Thu, 04 Feb 2016
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
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.

How predictable is technological progress?

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.

Fri, 29 Apr 2016
12:00
L6

Prandtl equations in Sobolev Spaces

Tong Yang
(City University of Hong Kong)
Abstract
The classical result of Oleinik and her collaborators in 1960s on the Prandtl equations shows that in two space dimensions, the monotonicity condition on the tangential component of the velocity field in the normal direction yields local in time well-posedness of the system. Recently, the well-posedness of Prandtl equations in Sobolev spaces has also been obtained under the same monotonicity condition. Without this monotonicity condition, it is well expected that boundary separation will be developed. And the work of Gerard-Varet and Dormy gives the ill-posedness, in particular in Sobolev spaces, of the linearized systemaround a shear flow with a non-degenerate critical point under when the boundary layer tends to the Euler flow exponentially in the normal direction. In this talk, we will first show that this exponential decay condition is not necessary and then in some sense it shows that the monotonicity condition is sufficient and necessary for the well-posedness of the Prandtl equations in two space dimensions in Sobolev spaces. Finally, we will discuss the problem in three space dimensions.
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