Cohomology and Rigidity:
- 4. First cohomology and rigidity for vector-valued cocycles
- 5. First cohomology and rigidity for general cocycles
- 6. Higher order cohomology
- References
- Index.

### Curriculum Vitae

**Research interests**- dynamical systems, rigidity of smooth group actions, ergodic theory
- extremal algebras
- operator algebras

**Education**- Ph.D. - Pennsylvania State University, 1995
- M.S. - University of Bucharest, Romania, 1986
- B.S. - University of Bucharest, Romania, 1985

**Professional History**- August 2010 - present: Professor, West Chester University, West Chester
- August 2006 - August 2010: Associate Professor, West Chester University, West Chester
- August 2006 - December 2006: Visiting Associate Professor, Pennsylvania State University, University Park
- August 2001 - August 2006: Assistant Professor, West Chester University, West Chester
- August 1998 - August 2001: Visiting Assistant Professor, University of Notre Dame, South Bend
- August 1997 - August 1998: Postdoctoral Fellow, Pennsylvania State University, University Park
- August 1995 - August 1997: Visiting Assistant Professor, Indiana University, Bloomington
- August 1990 - August 1995: Teaching Assistant, Pennsylvania State University, University Park

**Grants**- June 2005 - June 2008: NSF Grant DMS05-05539
- June 1999 - June 2002: NSF Grant DMS99-71826

### Publications

Acknowledgement: several publications were written while being funded by National Science Foundation, grants DMS-9971826 and DMS-0500832 and by Simons Foundation, Collaboration Grant for Mathematicians, 2011-2016.

