Christopher Eagle  Mathematics  University of Victoria
Christopher Eagle
Department of Mathematics and Statistics
University of Victoria
PO Box 1700 STN CSC
Victoria, British Columbia, Canada
V8W 2Y2
eaglec@uvic.ca
I am an Assistant Teaching Professor at the Department of Mathematics and Statistics at the University of Victoria. Here is a detailed CV.
Assistant Teaching Professor  University of Victoria. January 2016present.
Postdoctoral Fellow  University of Toronto at Mississauga. July 2015December 2015.
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
I am broadly interested in mathematical logic and its applications. My work has focused on the following more specific topics:
 Realvalued logic
 Infinitary logic
 Interactions between model theory and general topology
 Applications of model theory to C*algebras and other structures from functional analysis
 Omitting types and the Baire category theorem. (With F. Tall). Submitted. (Toggle abstract)
The Omitting Types Theorem in model theory and the Baire Category Theorem in topology are known to be closely linked. We examine the precise relation between these two theorems. Working with a general notion of logic we show that the classical Omitting Types Theorem holds for a logic if a certain associated topological space has all closed subspaces Baire. We also consider stronger Baire category conditions, and hence stronger Omitting Types Theorems, including a game version. We use examples of spaces previously studied in settheoretic topology to produce abstract logics showing that the game Omitting Types statement is consistently not equivalent to the classical one.
 Expressive power of infinitary [0, 1]valued logics. Chapter 1 in Beyond First Order Model Theory, CRC Press (2017). (Toggle abstract)
We consider modeltheoretic 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 rnw236 (2016). (Toggle abstract)
The only C*algebras that admit elimination of quantifiers in continuous logic are C, C^{2}, C(Cantor space) and M_{2}(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 AEaxiomatizable for any n ≥ 2.
 The pseudoarc is a coexistentially closed continuum. (with I. Goldbring and A. Vignati). Topology and its Applications 207 (2016), pp. 19. (Toggle abstract)
Answering a question of P. Bankston, we show that the pseudoarc is a coexistentially 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. 755773 (Toggle abstract)
We realize the JiangSu algebra, all UHF algebras, and the hyperfinite II_{1} 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
Ramseytheoretic results about the class of fullmatrix algebras.
 Saturation and elementary equivalence of C*algebras. (with A. Vignati). Journal of Functional Analysis 269 (2015), pp. 26312664. (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 σunital C*algebras; we extend their results by showing that some coronas of nonσunital C*algebras are countably degree1 saturated. We then relate saturation of the abelian C*algebra C(X), where X is 0dimensional, 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 0dimensional,
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 in infinitary [0, 1]valued logic. Annals of Pure and Applied Logic 165 (2014), pp. 913932. (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 twocardinal 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. 195206. (Toggle abstract)
We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subsetdifference scheme (SD). These are wellknown tree based broadcast encryption schemes. Park and Blake in:
Journal of Discrete Algorithms, vol. 4, 2006, pp. 215238, 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.
 Ph. D. thesis: Topological aspects of realvalued logic
. University of Toronto, 2015. (Toggle abstract)
We study interactions between general topology and the model theory of realvalued logic. This thesis includes both applications of topological ideas to obtain results in pure model theory, and a modeltheoretic approach to the study of compacta via their rings of continuous functions viewed as metric structures.
We introduce an infinitary realvalued extension of firstorder 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 0dimensional space. We show that the complete theories of C(X) (for X of any dimension) having quantifier elimination are exactly the theories of C, C^{2}, and C(Cantor set), and that if the theory of C(X) is model complete and X is connected then X is coelementarily equivalent to the pseudoarc. We use modeltheoretic forcing to answer a question of P. Bankston by showing that the pseudoarc is a coexistentially closed continuum. The results of the second part of the thesis were obtained jointly with several coauthors.

M. Math. thesis: The MordellLang Theorem from the Zilber Dichotomy. University of Waterloo, 2010. (Toggle abstract)
We present a largely selfcontained exposition of Ehud Hrushovski's proof of the function field MordellLang 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 Urank setting, as is in fact required for Hrushovski's proof of MordellLang to go through in positive characteristic.
For courses marked with * I was the coordinator of a multisection course.
University of Victoria
 Spring 2017:
 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
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