I recently finished giving a series of lectures at OPLSS16 on "Principles of Type Refinement"! You are welcome to check out the lecture notes I prepared for the course.
Research Interests
I am interested broadly in connections between logic, language, computation, and mathematics.
A longrunning project which I've been collaborating on seeks to develop a common mathematical language that unifies the disparate traditions of Hoare Logic, dependent type theory, and linear logic.
One conclusion reached from this project is that the established wisdom about the relationship between type theory and category theory needs to be revised, and that this has important implications for the way we look at type systems and other deductive formalisms.
You can find out more about this view in my paper with PaulAndré Melliès, "Functors are Type Refinement Systems" (POPL 2015).
Turning this vision into a practical reality will require a lot of work, but I'm convinced that the path ahead will be exciting!
Over the last few years I have also been immersed in exploring an unexpected world of connections between lambda calculus and the theory of graphs on surfaces, in particular the combinatorics of planar maps.
For a quick introduction to this world, you can look at my talk on Lambda Calculus and the 4CT, as well as some of the references mentioned there.
A different interest I have is in the notion of zeroknowledge proof from cryptography/complexity theory, but in particular how it can be reconciled with notions of knowledge from constructive logic. I gave a talk about these connections at HOPE 2012, which you can find in "Recent talks".
Recent papers and drafts
 A sequent calculus for a semiassociative law.
 In Proceedings of the Second International Conference on Formal Structures for Computation and Deduction (FSCD 2017). [doi] (Note: this mostly supersedes [arXiv:1701.02917], except for section 4 of the latter.)
 Connected chord diagrams and bridgeless maps.
 With Julien Courtiel and Karen Yeats. November 14, 2016. [arXiv:1611.04611]
 A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine.
 With PaulAndré Melliès. In Proceedings of the ThirtyFirst Annual ACM/IEEE Symposium on
Logic in Computer Science (LICS 2016). [doi] [arXiv:1601.06098]
 Towards a mathematical science of programming.
 Research statement, circa 2016.
 Linear lambda terms as invariants of rooted trivalent maps.
 Journal of Functional Programming, volume 26, 2016. [doi] [arXiv:1512.06751]
 Counting isomorphism classes of βnormal linear lambda terms.
 September 25, 2015. [arXiv:1509.07596]
 An Isbell Duality Theorem for Type Refinement Systems.
 With PaulAndré Melliès. Final version of January 31, 2017. To appear in Mathematical Structures in Computer Science. [doi] [arXiv:1501.05115]
 Balanced polymorphism and linear lambda calculus.
 Presented at TYPES 2015.
 Functors are Type Refinement Systems.
 With PaulAndré Melliès. In Proceedings of the 42nd ACM SIGPLANSIGACT Symposium on Principles of Programming Languages (POPL 2015). [doi]
 A correspondence between rooted planar maps and normal planar lambda terms.
 With Alain Giorgetti. Logical Methods in Computer Science, Vol. 11(3:22)2015, pp. 139. [doi] [arXiv:1408.5028]
 Type refinement and monoidal closed bifibrations.

With PaulAndré Melliès. October 1, 2013. [arXiv:1310.0263]
Recent talks
 Lambda calculus and the Four Colour Theorem.
 November 21, 2017, at the Edinburgh LFCS Seminar.
 Skewclosed objects, typings of linear lambda terms, and flows on trivalent graphs.
 September 9, 2017, at STRING 2017.
 A sequent calculus for a semiassociative law.
 September 4, 2017, at FSCD 2017.
 Some bridges between lambda calculus and graphs on surfaces.
 July 6, 2017, at the 26th British Combinatorial Conference.
 Some enumerative, topological, and algebraic aspects of linear lambda calculus.
 May 18, 2017, at CLA 2017.
 A Categorical Perspective on Type Refinement Systems.
 December 9, 2016, at the Logic and Semantics Seminar of the Cambridge Computer Laboratory.
 Principles of Type Refinement.
 Lectures given at OPLSS 2016, June 29July 2, 2016.
 Linear lambda calculus and the combinatorics of embedded graphs.

