Let $E/k$ be an elliptic curve over a number field and $K/k$ a Galois extension with group $G$. What can we say about $E(K)$ as a Galois module? Not just what complex representations appear, but its structure as a $\mathbb{Z}[G]$-module. We will look at some examples with small $G$.
The idea of isotropic localization is to substitute an algebro-geometric object (motive)
by its “local” versions, parametrized by finitely generated extensions of the ground field k. In the case of the so-called “flexible” ground field, the complexity of the respective “isotropic motivic categories” is similar to that of their topological counterpart. At the same time, new features appear: the isotropic motivic cohomology of a point encode Milnor’s cohomological operations, while isotropic Chow motives (hypothetically) coincide with Chow motives modulo numerical equivalence (with finite coefficients). Extended versions of the isotropic category permit to access numerical Chow motives with rational coefficients providing a new approach to the old questions related to them. The same localization can be applied to the stable homotopic category of Morel- Voevodsky producing “isotropic” versions of the topological world. The respective isotropic stable homotopy groups of spheres exhibit interesting features.
I will present a procedure for perturbatively constructing the field content of gravitational theories from a convolutive product of two Yang-Mills theories. A dictionary "gravity=YM * YM" is developed, reproducing the symmetries and dynamics of the gravity theory from those of the YM theories. I will explain the unexpected, yet crucial role played by the BRST ghosts of the YM system in the construction of gravitational fields. The dictionary is expected to develop into a solution-generating technique for gravity.
How long does a uniformly accelerated observer need to interact with a
quantum field in order to record thermality in the Unruh temperature?
In the limit of large excitation energy, the answer turns out to be
sensitive to whether (i) the switch-on and switch-off periods are
stretched proportionally to the total interaction time T, or whether
(ii) T grows by stretching a plateau in which the interaction remains
at constant strength but keeping the switch-on and switch-off
intervals of fixed duration. For a pointlike Unruh-DeWitt detector,
coupled linearly to a massless scalar field in four spacetime
dimensions and treated within first order perturbation theory, we show
that letting T grow polynomially in the detector's energy gap E
suffices in case (i) but not in case (ii), under mild technical
conditions. These results limit the utility of the large E regime as a
probe of thermality in time-dependent versions of the Hawking and
Unruh effects, such as an observer falling into a radiating black
hole. They may also have implications on the design of prospective
experimental tests of the Unruh effect.
Based on arXiv:1605.01316 (published in CQG) with Christopher J
Fewster and Benito A Juarez-Aubry.
I will define and describe in some details a large class of gauge theories in four dimensions. These theories admit a variational principle with the action a functional of only the gauge field. In particular, no metric appears in the Lagrangian or is used in the construction of the theory. The Euler-Lagrange equations are second order PDE's on the gauge field. When the gauge group is taken to be SO(3), a particular theory from this class can be seen to be (classically) equivalent to Einstein's General Relativity. All other points in the SO(3) theory space can be seen to describe "deformations" of General Relativity. These keep many of GR's properties intact, and may be important for quantum gravity. For larger gauge groups containing SO(3) as a subgroup, these theories can be seen to describe gravity plus Yang-Mills gauge fields, even though the associated geometry is much less understood in this case.
The geometric Langlands correspondence relates rank n integrable connections
on a complex Riemann surface $X$ to regular holonomic D-modules on
$Bun_n(X)$, the moduli stack of rank n vector bundles on $X$. If we replace
$X$ by a smooth irreducible curve over a finite field of characteristic p
then there is a version of the geometric Langlands correspondence involving
$l$-adic perverse sheaves for $l\neq p$. In this lecture we consider the
case $l=p$, proposing a $p$-adic version of the geometric Langlands
correspondence relating convergent $F$-isocrystals on $X$ to arithmetic
$D$-modules on $Bun_n(X)$.
This will be mainly a survey talk covering recently-resolved conjectures of Polya and Wiman for entire functions, and progress on extensions to meromorphic functions
An important problem in algebra is the study of algebraic objects
defined over fields and how they behave under field extensions,
for example the Brauer group of a field, Galois cohomology groups
over fields, Milnor K-theory of a field, or the Witt ring of bilinear
forms over
a field. Of particular interest is the determination
of the kernel of the restriction map when passing to a field extension.
We will give an overview over some known results concerning the
kernel of the restriction map from the Witt ring of a field to the
Witt ring of an extension field. Over fields of characteristic
not two, general results are rather sparse. In characteristic two,
we have a much more complete picture. In this talk, I will
explain the full solution to this problem for extensions that are
given by function fields of hypersurfaces over fields of
characteristic two. An important tool is the study of the
behaviour of differential forms over fields of positive
characteristic under field extensions. The result for
Witt rings in characteristic two then follows by applying earlier
results by Kato, Aravire-Baeza, and Laghribi. This is joint
work with Andrew Dolphin.