## ANNOUNCEMENTS

- Registration for Fall 2015-16 opens Thursday, May 21, 2015.
- Beginning of Instruction for the Fall term is Monday, September 28, 2015.
- This page was last updated: .

## COURSE DESCRIPTION

Continuous logic is a generalization of the usual first order logic, suitable to the study of metric structures. Natural fields of application include functional analysis (Banach spaces, operator algebras) and metric geometry. This course will provide an introduction to continuous logic and an overview of some of its most notable applications. No specific background knowledge in model theory will be required.

## PREREQUISITES

There are no prerequisites. Some previous knowledge of logic can be useful but not necessary.

## OFFICE HOURS

Tuesday and Thursday 5-6 pm

You can also email me to schedule an appointment outside the offical office hour.

## TOPICS COVERED

Lecture 1: Fundamental concepts from the usual first order "discrete" logic: language, structure, formula, model, ultraproduct, Los' theorem. (See Chapter 1 and Exercise 2.5.19 of Marker's book.)

Lecture 2: Compactness theorem and applications, consistent and complete theories, elementary and atomic diagram. (See Chapter 2 of Marker's book.)

Lecture 3: Upward and Downward Lowenheim-Skolem theorem, completenetss of the theory of algebraically closed fields of fixed characteristic, Ax's theorem on injective polynomial functions (See Chapter 2 of Markers' book.) Introduction to the syntax and semantics for the logic for metric structures, and ultraproducts. (See Sections 2,3,5 of "Model theory for metric structures".)

Lecture 4: Compactness theorem for the logic for metric structures. (See Section 5 of "Model theory for metric structures".) Introduction to sofic groups. Characterization of soficity in terms of formulas and embedding into ultraproducts. (See Section I.4 and Chapter II of "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture".)

Lecture 5: Kaplansky's direct finiteness conjecture. The Gottschalk conjecture implies Kaplansky's direct finiteness conjecture. Ranked rings as metric structures. Sofic groups satisfy Kaplansky's direct finiteness conjecture. (See Section I.5 and Section II.10 of "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture".)

Lecture 6: Hyperlinear group: definition and chracterizations. The Kervaire-Laudenbach conjecture for hyperlinear groups. (See Section II.2 and II.8 of "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture".) Bounded operator on Hilbert spaces. (See Section 3.1 and 3.2 of "Analysis now".)

Lecture 7: Positive operators, unitary operators, and partial isometries. The polar decomposition. The canonical trace on B(H). The sigma-weak and sigma strong topologies on B(H) . Von Neumann's double commutant theorem and Kaplansky's density theorem. Equivalent definition of von Neumann algebras. (See Section 3.2, 3.4, 4.6 of "Analysis now" and Section 1 and 2.1 of "A gentle introduction to von Neumann algebras for model theorists".)

Lecture 8: Continuity of operations in B(H) with respect to the sigma-weak and sigma-strong topologies. States, tracial states, tracial von Neumann algebra and GNS construction. The predual of a von Neumann algebra. The standard form of a von Neumann algebra. (See Section I.3 of "Operator algebras", Section 1 of "A gentle introduction to von Neumann algebras for model theorists", Chapter 9 of "Von Neumann algebras".)

Lecture 9: Tensor product of Hilbert spaces. Comparison between the weak operator topology (WOT), the strong operator topology (SOT), the sigma-weak operator topology, and the sigma-strong operator topology. The 2-norm induces the sigma-strong topology on the unit ball. Axiomatization of C*-algebras and von Neumann algebras in the logic for metric structures. (See Section 2.4 and Section 9.1 of "Von Neumann algebras", Section I.1 and II.11 of "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture", Section I.3 of "Operator algebras".)

Lecture 10: Group von Neumann algebras. (See pages 6,7 of "An invitation to von Neumann algebras".) Kaplansky's direct finiteness conjecture for complex group algebras. (See Section I.5 of "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture".) Spectrum of an operator. Continuous and Borel functional calculus. (See I.2.4, I.6, I.4.3 of "Operator algebras".)

Lecture 11: The algebraic eigenvalues conjecture for sofic groups. (See Section II.12 of "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture".) Erenfeucht-Fraisse games. (See Section 2.4 of Marker's book.) The number of universal sofic and hyperlinear groups. (See ""Logic for metric structures and the number of universal sofic and hyperlinear groups".)

Lecture 12: Tarski-Vaught test. Elementary chains. Consistent set of conditions. Saturated structures. Every structure has a k-saturated elementary extension. (See Section 7 of "Model theory for metric structures", and Section 4.3 of Marker's book for the same topics in the discrete setting.)

Lecture 13: Strongly homogeneous structures. Back-and-forth method. Every structure has a k-saturated and k-homogeneous elementary extension. (See Section 7 of "Model theory for metric structures", and Section 4.2 of Marker's book for the same topics in the discrete setting.)

