L3

7 years ago, also in Oxford, Sylvy Anscombe and I asked this question, which is part of the general effort to try and understand the model theory of henselian valued fields through dividing lines. In 2024, Sylvy Anscombe and Franziska Jahnke completely classified NIP henselian valued fields. Their methods can be extended, with the help of works of Chernikov, Kaplan and Simon and of Kuhlmann and Rzepka, to NTP2 henselian valued fields, obtaining the following:

• if a henselian valued field is NTP2, then it is semitame and its residue field is NTP2;
• if a henselian valued field is separably algebraically maximal Kaplansky and its residue field is NTP2, then it is NTP2.

This covers a large class of fields, but there is still a gap. Notably, Fp((Q)) is in the middle: it is semitame but not Kaplansky.

To answer this question, we studied so called tame henselian fields with finite residue field, and derived quantifier elimination results, namely, we prove that any formula in the language of valued fields reduces to a formula of the form (∃y f(x,y)=0) ∧ φ(v(x)) ∧ ψ(res(x)), where φ and ψ are formulas in the language of ordered groups and of rings, respectively.

In Fp((Q)) specifically, the valuation ring itself is definable with a diophantine formula (ie of the form ∃y f(x,y)=0), reducing further our quantifier elimination result.

Finally, a large chunk of these formulas are known to be NTP2: when f(x,y) is additive in y, the formula ∃y f(x,y)=z is NTP2 (with respect to x and z). Unfortunately, that does not cover all formulas, so the answer to the titular question is still unknown.

I will start the talk by discussing renormlisation group in perturbative algebraic quantum field theory (pAQFT) and its non-perturbative incarnation acting on the Buchholz-Fredenhagen dynamical C*-algebra. I will also explain how pAQFT can be used to derive functional renormlisation group (FRG) equations that generalize Wetterich equations to globally hyperbolic Lorentzian manifolds and arbitrary states (beyond the usual FRG in the vacuum).

We study the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

TBA

We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven  by sub-diffusions. The new framework can capture the less financial activity phenomenon during the bear markets while having the classical Black-Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is in general incomplete. We investigate the pricing for European-style contingent claims under this new model. For this, we study the Girsanov transform for sub-diffusions and use it to find risk-neutral probability measures for the new Black-Scholes model. Finally, we derive the explicit formula for the price of European call options and show that it can be determined by a partial differential equation (PDE) involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE.

TBA

This is a joint OxPDE and Numerical Analysis seminar.