Christopher Eagle

Department of Mathematics and Statistics
University of Victoria
PO Box 1700 STN CSC
Victoria, British Columbia, Canada
V8W 2Y2
eaglec@uvic.ca

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Employment

Associate Teaching Professor - University of Victoria. July 2023-Present.
Assistant Teaching Professor - University of Victoria. January 2016-June 2023.
Postdoctoral Fellow - University of Toronto at Mississauga. July 2015-December 2015.

Education

Ph. D. - Mathematics - University of Toronto, 2015
M. Math. - Pure Mathematics - University of Waterloo, 2010
M. Litt. - Philosophy - University of St. Andrews, 2008
B. Math. - Pure Mathematics - University of Waterloo, 2007

Research Interests

I am broadly interested in mathematical logic and its applications. My work has focused on the following more specific topics:

• Real-valued logic
• Infinitary, higher-order, and other non-first-order logics
• Applications of logic to C*-algebras and other structures from functional analysis
• Interactions between model theory and topology

Publications and preprints

• Computable K-theory for C*-algebras: UHF algebras
  • With I. Goldbring, T. McNicholl, and R. Miller
  • Submitted
  • Toggle abstract
    We initiate the study of the effective content of K-theory for C*-algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a C∗-algebra A, computably enumerable presentations of the abelian groups K_0(A) and K_1(A). When A is stably finite, we show that the positive cone of K_0(A) is computably enumerable. We strengthen the results in the case that A is a UHF algebra by showing that the aforementioned presentation of K_0(A) is actually computable. In the UHF case, we also show that A has a computable presentation precisely when K_0(A) has a computable presentation, which in turn is equivalent to the supernatural number of A being lower semicomputable; we give an example that shows that this latter equivalence cannot be improved to requiring that the supernatural number of A is computable. Finally, we prove that every UHF algebra is computably categorical.
  
  
• Computable Gelfand duality
  • With P. Burton, A. Fox, I. Goldbring, M. Harrison-Trainor, T. McNicholl, A. Melnikov, and T. Thewmorakot
  • Submitted
  • Toggle abstract
    We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative C*-algebras.
  
  
• Relative topological principality and the ideal intersection property for groupoid C*-algebras
  • With G. Goerke and M. Laca
  • Integral Equations and Operator Theory 96 (2024) Article 30
  • Toggle abstract
    We introduce the notion of relative topological principality for a family {Hα} of open subgroupoids of a Hausdorff étale groupoid G. The C*-algebras C∗r(Hα) of the groupoids Hα embed in C∗r(G) and we show that if G is topologically principal relative to {Hα} then a representation of C∗r(G) is faithful if and only if its restriction to each of the subalgebras C∗r(Hα) is faithful. This variant of the ideal intersection property potentially involves several subalgebras, and gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As applications we prove a uniqueness theorem for Toeplitz C*-algebras of left cancellative small categories that generalizes a recent result of Laca and Sehnem for Toeplitz algebras of group-embeddable monoids, and we also discuss and compare concrete examples arising from integer arithmetic.
  
  
• K-theory of co-existentially closed continua
  • With J. Lau
  • Journal of Analysis and Logic 16 (2024) Article 1
  • Toggle abstract
    We describe the possible values of K-theory for C(X) when X is a co-existentially closed continuum. As a consequence we also show that all pseudo-solenoids, except perhaps the universal one, are not co-existentially closed.
  
  
• An undecidable extension of Morley's theorem on the number of countable models
  • With C. Hamel, S. Müller, and F. Tall
  • Annals of Pure and Applied Logic 174 (2023) Article 103317
  • Toggle abstract
    We show that Morley’s theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of σ-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.
  
  
• Model theoretic properties of dynamics on the Cantor set
  • With A. Getz
  • Notre Dame Journal of Formal Logic 63 (2022) pp. 357-371
  • Toggle abstract
    We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of ``generic" used by Akin, Glasner, and Weiss is distinct from the notion of ``generic" encountered in Fraïssé theory.
  
  
• Two applications of topology to model theory
  • With C. Hamel and F. Tall
  • Annals of Pure and Applied Logic 172 (2021) Article 102907
  • Toggle abstract
    By utilizing the topological concept of pseudocompactness, we simplify and improve a proof of Caicedo, Duenez, and Iovino concerning Terence Tao's metastability. We also poinpoint the exact relationship between the Omitting Types Theorem and the Baire Category Theorem by developing a machine that turns topological spaces into abstract logics.
  
