I will be talking of recent results by Ayse Berkman and myself, as well as about a more general program of research in this area.
I will present joint work with A. Pillay on the theory of NIP formulas in arbitrary groups, which exhibit a local formulation of the notion of finitely satisfiable generics (as defined by Hrushovski, Peterzil, and Pillay). This setting generalizes ``local stable group theory" (i.e., the study of stable formulas in groups) and also the case of arbitrary NIP formulas in pseudofinite groups. Time permitting, I will mention an application of these results in additive combinatorics.
Applied mathematics provides a collection of methods to allow scientists and engineers to make the most of experimental data, in order to answer their scientific questions and make predictions. The key link between experiments, understanding, and predictions is a mathematical model: you can find many examples in our case-studies. Experimental data can be used to calibrate a model by inferring the parameters of a real-world system from its observed behaviour.
I will elaborate on some recent developments on the theory of special functions which are relevant to the calculation of Feynman integrals in perturbative quantum field theory, highlighting the connections with some recent ideas in pure mathematics.
The smooth mapping class group of the 4-sphere is pi_0 of the space of orientation preserving self-diffeomorphisms of S^4. At the moment we have no idea whether this group is trivial or not. Watanabe has shown that higher homotopy groups can be nontrivial. Inspired by Watanabe's constructions, we'll look for interesting self-diffeomorphisms of S^4. Most of the talk will be an outline for a program to find a nice geometric generating set for this mapping class group; a few small steps in the program are actually theorems. The point of finding generators is that if they are explicit enough then you have a hope of either showing that they are all trivial or finding an invariant that is well adapted to obstructing triviality of these generators.
The second in our fascinating collaboration with the Orchestra of the Age of Enlightenment (OAE) and Music at Oxford combines the muscial intelligence of the eighteenth century with the artificial intelligence of the twenty-first. Come along and hear the beauty of Bach's Nun komm, der Heiden Heiland (Now come, Saviour of the Gentiles) and the modern beauty of machine learning which may itself be the musical choice of audiences in 300 years' time.
The OAE provide the music (you even get to join in), Marcus delivers the sermon. Maths and Music; saying everything.
We continue with our series of Student Lectures with this first lecture in the 2nd year Course on Differential Equations. Professor Philip Maini begins with a recap of the previous year's work before moving on to give examples of ordinary differential equations which exhibit either unique, non-unique, or no solutions. This leads us to Picard's Existence and Uniqueness Theorem...