*On tilings of quadrants and rectangles and rectangular pattern*, Open Journal of Discrete Mathematics 6 (2016) 351--371- (with K. Gill
*) Signed tilings by ribbon L n-ominoes, n even, via Grobner bases*, Open Journal of Discrete Mathematics 6 (2016) 185--206 *Signed tilings by ribbon L n-ominoes, n odd, via Grobner bases,*Open Journal of Discrete Mathematics 6 (2016) 297-313- (with D. Ehrmann, Z. Higgins)
*The geometric structure of max-plus hemispaces*, Linear Algebra and its Applications,**489**(2016) 144-166 *Every tiling of the first quadrant by L n-ominoes follows the rectangular pattern*, Open Journal of Discrete Mathematics,**5**(2015) 11-25.- (with K. Lui, S. Venkatesh),
*The semigroup problem for central semidirect product of R^n with R^m*, Topology Proceedings**45**(2015) 9—29. - (with S. Sergeev),
*The tropical dimension of max-min convex sets,*Fuzzy Sets and Systems**271**(2015) 81—101. - (with A. Torok) ,
*Open and dense topological transitivity of extensions by non-compact fiber of hyperbolic systems: a review*, Axioms**4**(2015) 84—101. - (with S. Sergeev),
*Tropical convexity over max-min semiring,*In*Tropical and Idempotent Mathematics and Applications*, vol. 616, pages 241--260, Contemporary Mathematics AMS, Providence, 2014. - (with J. Eskeldson, M. Jaffe),
*A metric on max-min algebra*, In*Tropical and Idempotent Mathematics and Applications*, vol. 616, pages 101--114, Contemporary Mathematics, AMS, Providence, 2014. - (with A. Torok),
*Stable transitivity of Heisenberg group extensions of hyperbolic systems*, Nonlinearity**27**(2014) 661–-683. - (with I. Singer),
*Extended-valued topical and anti-topical functions on semimodules*, Lin Alg Appl**446**(2014) 25-70. - (with R. D. Katz and S. Sergeev),
*Characterization of tropical hemispaces by (P,R)-decompositions*, Lin Alg Appl**440**(2014) 131--163. - (with M. Chao, D. Levenstein, R. Sharp),
*A coloring invariant for ribbon L-tetrominoes,*Discrete Mathematics**313**(2013) 611—621. - (with I. Singer),
*Topical functions on semimodules and generalizations,*Lin Alg Appl**437**(2012) 2471—2488. - (with S. Sergeev),
*Semispaces**in the max-min convexity*, Proceedings of the International Workshop Tropical and Idempotent Mathematics, Moscow, Russia, August 26--31, 2012, 208--214, ISBN 978-5-91910-143-7. - (with S. Sergeev),
*The structure of max-plus hemispaces*, Proceedings of the International Workshop Tropical and Idempotent Mathematics, Moscow, Russia, August 26--31, 2012, 199--207, ISBN 978-5-91910-143-7. - (with I. Melbourne, A. Torok),
*Transitivity of Heisenberg group extensions of hyperbolic systems*, Ergodic Theory and Dynamical Systems**32**(1) (2012) 223—235. - (with M. Akian, S. Gaubert, I. Singer),
*Best approximation in max-plus semimodules,*Lin Alg Appl**435**(2011) 3261-3296. *Stably transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups*, Discrete and Continuous Dynamical Systems**29**(3) (2011) 1197—1203.- (with S. Sergeev),
*An interval version of separation by semispaces in max-min convexity,*Lin Alg Appl**435**(2011) 1637–-1648. - (with S. Sergeev),
*On hyperplanes and semispaces in max-min convex geometry*, Kybernetika**46**(2010) 548-557. *The structure of max-min hyperplanes*, Linear Algebra Appl.**432**(2010) 402--429.- (with I. Melbourne, A. Torok)
*Transitivity of Euclidean-type extensions of hyperbolic systems,*Ergodic Theory and Dynamical Systems**29**(2009) 1585--1602. PDF - (with C. Nitica)
*Tiling a deficient rectangle by P-pentominoes*, Geombinatorics**19**(2009) 18--27. - (with C. Nitica)
*Tiling a deficient board by P-pentominoes*, Geombinatorics**18**(2009) 175--184. *Ergodic Theory: Rigidity,*in the volume Encyclopedia of Complexity and Systems Science, Springer 2009.*Journ**\'e's theorem for C^{n,omega} regularity,*Discrete and Continuous Dynamical Systems**22**(2008) 413--425.- (with I. Singer)
*Contributions to max-min convex geometry II. Semispaces and convex sets,*Linear Algebra and its Applications**428**(2008) 2085-2115. - (with I. Singer)
*Contributions to max-min convex geometry I. Segments,*Linear Algebra and its Applications**428**(2008) 1439--1459. - (with I. Singer)
*The structure of max-plus hyperplanes,*Linear Algebra and its Applications**426**(2-3) (2007) 382--414. - (with A. Katok)
*Rigidity of cocycles with values in diffeomorphism groups, ,*Geometriae Dedicata**124**(2007) 109--131. *Tiling a deficient rectangle by L-tetrominoes,*Journal of Recreational Mathematics**33**(4) (2004-2005) 259--271.- (with I. Singer)
*Max-plus convex sets and max-plus semispaces II,*Optimization**56**(2007) 293--303. - (with I. Singer)
*Max-plus convex sets and max-plus semispaces I,*Optimization**56**(1--2) (2007) 171--205. - (with I. Melbourne, A. Torok)
*A note about stable transitivity of noncompact extensions of hyperbolic systems,*Discrete and Contin. Dynam. Systems**14**(2006) 355--363. - (with I. Melbourne, A. Torok)
*Stable transitivity of certain noncompact extensions of hyperbolic systems,*Annales Henri Poincare**6**(2005) 725--746. - (with M. Durante)
*Tiling the chair,*Geombinatorics 13 (2004), 117--122. - (with A. Torok)
*Cocycles**over TNS actions*, Geometriae Dedicata**102**(2003) 65-90. - (with M. Pollicott)
*Transitivity of Euclidean extensions of Anosov diffeomorphisms*, Ergod. Th. and Dynam. Sys.**25**(2005), 257--269. *Reptiles revisited,*in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics, American Mathematical Society, 2003.- (with A. Torok)
*On the cohomology of Anosov actions*, in Rigidity and Geometry in Dynamics, Springer - Verlag, 2002. - (with M. Field)
*Stable topological transitivity of skew and principal extensions,*Nonlinearity**14**(2001), 1055-1069. - (with F. Xavier)
*Schrodinger operators and topological pressure on manifolds of negative curvature,*in Proceedings of Symposia in Pure Mathematics**69,**(2001). *A note about topologically transitive cylindrical cascades,*Israel Journal of Mathematics**126**(2001), 141-156.*Examples of topologically transitive skew-products*, Discrete and Continuous Dynamical Systems**6**(2000), 351-360.- (with A. Torok)
*An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one*, Topology**40**(2001), 259-278. - (with A. Torok)
*Local rigidity of certain partially hyperbolic actions of product type*, Ergodic Theory and Dynamical Systems**21**(2001), 1213-1237. - (with M. Gerber)
*Holder exponents for horocycle foliations on surfaces,*Ergodic Theory and Dynamical Systems**19**(1999), 1247-1254. - (with H. Bercovici)
*A Banach algebra version of the Livsic theorem,*Discrete and Continuous Dynamical Systems**4**(1998), 523-534. - (with H. Bercovici)
*Cohomology**of higher rank abelian Anosov actions for Banach algebra valued cocycles ,*Proceedings of the International Conference on Dynamical Systems and Differential Equations, Kennesaw State University, Georgia, 2000, 50-55. - (with A. Katok and A. Torok)
*Nonabelian**cohomology of abelian Anosov actions*, Ergodic Theory and Dynamical Systems**20**(2000), 259-288. - (with A. Torok)
*Regularity of the transfer map for cohomologous cocycles*, Ergodic Theory and Dynamical Systems**18**(1998), 1187-1209. - (with A. Torok)
*Regularity results of the solutions of the Livsic cohomology equation with values in diffeomorphism groups*, Ergodic Theory and Dynamical Systems**16**(1996), 325-333. - (with A. Torok)
*Cohomology**of dynamical systems and rigidity of partially hyperbolic actions of higher rank lattices*, Duke Math J.**79**(1995), 751-810, - (with A. Torok)
*Maximal abelian and singular subalgebras in L(*, J. Operator Theory**F**_{N})**30**(1993), 3-19. - (with A. Torok)
*The automorphism group of a free product of groups and simple C*-algebras*, J. Operator Theory**28**(1992), 187-199. - (with A. Torok)
*About the automorphism group of a free products of groups*, Archiv der Mathematik**58**(1992), 214-219. - (with F. Boca)
*Combinatorial properties of groups and simple C*-algebras with a unique trace,*J. Operator Theory**20**(1988), 183-196. - (with F. Boca)
*Extensions of groups and simple C*-algebras,*in Linear operators in function space (Timisoara 1988), Birkhauser, Basel.

Last Updated: January 2017.

### Recreational Mathematics

- (with C. Nitica)
*Tiling a deficient rectangle by P-pentominoes*, Geombinatorics**19**(2009) 18--27. - (with C. Nitica)
*Tiling a deficient board by P-pentominoes*, Geombinatorics**18**(2009) 175--184. *Tiling a deficient rectangle by L-tetrominoes,*Journal of Recreational Mathematics**33**(4) (2004-2005) 259--271.- (with M. Durante)
*Tiling the chair,*Geombinatorics 13 (2004), 117--122. *Reptiles revisited*, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics, American Mathematical Society, 2003.