  
• Concrete barries to quantifier elimination in finite dimensional C*-algebras
  • With T. Schmid
  • Mathematical Logic Quarterly 65 (2019) pp. 490-497
  • Toggle abstract
    Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*-algebras that admit quantifier elimination in continuous logic are C, C², M₂(C), and the continuous functions on the Cantor set. We show that, among finite dimensional C*-algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections, and one for pairs of unitarily conjugate projections. Both of these predicates are definable, but not quantifier-free definable, in the usual language of C*-algebras. We also show that adding just the predicate for minimal projections is sufficient in the case of full matrix algebras, but that in general both new predicate symbols are required.
  
  
• Expressive power of infinitary [0, 1]-valued logics
  • Chapter 1 in Beyond First Order Model Theory, CRC Press (2017)
  • Toggle abstract
    We consider model-theoretic properties related to the expressive power of three analogues of L_{ω1, ω} for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for complete separable metric structures.
  
  
• Quantifier elimination in C*-algebras
  • With I. Farah, E. Kirchberg, and A. Vignati
  • International Mathematics Research Notices 2017(24) (2017), pp. 7580-7606
  • Toggle abstract
    The only C*-algebras that admit elimination of quantifiers in continuous logic are C, C², C(Cantor space) and M₂(C). We also prove that the theory of C*-algebras does not have model companion and show that the theory of M_n(O_n+1) is not AE-axiomatizable for any n ≥ 2.
  
  
• The pseudoarc is a co-existentially closed continuum
  • With I. Goldbring and A. Vignati
  • Topology and its Applications 207 (2016), pp. 1-9
  • Toggle abstract
    Answering a question of P. Bankston, we show that the pseudoarc is a co-existentially closed continuum. We also show that C(X), for X a nondegenerate continuum, can never have quantifier elimination, answering a question of the the first and third named authors and Farah and Kirchberg.
  
  
• Fraïssé limits of C*-algebras
  • With I. Farah, B. Hart, B. Kadets, V. Kalashnyk, and M. Lupini
  • Journal of Symbolic Logic 81 (2016), pp. 755-773
  • Toggle abstract
    We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II₁ factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
  
  
• Saturation and elementary equivalence of C*-algebras
  • With A. Vignati
  • Journal of Functional Analysis 269 (2015), pp. 2631--2664
  • Toggle abstract
    We study the saturation properties of several classes of C*-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of σ-unital C*-algebras; we extend their results by showing that some coronas of non-σ-unital C*-algebras are countably degree-1 saturated. We then relate saturation of the abelian C*-algebra C(X), where X is 0-dimensional, to topological properties of X, particularly the saturation of CL(X). We also characterize elementary equivalence of the algebras C(X) in terms of CL(X) when X is 0-dimensional, and show that elementary equivalence of the generalized Calkin algebras of densities א_α and א_β implies elementary equivalence of the ordinals α and β.
  
  
• Omitting types for infinitary [0, 1]-valued logic
  • Annals of Pure and Applied Logic 165 (2014), pp. 913-932
  • Toggle abstract
    We describe an infinitary logic for metric structures which is analogous to L_{ω1, ω}. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.
  
  
• Distribution of the number of encryptions in revocation schemes for stateless receivers
  • With Z. Gao, M. Omar, D. Panario, and B. Richmond
  • Fifth Colloquium on Mathematics and Computer Science, Discrete Mathematics and Theoretical Computer Science (2008), pp. 195-206
  • Toggle abstract
    We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215-238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.

Theses

• Ph. D. thesis: Topological aspects of real-valued logic . University of Toronto, 2015. (Toggle abstract)
  We study interactions between general topology and the model theory of real-valued logic. This thesis includes both applications of topological ideas to obtain results in pure model theory, and a model-theoretic approach to the study of compacta via their rings of continuous functions viewed as metric structures. We introduce an infinitary real-valued extension of first-order continuous logic for metric structures which is analogous to the discrete logic L_{ω1, ω}, and use topological methods to develop the model theory of this new logic. Our logic differs from previous infinitary logics for metric structures in that we allow the creation of formulas inf_n φ_n and sup_n φ_n for all countable sequences (φ_n)_{n < ω} of formulas. Our more general context allows us to axiomatize several important classes of structures from functional analysis which are not captured by previous logics for metric structures. We give a topological proof of an omitting types theorem for this logic, which gives a common generalization of the omitting types theorems of Henson and Keisler. Consequently, we obtain a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces. We show that continuous functions on separable metric structures are definable in our L_{ω1, ω} if and only if they are automorphism invariant. The second part of this thesis develops the model theory of the C*-algebras C(X), for X a compact Hausdorff space. We describe all complete theories of these algebras for X a compact 0-dimensional space. We show that the complete theories of C(X) (for X of any dimension) having quantifier elimination are exactly the theories of C, C², and C(Cantor set), and that if the theory of C(X) is model complete and X is connected then X is co-elementarily equivalent to the pseudoarc. We use model-theoretic forcing to answer a question of P. Bankston by showing that the pseudoarc is a co-existentially closed continuum. The results of the second part of the thesis were obtained jointly with several co-authors.
  