October 14, 2015, at Journées Nationales GEOCALLACLTP 2015. [pdf]
 Balanced polymorphism and linear lambda calculus.

May 18, 2015, at TYPES 2015.
 A connection between lambda calculus and maps.

January 18, 2015, at OBT 2015.
 Functors are Type Refinement Systems.

January 15, 2015, at POPL 2015
 Polarity in Proof Theory and Programming.

August 30, 2013, at the Summer School on Linear Logic and Geometry of Interaction in Torino, Italy.
 HOPE for a typetheoretic understanding of zeroknowledge.

September 9, 2012, at the 1st ACM SIGPLAN workshop on HigherOrder Programming with Effects. (Note: the slides seem to render funny with Firefox  best viewed in Chrome or Safari.)
BONUS: slides for
the "15 minute" version I gave October 4th at the IAS postdoc seminar series.
Older publications
 Polarity and the logic of delimited continuations.

In Proceedings of the TwentyFifth Annual IEEE Symposium on Logic in Computer Science (LICS 2010). [twelf code] [slides]
 Defunctionalizing focusing proofs.

Presented at the 2009 International Workshop on ProofSearch in Type Theories.
[twelf code] [more twelf]
 Refinement types and computational duality.

In Proceedings of the 2009 Workshop on Programming Languages meets Program Verification (PLPV 09). [agda code]
 Focusing on binding and computation.

With Dan Licata and
Bob Harper.
In Proceedings of the TwentyThird Annual IEEE Symposium on Logic in Computer Science (LICS 08).
[tech report]
 Focusing and higherorder abstract syntax.

In Proceedings of the 35th Annual ACM SIGPLANSIGACT Symposium on Principles of Programming Languages (POPL 08).
[coq code] [notes]
 On the unity of duality.

Annals of Pure and Applied Logic 153:1 (2008), special issue on "Classical logic and computation". [doi]
Dissertation
PhD in computer science, 2009, Carnegie Mellon University
The Logical Basis of Evaluation Order and PatternMatching.
Committee:
Peter Lee (coadvisor),
Frank Pfenning (coadvisor),
Robert W. Harper,
PaulAndré Melliès (external member)
Techniques from linear logic and infinitary proof theory (connected to the old
idea of a "prooftheoretic semantics" of logic) yield new insights
into seemingly extralogical features of modern programming languages.
By applying the CurryHoward correspondence to focusing proofs,
we develop a polarized type theory in which evaluation order is
explicitly reflected at the level of types, and which has builtin support for
patternmatching. This framework provides an elegant, uniform account of
both untyped and intrinsically welltyped computation, and moreover can
be extended with an extrinsic (Currystyle) type system to express
and enforce more refined semantic properties of programs. We
apply these ideas to explore the theory of typing and subtyping for
intersection and union types in the presence of effects, giving a
simplified explanation of some of the unusual artifacts of existing systems.
Other papers
 Nathack: a natural language interface for nethack.

January, 2003. With Cassia Martin, David Molnar, and Dev Purkayastha. As the title suggests,
this was a natural language interface for nethack! Done with a mix of prolog, embedded lua, and
scary hacking within nethack's internal C source. Our code is lying around
somewhere, and I could dig it up upon request.
Et cetera
 the nLab
 Knowledge is a collaborative effort.
 And Quiet Flows the Mon
 A photography project from the dark days after the 2004 U.S. Presidential Elections. 2017 update: whoops, it seems that my dystopian vision of US politics was a little bit too optimistic!
 Archive of papers by John Reynolds
 Direct link to the CMU AFS directory, since the ftp server is sometimes down.
 In Tune With Fun
 A truelife story about learning the accordion.
We may just be cockroaches at the base of a very large garbage mountain.
Dana Scott (on mathematics)
And tell me Margaret, when Iâ€™m gone, what will I want,
To be left at the bottom of a garbage bin, or dusted off and pulled up onto stage?
Jason Webley