# Abstracts

**Colin McLarty: **What do large scale structures add, and not add, to Cohomological Number Theory?

(1) From "a different miracle in each case" to geometrically structured number theory: Weil, Serre, and Grothendieck

A look at the geometrizing achievements of Weil and the advances by Serre, Grothendieck, and of course many others, including how such ideas led Grothendieck to his approach, and their relation to logical foundations.

(The quote in the title is Serre on Diophantine equations before Weil.)

(2) What do large scale structures add, and not add, to Cohomological Number Theory: what does it take to prove Fermat's Last Theorem?

A more detailed look at Grothendieck's influence.

(3) "Unity represents the profound aspect, and generality the superficial aspect"

This talk shows how even the largest organizing tools of Grothendieck's cohomology theory do not require the strength of Grothendieck universes. It has long been known that abstract topos cohomology is a first order theory, and scheme theory is close to first order arithmetic. Grothendieck adopted universes as a quick, yet fully rigorous way of joining the two together. We can join them at the vastly lower logical strength of finite order arithmetic.

(The quote in the title is Grothendieck on his method.)

(4) Proving Fermat's Last Theorem in Peano Arithmetic and in Elementary Function Arithmetic: one view of the situation and prospects

A survey of prospects for proving Fermat's Last Theorem in second order arithmetic, Peano arithmetic, and weak fragments of Peano arithmetic.

**Angus Macintyre: ** How much arithmetic can one derive without appeal to set quantification?

Although the Gödel phenomenon shows that no computable axiom system can derive all true unsolvability statements in arithmetic (even if set-theoretic principles are allowed), no natural example has even been found of such a statement which is unprovable in PA, first-order Peano arithmetic. In these lectures I will explain the initial moves in a programme to show that Fermat's Last Theorem, and the related Modularity Theorem, are theorems of PA. In view of the exceptionally wide range of techniques, from beyond arithmetic, that go into the unconstrained proof, this is a very difficult undertaking, and can be construed as a contribution to a revived Hilbert Programme. This should give the enterprise philosophical, as well as mathematical interest. In this connection I will also pay attention to the issue of what more one may know once one has a proof in PA.

(1) PA, its proof theory and model theory

How to transcribe the content of the Hardy and Wright book (including the analytic number theory) into PA, and prove the theorems.

(2) Primes in PA

RIemann Hypotheisis for curves (Bombieri's elementary proof). Algebraic geometry in PA: Riemann-Roch, genus, etc. (for availability when dealing with nonstandard versions). How to arithmetize etale cohomology.

(3) Analysis in PA

Real, p-adic, complex, adelic, complete and/or compact structures arising in algebra, such as the profinite Galois groups and the deformation rings. Early work by Kreisel.

(4) Modular forms and modular curves

Why the Modularity Conjecture, despite overt appeal to much higher-order quantification, is really a basic unsolvability statement.