This page is three and a half years out-of-date.

To everyone who follows my research - I apologise! I will try to update this webpage (or build an entirely new page) ASAP, around Easter 2010. New papers, new preprints, totally new explanations and "programmatic" texts (explanations of dreams) will be added. If you want to see my recent preprints - just send me an email.


My research motivation and my explanations have evolved very much, so the new page will be completely different. I'll add all the articles/preprints that accummulated in the last three years:

I'll aslo write some explanations about the four other ongoing projects:

everything below is a mixture of very old material (a lot is out-of date) from three-four years ago and recent edits.

I updated some places where some papers got published, but the whole thing needs total rewriting!!! Several files were later superceded by better versions. There is just too much material and I am working on teh problem of arranging it right...

Unprovability and logical strength in mathematics

Finished pieces from the past:
Unprovability threshold for planar graph minor theorem. (2010). Annals of Pure and Applied Logic, 162 (3), pp. 175 - 181. available online here
Unprovability, phase transitions and the Riemann zeta-function. (With Andreas Weiermann.) New directions in value-distribution theory of zeta and L-functions, pp. 19-37. Shaker-Verlag, 2009.
Unprovable Ramsey-type statements reformulated to talk about primes. (2009). in Proofs, Categories and Computations, Essays in honor of Grigori Mints. Tributes Series, College Publications, pp. 71 -80.
Exact unprovability results for compound combinatorial classes. (2009). Annals of Pure and Applied Logic, 157, pp. 77-84.
Unprovability of sharp versions of Friedman's sine-principle. (2007). Proceedings of the American Mathematical Society, 135, pp. 2967 - 2973. available online.
The strength of infinitary ramseyan principles can be accessed by their densities (With Andreas Weiermann.) (2005). Accepted for publication in Annals of Pure and Applied Logic. A reworked final version is coming in summer 2010.
Brief introduction to unprovability. (2009). Logic Colloquium 2006, Lecture Notes in Logic, pp. 38 -64.
Several proofs of PA-unprovability. (2005). Contemporary Mathematics Series of the AMS, 380, pp. 29-43.

Model theory and models of arithmetic

I originally studied Model Theory (models of arithmetic) as the main subject of my PhD research. My PhD results (on order-types of models of arithmetic) are unpublished but you can download the thesis file below.

My modest contributions to this subject are: a body of results about the linear order structure of models of Peano Arithmetic, including nonclassifiability results (non-structure arguments), ultimate results on order-types of models of PA strongly interpreted in other models of PA, partial results on the famous problem of whether order-types of all nonstandard models of PA are the same for all completions of PA, a study of arithmetically saturated models, and, recently, a study of first-order theories, whose models are treatable or categorically treatable (intuitively: resembling unique countable models (of omega-categorical theories) or unique countable recursively saturated models) and my recent article below that uses internalized constructions to give sufficient and necessary Sigma_1^1 conditions for a structure to be resplendent or chronically resplendent. The main idea here is to prove arithmetized versions of constructions of resplendent and recursively saturated models (versions of the arithmetized completeness theorem) and then infer results about models "in the real world".

Resplendent models and Sigma_1^1 definability with an oracle. (2008). Archive for Mathematical Logic, 47, pp. 604 - 621. , available online at Springer
Resplendent linear orders. (with Richard Kaye) (2002). Unpublished Article.
On order-types of models of arithmetic. (with R.Kaye) (2001). Contemporary Mathematics Series of the AMS, 302, pp. 275-285.
Order-types of models of arithmetic and a connection with arithmetic saturation. (2004). Lobachevskii Journal of Mathematics, 16, pp. 3-15. available online
On order-types of models of arithmetic. PhD Thesis. University of Birmingham. (2000).

Drafts and other unpublished and as-yet-unpublished things.

Here are some (a small portion of) drafts and sketches (usually totally out-of-date). This section is intended for colleagues who are closely following this research and need the actual texts, not the official versions that appear some 3-5 years later. This dungeon is constantly changing and its inhabitants eventually migrate upstairs.
Polynomial equations with quantifier-prefixes. 2010. Joint with Michiel De Smet. I removed the old version because the new, much better one, is coming shortly.
The Atlas of Truth and Strength.   2010. no draft yet.
Notes of my Bristol Logic Lectures started growing into something like a coherent text. the zeroth draft will be posted here eventually
Unprovability theory of the "thin implies Ramsey" template in Nash-Williams Theory. 2010. Joint with Michiel De Smet. A tentative 25-page draft is ready, but will change entirely this summer. But do ask us for a copy if you are interested in the current version.
Foundational theories and Arithmetical Bifurcation. A "programmatic" article (an explanation of a dream). 2009 - 2010. In progress. a tentative draft is ready. ask me for a copy.
Unprovable statements based on diophantine approximation and distribution of values of zeta-functions. (2007). Joint with Andreas Weiermann. Preprint. (this is a short version). This old rotting file will be totally disassembled and developed to form parts of other papers in the future.
Long sequences of braids. (2007). With Lorenzo Carlucci. Will not be published. Was subsumed by a later paper by Carlucii, Dehornoy and Weiermann.

Non-papers. Maybe I'll erase this whole section soon (no point keeping it).


Academic life

Current teaching