# Kechris Classical Descriptive Essay

## ALEXANDER S. KECHRIS

Professor of Mathematics

Diploma, Mechanical & Electrical Engineering, National Technical University of Athens, 1969

Ph.D., Mathematics, UCLA, 1972

CV

Publications#### Research Interests

*Foundations of mathematics; mathematical logic and set theory; their interactions with analysis and dynamical systems*. Recent projects include the study of foundational and set theoretic questions, and the application of the methodology and results of descriptive set theory, in classical real analysis, harmonic analysis, dynamical systems (especially ergodic theory and topological dynamics), model theory, and combinatorics.

**Teaching**2017-2018

Fall: Ma 191a, Topics in Set Theory and its Applications

Winter: Ma 1b Practical, Linear Algebra

#### Selected Publications

*The theory of countable Borel equivalence relations*, preprint, 2018. Click here for the .pdf file*Quasi-invariant measures for continuous group actions*, preprint, 2018. Click here for the .pdf file- (with R. Chen)
*Structurable equivalence relations*, Fund. Math. (2018), Click here for the .pdf file. arXiv expanded version https://arxiv.org/abs/1606.01995v3 - (with A. Nies and K. Tent)
*The complexity of topological group isomorphism,*preprint, 2017. https://arxiv.org/abs/1705.08081 *The spaces of measure preserving equivalence relationsand graphs*, preprint, 2017. Click here for the .pdf file- (with P.J. Burton)
*Weak con**tainment of measure preserving group actions*, preprint, 2017. Click here for the .pdf file - (with P.J. Burton
*) Invariant random subgroups and action versus represe**ntation maximality*, Proc. Amer. Math. Soc.,**145(9)**(2017), 3961--3971. Click here for the .pdf file - (with A.S. Marks)
*Descriptive graph combinatorics,*preprint, 2016. Click here for the .pdf file - (Co-edited with B. Loewe and J.R. Steel)
*,**Ordinal Definability and Recursion Theory: The Cabal Seminar, Volume III,*Lecture Notes in Logic**43**, Cambridge University Press, 2016. - (with D.A. Martin)
*On the theory of $\Pi^1_3$ sets of reals, II*, in: Ordinal Definability and Recursion Theory: The Cabal Seminar, Volume III, Lecture Notes in Logi**c****43****,**Cambridge University Press, 2016, 200-219. - (with H.L. Macdonald)
*Borel equivalence relations and cardinal algebras*, Fund. Math.,**235**(2016), 183--198. Click here for the .pdf file - (with M. Sokic and S. Todorcevic)
*Ramsey properties of finite measure algebras and topological dynamics of the group of measure preserving automorphisms: some results and an open problem*, in: Foundations of Mathematics, Essays in Honor of W. Hugh Woodin's 60th Birthday, Ed. by A.E. Caicedo et al., Contemp. Math.,**690**, 69-85, Amer. Math. Soc., 2017. Click here for the .pdf file *Commentary on Problem 50 (of Banach) in the Scottish Book*, in: R. D. Mauldin, The Scottish Book, Second Edition, Birkhäuser, 2015, 127-129. Click here for the .pdf file- (with O. Angel and R. Lyons)
*Random orderings and unique ergodicity of automorphism g**roups*, J. European Math. Society,**16 (**2014),2059-2095. Click here for the .pdf file *Dynamics of non-archimedean Polish groups*, European Congress of Mathematics, Krakow, 2-7 July, 2012, 375-397, R. Latala et al., Eds., European Math. Society, 2014.- (with C. T. Conley and R.D. Tucker-Drob)
*Ultraproducts of measure preserving actions and graph combinatorics*, Erg. Theory and Dynam. Systems,**33 (**2013),334-373. - (with C.T. Conley)
*Measurable chromatic and independence numbers for ergodic graphs and group actions*, Groups, Geometry, and Dynamics,**7**(2013), 127-180. - (with R.D. Tucker-Drob)
*The complexity of classification problems in ergodic theory*, Appalachian Set Theory, Ed. by J. Cummings and E. Schimmerling, London Math. Society Lecture Note Series,**406,**265-299, Cambridge University Press, 2013. - (with C.T. Conley and B.D. Miller)
*Stationary probability measures and topological realizations*, Israel. J. Math.,**198 (1)**(2013), 333-345. *Trigonometric series and set theory*, Wiadomosci Matematyczne, 48(2) (2012), 109–118. Click here for the .pdf file- (Co-edited with B. Loewe and J.R., Steel)
*, Wadge degrees and projective ordinals: The Cabal Seminar, Volume II*, Lecture Notes in Logic**37**, Cambridge University Press, 2012. - (with M. Sokic)
*Dynamical properties of the automorphism groups of the random poset and random distributive lattice*, Fund. Math.**218**(2012), 69-94. *In memoriam: Gregory Hjorth (1963-2011)*, Bulletin of Symbolic Logic**17(3)**(2011), 471-477.*Global aspects of ergodic group actions*, Mathematical Surveys and Monographs,**160**, American Mathematical Society, 2010.- Corrections and Updates: [pdf].
*Weak containment in the space of actions of a free group, Israel J. Math., 189 (2012), 461--507.*- (with A. Ioana and T. Tsankov)
*Subequivalence relations and positive-definite functions*, Groups, Geometry, and Dynamics**3(4)**(2009), 579-625. *Set theory and dynamical systems*, Logic, Methodology and Philosophy of Science, Proc. of the Thirteenth International Congress, Ed. C. Glymour et al., College Publ., London, 2009, 97-107.- (Co-edited with B. Loewe and J.R. Steel)
*Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I,*Lecture Notes in Logic**31**, Cambridge University Press, 2008. - (with W.H. Woodin)