Lecture 14: Partial and complete n-types. Definable predicates. The logic topology on the space of n-types. (See Section 8 and 9 of "Model theory for metric structures", and Section 4.2 of Marker's book for the same topics in the discrete setting.) The space of complete n-types as the spectrum of the abelian C*-algebra of definable predicates. (See the lecture notes below on definable predicates by Ronnie Chen and Connor Meehan.)

Lecture 15: The metric on the space of types. (Section 8 of "Model theory for metric structures".) Topometric spaces. Definable sets: definition and characterization. (Section 8 of "Model theory for metric structures".) Definition of principal (or isolated) types. (Section 12 of "Model theory for metric structures".)

Lecture 16: A characterization of principal types. (Section 12 of "Model theory for metric structures".) Statement of the omitting types theorem for the logic for metric structures. (The statement considered in class is folklore knowledge, but I could not find it written in that form anywhere. A particular case is stated in Theorem 2.12 of "Generic orbits and type isolation in the Gurarij space". The classical discrete case is considered in Section 4.2 of Marker's book.)

Lecture 17: The topological proof of the omitting types theorem. Atomic and prime models. (Section 12 of "Model theory for metric structures".)

Lecture 18: Characterization of complete theories with a prime models. Separably categorical theories and the Ryll-Nardzewski theorem. (See the notes on the Ryll-Nardzewski theorem below, and Section 12 of "Model theory for metric structures".) Approximate homogeneity and approximate ultrahomogeneity. Examples of separable structures with separably categorical theories: countable dense linear order with no endpoints, the random graph, the Hilbert space, the measure algebra, the Gurarij Banach space, the Urysohn metric space (See Section 2.4 of Marker's book, and Sections 15, 16 of "Model theory for metric structures.) Roelcke precompactness for topological groups. (Section 2.1 of "Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups".)

Lecture 19: Separably categorical theories and Roelcke precompact automorphism groups. (See Section 2.1 of "Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups" and the notes on the Ryll-Nardzewski theorem below.) Description of the Roelcke compactification as the space of joint types of two copies of the structures. (See Section 2.2 of "Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups".)

## TEXTBOOKS

There is no official textbook. We list below some recommended reading containing most of the topics covered.

For the usual "discrete" first order logic:

D. Marker, "Model Theory: An Introduction", Graduate Texts in Mathematics 217, Springer-Verlag, 2002 (The Caltech library " SFL Basement" has two copies.)

C. C. Chang, H. J. Keisler,
"Model theory", Studies in
Logic and the Foundations of Mathematics, Vol. 73. *North-Holland
Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc.,
New York,* 1973. (This has more details about ultraproducts.)

For the logic for metric structures:

I. Ben Yaacov, A. Berenstein, C. W. Henson, A. Usvyatsov, "Model theory for metric structures", in "Model theory with applications to algebra and analysis. Vol. 2", London Mathematical Society Lecture Note Series 350, 2008. Available at www.math.uiuc.edu/~henson/cfo/mtfms.pdf

For the logic for metric structures applied to sofic and hyperlinear groups:

V. Pestov, "Hyperlinear and sofic groups: A brief guide", Bulletin of Symbolic Logic 14 (4), 2008.

V. Capraro, M. Lupini, "Introduction to sofic and hyperlinear groups and Connes' embedding conjecture", Lecture Notes in Mathematics vol. 2136, Springer. Available at http://arxiv.org/abs/1309.2034v6

M. Lupini, "Logic for metric structures and the number of universal sofic and hyperlinear groups", Proceedings of the American Mathematical Society 142(10), 2014

For functional analysis and operator algebras:

G. K. Pedersen, "Analysis now", Graduate Texts in Mathematics 118, Springer-Verlag (1989).

B. Blackadar, "Operator algebras", Encyclopaedia of Mathematical Sciences 122, Springer-Verlag (2006). Available here.

For von Neumann algebras:

V. F.R. Jones, "Von Neumann algebras", course notes. Available at http://www.math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf

I. Goldbring, "A gentle introduction to von Neumann algebras for model theorists", course notes. Available at http://homepages.math.uic.edu/~isaac/vNanotes.pdf

C. Houdayer, "An invitation to von Neumann algebras", course notes. Available at http://cyrilhoudayer.files.wordpress.com/2014/09/paris-2013.pdf

For applications of logic to Banach spaces:

W. Henson and I. Ben Yaacov, "Generic orbits and type isolation in the Gurarij space". Available at http://arxiv.org/abs/1211.4814v3

For applications of logic to dynamics:

I. Ben Yaacov, T. Tsankov, "Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups". Available at http://arxiv.org/abs/1312.7757

## LECTURE NOTES

Date | Description |
---|---|

November 25th | Notes on definable predicates by Ronnie Chen and Connor Meehan |

November 24th | Notes on the omitting types theorem by Jalex Stark and Michael Wheeler |

December 3rd | Notes on the Ryll-Nardzewski Theorem by Martino Lupini |

## HOMEWORK

Due Date | Homework | Solutions |
---|---|---|

October 13th | Assignment 1 | |

October 20th | Assignment 2 | |

October 27th | Assignment 3 | |

November 3rd | Assignment 4 | |

November 17th | Assignment 5 | |

November 24th | Assignment 6 | |