Thu, 18 Feb 2016
16:00
16:00
L5
Joint Number Theory/Logic Seminar: On a modular Fermat equation
Jonathan Pila
((Oxford University))
Abstract
`I will describe some diophantine problems and results motivated
by the analogy between powers of the modular curve and powers of the
multiplicative group in the context of the Zilber-Pink conjecture.
Thu, 04 Feb 2016
16:00
16:00
L5
Joint Number Theory/Logic Seminar: Strongly semistable sheaves and the Mordell-Lang conjecture over function fields
Damian Rössler
((Oxford University))
Abstract
We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.
Thu, 10 Mar 2016
12:00
12:00
L6
Sharp decay estimates for waves on black holes and Price's law
Dejan Gajic
(Cambridge)
Abstract
Price’s law postulates inverse-power polynomial decay rates for solutions to the wave equation on Schwarzschild backgrounds with respect to appropriately normalized null coordinates. Polynomial decay rates as a lower bound are known in the physics literature as “late-time power law tails”. I will discuss new physical space methods for proving sharp decay rates for solutions to the wave equation on a class of asymptotically flat, stationary, spherically symmetric spacetimes, establishing in particular the upper bounds and lower bounds in Price’s law on Schwarzschild. This work has been done jointly with Yannis Angelopoulos and Stefanos Aretakis.
Everyone knows that Moore’s law says that computers get cheaper at an exponential rate. What is not as well known is that many other technologies that have nothing to do with computers obey a similar law. Costs for DNA sequencing, some forms of renewable energy, chemical processes and consumer goods have also dropped at an exponential rate, even if the rates vary and are typically slower than for computers.
Mon, 18 Jan 2016
16:00 -
17:00
L3
4th moment of quadratic Dirichlet L-functions in function fields
Alexandra Florea
(Stanford University)
Abstract
We discuss moments of $L$-functions in function fields, in the hyperelliptic ensemble, focusing on the fourth moment of quadratic Dirichlet $L$-functions at the critical point. We explain how to obtain an asymptotic formula with some of the secondary main terms.