*The equivalence of partition properties and determinacy*, in: Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I, Lecture Notes in Logic**31**, Cambridge University Press, 2008, 355-378. - (with W.H. Woodin)
*Generic codes for uncountable ordinals, partition properties, and elementary embeddings*,in: Games, Scales, and Suslin Cardinals: The Cabal Seminar, Volume I, Lecture Notes in Logic**31**, Cambridge University Press, 2008, 379-397. - (with B.D. Miller)
*Means on equivalence relations*, Israel. J. Math.**163**(2008), 241-262. - (with T. Tsankov)
*Amenable actions and almost invariant sets*, Proc. Amer. Math. Soc.**136(2)**(2008), 687-697. - (with C. Rosendal)
*Turbulence, amalgamation and generic automorphisms of homogeneous structures*, Proceedings of the London Math. Society,**94(3)**(2007), 302–350. *Unitary representations and modular actions*, Journal of Math. Sciences,**140(3)**(2007), 398-425.- (with G. Hjorth)
*Rigidity theorems for actions of product groups and countable Borel equivalence relations*, Memoirs of the Amer. Math. Soc.,**177**, No. 833, 2005. - (with V.G. Pestov and S. Todorcevic)
*Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups*, Geometric and Functional Analysis**15**(1) (2005), 106-189. - (with B.D. Miller)
*Topics in Orbit Equivalence*, Lecture Notes in Mathematics, Springer, Vol. 1852, 2004 - (with S. Gao)
*On the classification of Polish metric spaces up to isometry*, Memoirs of Amer. Math. Soc.,**766**, Amer. Math. Soc., 2003. - (with C.W. Henson, J. Iovino and E. Odell)
*Analysis and Logic*, London Math. Soc. Lecture Notes Series,**262**, Cambridge University Press, 2002. - (with S. Jackson and A. Louveau)
*Countable Borel equivalence relations*, J. Math. Logic**2(1)**(2002), 1-80. - (with S.R. Buss, A. Pillay, and R.A. Shore)
*The prospects for mathematical logic in the 21st century*, Bull. Symb. Logic**7(2)**(2001), 169-196. - (with J.D. Clemens and S. Gao)
*Polish metric spaces: Their classification and isometry groups*, Bull. Symb. Logic**7(3)**(2001), 361-375. - (with G. Hjorth)
*Recent developments in the theory of Borel reducibility*, Fund. Math.**170(1)**(2001), 21-52 (volume dedicated to the memory of J. Los). - (with N. Sofronidis)
*A strong generic ergodicity property of unitary and self-adjoint operators*, Ergodic Theory and Dynam. Systems**21**(2001), no. 5, 1459-1479. - (co-edited with M. Foreman, A. Louveau, and B. Weiss) Descriptive Set Theory and Dynamical Systems, London Math. Soc. Lecture Note Series
**277**, 231-259, Cambridge University Press, 2000. *Descriptive Dynamics*, in: Descriptive Set Theory and Dynamical Systems, Ed. by M. Foreman et al., London Math. Soc. Lecture Note Series**277**, 231-259, Cambridge University Press, 2000.*On the classification problem for rank 2 torsion-free abelian groups*, J. London Math. Soc.**62(2)**(2000), 437-450.- (with S. Adams)
*Linear algebraic groups and countable Borel equivalence relations*, J. Amer. Math. Soc.**13(4)**(2000), 909-943. - (with R. Camerlo)
*Countable structures with a fixed group of automorphisms*, Israel J. Math.**117**(2000), 105-124. - (with R. Dougherty)
*How many Turing degrees are there?*in: Computability Theory and Its Applications, Current Trends and Open Problems, Ed. by P.A. Cholak et al., Contemp. Math.**257**, 83-95, Amer. Math. Soc., 2000. - (with G. Hjorth)
*The complexity of the classification of Riemann surfaces and complex manifolds*, Ill. J. Math.**44(1)**(2000), 104-137. *New directions in descriptive set theory*, Bull. Symb. Logic**5(2)**(1999), 161-174.- (with A. Gordon)
*Measurable enumeration of eigenelements*, Applicable Analysis**71**(1999), 41-62. - (with S. Solecki and S. Todorcevic)
*Borel chromatic numbers*, Adv. in Math.**141**(1999), 1-44. *The descriptive classification of some classes of C*-algebras*, Proc. 6th Asian Logic Conference (C.T. Chong, et al., eds.) World Scientific, Singapore (1998), 121-149.*Rigidity properties of Borel ideals on the integers*, Topology and Its Appl.**85**(1998), 195-205.- (with G. Hjorth and A. Louveau)
*Borel equivalence relations induced by actions of the symmetric group*, Ann. Pure and Appl. Logic**92**(1998), 63-112. *On the concept of coanalytic-completeness*, Proc. Amer. Math. Soc.**125(6)**(1997), 1811-1814.- (with G. Hjorth)
*New dichotomies for Borel equivalence relations*, Bull. Symb. Logic**3(3)**(1997), 329-346. - (with A. Louveau)
*The classification of hypersmooth Borel equivalence relations*, J. Amer. Math. Soc.**10(1)**(1997), 215-242. *Set theory and uniqueness for trigonometric series*, preprint, 1997. Click here for the .pdf file- (with H. Becker)
*The Descriptive Set Theory of Polish Group Actions*, London Math. Soc. Lecture Note Series,**232**, Cambridge University Press, 1996. - Corrections and Updates: click here for the .pdf file
- (with G. Hjorth)
*Borel equivalence relations and classifications of countable models*, Ann. Pure and Appl. Logic**82**(1996), 221-272. - (with G. Hjorth)
*Analytic equivalence relations and Ulm-type classifications*, J. Symb. Logic**60**(1995), 1273-1300. - Classical Descriptive Set Theory, Graduate Texts in Mathematics,
**156**, Springer, 1995.- Corrections and Updates: Click here for the .pdf file

