Oxford

 We consider two decision problems for linear recurrence sequences (LRS) 
over the integers, namely the Positivity Problem (are all terms of a given 
LRS positive?) and the Ultimate Positivity Problem (are all but finitely 
many terms of a given LRS positive?). We show decidability of both 
problems for LRS of order 5 or less, and for simple LRS (i.e. whose 
characteristic polynomial has no repeated roots) of order 9 or less. Our 
results rely on on tools from Diophantine approximation, including Baker's 
Theorem on linear forms in logarithms of algebraic numbers. By way of 
hardness, we show that extending the decidability of either problem to LRS 
of order 6 would entail major breakthroughs on Diophantine approximation 
of transcendental numbers.

This is joint with work with Joel Ouaknine and Matt Daws.

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen, Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

We review the theory of $E_n$-algebras (roughly, algebras with $n$ compatible multiplications) and discuss $E_n$-deformation theory in

the sense of Lurie. We then describe, to the best of our ability, the use of $E_n$-deformation theory in the on-going work of Calaque, Pantev, Toen,

Vezzosi, and Vaquie about deformation quantization of derived stacks with shifted Poisson structure.

A program for Geometric Unity is presented to argue that the seemingly baroque features of the standard model of particle physics are in fact inexorable and geometrically natural when generalizations of the Yang-Mills and Dirac theories are unified with one of general relativity.