Formal truth theory sits between mathematical logic and philosophy. In this talk, I will try to give a partial overview of formal truth theory, from my particular perspective and research, in connection to some areas of mathematical logic.
A pseudofinite group is an infinite model of the theory of finite groups. I will discuss what can be said about pseudofinite groups under various tameness assumptions on the theory (e.g. NIP, supersimplicity), structural results on pseudofinite permutation groups, and connections to word maps and generalisations.
Several number theorists have stressed that the proofs of FLT focus on small concrete arithmetically defined groups rings and modules, so the steps can be checked by direct calculation in any given case. The talk looks at this in relation both to Hilbert's idea of contentual (inhaltlich) mathematics, and to formal provability in Peano arithmetic and other stronger and weaker axioms.
For Logic Seminar: Note change of time and location!
It is well known that infinite perfect information two person games at low levels in the arithmetic hierarchy of sets have winning strategies for one of the players, and moreover this fact can be proven in analysis alone. This has led people to consider reverse mathematical analyses of precisely which subsystems of second order arithmetic are needed. We go over the history of these results. Recently Montalban and Shore gave a precise delineation of the amount of determinacy provable in analysis. Their arguments use concretely given levels of the Gödel constructible hierarchy. It should be possible to lift those arguments to the amount of determinacy, properly including analytic determinacy, provable in stronger theories than the standard ZFC set theory. We summarise some recent joint work with Chris Le Sueur.
In joint work with T. Tsankov we study a (yet other) point at which model theory and dynamics intersect. On the one hand, a (metric) aleph_0-categorical structure is determined, up to bi-interpretability, by its automorphism group, while on the other hand, such automorphism groups are exactly the Roelcke precompact ones. One can further identify formulae on the one hand with Roelcke-continuous functions on the other hand, and similarly stable formulae with WAP functions, providing an easy tool for proving that a group is Roelcke precompact and for calculating its Roelcke/WAP compactification. Model-theoretic techniques, transposed in this manner into the topological realm, allow one to prove further that if R(G) = W(G); then G is totally minimal.
Motivated by integrals of the Calculus of Variations considered in
Nonlinear Elasticity, we study mathematical models which do not fit in
the classical existence and regularity theory for elliptic and
parabolic Partial Differential Equations. We consider general
nonlinearities with non-standard p,q-growth, both in the elliptic and
in the parabolic contexts. In particular, we introduce the notion of
"variational solution/parabolic minimizer" for a class of
Cauchy-Dirichlet problems related to systems of parabolic equations.
In this talk, we will discuss low Weissenberg number
effects on mathematical properties of solutions for several PDEs
governing different viscoelastic fluids.
An old conjecture by A. Zygmund proposes
a Lebesgue Differentiation theorem along a
Lipschitz vector field in the plane. E. Stein
formulated a corresponding conjecture about
the Hilbert transform along the vector field.
If the vector field is constant along
vertical lines, the Hilbert transform along
the vector field is closely related to Carleson's
operator. We discuss some progress in the area
by and with Michael Bateman and by my student
Shaoming Guo.
We consider equations with the simplest hysteresis operator at
the right-hand side. Such equations describe the so-called processes "with
memory" in which various substances interact according to the hysteresis
law. The main feature of this problem is that the operator at the
right-hand side is a multivalued.
We present some results concerning the optimal regularity of solutions.
Our arguments are based on quadratic growth estimates for solutions near
the free boundary. The talk is based on joint work with Darya
Apushkinskaya.