- Corrections and Updates: Click here for the .pdf file

#### RTG Logic Meetings at Caltech

Logic in Southern California 2015-16: February 20, 2016**Logic in Southern California 2014-15: March 7, 2015**

**Logic in Southern California 2013-2014: March 8, 2014**

**Logic in Southern California 2012-13: November 17, 2012**

**Logic in Southern California 2011-12: February 18, 2012**

Go to the Directory for mailing address, fax and telephone numbers, and e-mail address.

Click here to get a listing of A. Kechris's papers from the AMS MathSciNet with links to Mathematical Reviews.

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The centenary of P.S. Novikov's birth provides an inspiring motivation to present, with full proofs and from a modern standpoint, the presumably definitive solutions of some classical problems in descriptive set theory which were formulated by Luzin [Lusin] and, to some extent, even earlier by Hadamard, Borel, and Lebesgue and relate to regularity properties of point sets. The solutions of these problems began in the pioneering works of Aleksandrov [Alexandroff], Suslin [Souslin], and Luzin (1916-17) and evolved in the fundamental studies of Gödel, Novikov, Cohen, and their successors. Main features of this branch of mathematics are that, on the one hand, it is an ordinary mathematical theory studying natural properties of point sets and functions and rather distant from general set theory or intrinsic problems of mathematical logic like consistency or Gödel's theorems, and on the other hand, it has become a subject of applications of the most subtle tools of modern mathematical logic.

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