  
• M. Math. thesis: The Mordell-Lang Theorem from the Zilber Dichotomy. University of Waterloo, 2010. (Toggle abstract)
  We present a largely self-contained exposition of Ehud Hrushovski's proof of the function field Mordell-Lang conjecture beginning from the Zilber Dichotomy for differentially closed fields and separably closed fields. Our account is based on notes from a series of lectures given by Rahim Moosa at a MODNET workshop at Humboldt Universitat in Berlin in September 2007. We treat the characteristic 0 and characteristic p cases uniformly as far as is possible, then specialize to characteristic p in the final stages of the proof. We also take this opportunity to work out the extension of Hrushovski's "Socle Theorem" from the finite Morley rank setting to the finite U-rank setting, as is in fact required for Hrushovski's proof of Mordell-Lang to go through in positive characteristic.
  
  

Teaching

University of Victoria

• Spring 2025:
  • MATH 365 - Introduction to Topology
• Summer 2024:
  • MATH 580 - Topics in Pure Mathematics: Structural Ramsey Theory
• Spring 2023:
  • MATH 435 - Real Analysis II
• Fall 2022:
  • MATH 110* - Matrix Algebra for Engineers
  • MATH 493/580 - Topics in Pure Mathematics: Mathematical Logic (Set Theory)
• Spring 2022:
  • MATH 365 - Introduction to Topology
• Fall 2021:
  • MATH 110* - Matrix Algebra for Engineers
  • MATH 311 - Linear Algebra
• Spring 2021:
  • MATH 122 - Logic and Foundations
  • MATH 211* - Matrix Algebra I
• Fall 2020:
  • MATH 110* - Matrix Algebra for Engineers
  • MATH 435 - Real Analysis II
• Summer 2020:
  • MATH 490 - Directed Studies in Mathematics (Topic: Model Theory)
  • MATH 490 - Directed Studies in Mathematics (Topic: Set Theory)
• Spring 2020:
  • MATH 122 - Logic and Foundations
  • MATH 493/580 - Topics in Pure Mathematics: Mathematical Logic (Model Theory)
• Fall 2019:
  • MATH 110* - Matrix Algebra for Engineers
  • MATH 312 - Abstract Algebra I
• Spring 2019:
  • MATH 122* - Logic and Foundations
  • MATH 236 - Introduction to Real Analysis
• Fall 2018:
  • MATH 110* - Matrix Algebra for Engineers
  • MATH 236 - Introduction to Real Analysis
• Spring 2018:
  • MATH 122 - Logic and Foundations
  • MATH 311 - Linear Algebra
• Fall 2017:
  • MATH 110* - Matrix Algebra for Engineers
  • MATH 236 - Introduction to Real Analysis
• Spring 2017:
  • MATH 100 - Calculus I
  • MATH 101* - Calculus II
• Fall 2016:
  • MATH 100* - Calculus I
  • MATH 110* - Matrix Algebra for Engineers
• Summer 2016:
  • MATH 490 - Directed Studies in Mathematics (Topic: Model Theory)
• Spring 2016:
  • MATH 102* - Calculus for Students in the Social and Biological Sciences
  • MATH 151 - Finite Mathematics

University of Toronto - Mississauga Campus

• Fall 2015:
  • MAT135H5Y - Calculus I

University of Toronto - St. George Campus

• Summer 2015:
  • MAT136H1F - Calculus I (B)
• Fall 2014:
  • MAT224H1F - Linear Algebra II
• Summer 2014:
  • MAT136H1F - Calculus I (B)
• Fall 2013:
  • MAT224H1F - Linear Algebra II
• Summer 2013:
  • MAT224H1F - Linear Algebra II