Selected preprints

Here are links to preprints that represent my main contributions to the various areas in which I have worked,  They are organized in roughly reverse chronological order.  

Effective metric structure theory 

Since 2013, after attending an inspiring talk by A.G. Melnikov at a CCA meeting, I have been thinking mostly about how to do computability theory on uncountable but separable structures.  Classical computabilty theory deals with countable algebraic structures.  However, a significant amount of scientific work takes place in uncountable structures such as Banach spaces and operator algebras.  Effective metric structure theory seeks to fill in this gap in our knowledge.

Operator Algebras 

• Evaluative presentations This paper built on the technology developed in the paper below.  Among other things, it demonstrated the full computable functoriality of the Gelfand duality.  This is accomplished by considering presentations of C^*(X) for which the evaluation map is computable.
• Computable Gelfand duality (with C. Eagle, A. Fox, I. Goldbring, M. Harrison-Trainor, A.G. Melnikov, T. Thewmorakot) This paper grew out of the 2023 BIRS meeting organized by Johanna Franklin and myself.  The main result is that C^*(X) is computably presentable if and only if X has a computably compact presentation.  

Complexity of theories

• Hyperarithmetic numerals (with C. Camrud) Appeared in proceedings of the 2014 Computability in Europe Conference, Lecture Notes in Computer Science, no. 14773.  A short paper in which we showed the truth values realized by a computable model of a metric language are precisely the hyperarthimetic reals.
• On the complexity of the theory of computably presented metric structures (with C. Camrud and I. Goldbring) Appeared in Archive for Mathematical Logic, vol 62 no. 7 - 8.   A short paper with my former graduate student Caleb Camrud and Isaac Goldbring.  We pinned down the relation between the syntactic complexity of a wff of continuous logic and the computational complexity of its truth values.  

Lp spaces

• Continuous logic and embeddings of Lebesgue spaces  Appeared in Archive for Mathematical Logic, vol. 60, no. 1 - 2.  Not a computability paper, but it was inspired by, and used some of the technology from, the papers below on Lebesgue spaces.
• Computing the exponent of a Lebesgue space. Appeared in Journal of Logic and Analysis, no. 12.  Considers whether p needs to be computable in order for a non-trivial L^p space to be computably presentable.  For p > 2, the answer is `yes'.  For p < 2, the question is still open.  
• On the complexity of classifying Lebesgue spaces (with T. Brown and A.G. Melnikov). Appeared in Journal of Symbolic Logic, vol. 85, no. 3.  The classification problem for L^p spaces is considered within the framework of computability theory.
• The isometry degree of a computable copy of l^p (with D. Stull) Appeared in Computability, vol. 8, no 2.  We introduced the concept of a degree of isometry which is the least powerful Turing degree (if there is one) that computes an isometric isomorphism between two structures.  We showed that with respect to the standard presentation of l^p these are precisely the c.e. degrees.
• Analytic computable structure theory and L^p space part 2 (with T. Brown) Appeared in Archive for Mathematical Logic, vol. 59, no. 3-4.  Completed the work is the papers below by determining the degrees of categoricity for atomic L^p spaces.
• Analytic computable structure theory and L^p spaces (with J. Clanin and D. Stull) . Appeared in Fundamenta Mathematicae, vol. 244, no. 3.  We showed L^p[0,1] is computably categorical.  
• Computable copies of l^p Appeared in Computability, vol. 6, no. 4.  Completed the work in the paper below by showing that when p is a computable real other than 2, the degree of categoricity of l^p is 0'.  Along the way, two useful devices are introduced: distintegrations and a stable version of a theorem of Lamperti.
• A note on the computable categoricity of l^p spaces  Appeared in Lecture Notes in Computer Science, no. 9136.  Used an old result of Banach and Lamperti to show l^p is not computably categorical when p is not 2.

Computable complex analysis

• Algorithmic randomness and Fourier analysis (with J. Franklin and J. Rute). Appeared in Theory of Computing Systems, vol. 63, no. 3.  The exceptional set of Carleson's Theorem is used to characterize Schnorr randomness.
• Computing boundary extensions of conformal maps part 2 
• Computing boundary extensions of conformal maps Appeared in London Mathematical Society Journal of Computational Science, vol. 17, no. 1.  Proves that the boundary extension can be computed if the boundary is effectively locally connected.
• Computing conformal maps of multiply connected domains onto canonical slit domains (with V. Andreev). Appeared in Theory of Computing Systems, vol. 50, no. 2.
• Computing interpolating sequences (with V. Andreev). Appeared in Theory of Computing Systems, vol. 46, no. 2.  We proved a computable version of Naftalevich's Theorem.
• Uniformly computable aspects of inner functions: estimation and factorization. Appeared in Mathematical Logic Quarterly, vol. 54, no. 8.  Completed the work in the paper below by showing that the Blaschke sum provides the correct parameter.
• Computable analysis and Blaschke products (with A. Matheson). Appeared in Proceedings of the American Mathematical Society, vol. 136, no. 1.  My first paper in computable complex analysis.  We showed that a computable Blaschke product has a computable Blashke sequence, but that some incomputable Blaschke products nevertheless have computable Blaschke sequences.

Computable topology

• The power of backtracking and the confinement of length. Appeared in Proceedings of the American Mathematical Society, vol. 141, no. 3.  A fun little paper inspired by conversations with Jack Lutz about points that avoid various kids of computable curves.  
• Computing links and accessing arcs. Appeared in Mathematical Logic Quarterly, vol. 59, no. 1- 2.  
• Computing space-filling curves (with P.J. Couch and D.Daniel) Appeared in Theory of Computing Systems, vol. 50, no. 2
• Effective local connectivity properties (with D. Daniel) Appeared in Theory of Computing Systems, vol. 50, no. 4.  This set the stage for the papers above.  It built on Joe Miller's definition of effective local connectedness.  

Complex analysis

While working on computable complex analysis, I produced a few papers in pure complex analysis.  

• Carathadory's Theorem and moduli of local connectivity Appeared in Complex Variables and Elliptical Equations, vol. 61, no. 1
• An operator-theoretic proof of the existence of solutions to planar Dirichlet problems Appeared in 
• A potential-theoretic construction of the Schwarz-Christoffel map for finitely connected domains (with V. Andreev). Appeared in Complex Variables and Elliptic Equations, vol. 58, no. 2.  
• Estimating the error in the Koebe construction (with V. Andreev). Appeared in Computational Methods and Function Theory, vol. 11, no. 2.  This used some recent developments on the computation of capacity by Ransford to find an error estimate on the Koebe construction that did not presume knowledge of the target domain.  It answered a long-standing question in computational complex analysis.

Asymptotic density and coarse computability

A side project that arose while I was working on computable topology and complex analysis.  It used asymptotic density to build a framework for approximation of incomputable sets by computable sets.

• Asymptotic density and the coarse computability bound (with D. Hirschfeldt, C. Jockusch, and P. Schupp) Appeared in Computability vol. 5, no. 1
• Asymptotic density and the Ershov hierarchy (with R.Downey, C. Jockusch, and P. Schupp). Appeared in Mathematical Logic Quarterly, vol. 61, no. 3

Bounded queries

After finishing my thesis, I worked on bounded queries for a brief time.  The general idea in these papers is to understand which sets can be computed from a given oracle with a fixed number of queries.  

• On the convergence of query-bounded computations and logical closure properties of c.e. sets.  No preprint available.  Appeared in Journal of Symbolic Logic , vol 66, no. 4
• On the commutativity of jumps.  No preprint available.  Appeared in Journal of Symbolic Logic, vol. 65, no. 4