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Abstracts for Number theory lunch seminar

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The least common multiple of sets of positive integers

Posted in
Speaker: 
Ana Zumalacarregui
Affiliation: 
U. Autónoma de Madrid
Date: 
Wed, 2011-12-07 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

It is well known that the classical Chebyshev's function $\psi(n)=\sum_{m<n}\Lambda(m)$ has an alternative expression in terms of the least common multiple of the first n integers:$\psi(n)=\text{log lcm} (1,2,\dots, n)$.

Here we generalize this function by considering, for a set $\mathcal A\subseteq  [1,n]$, the quantity $\psi(\mathcal A):=\text{log lcm} \{a\,:\, a\in\mathcal A\}$ and we ask ourselves about its asymptotic behavior.

On the least prime in an arithmetic progression (d'apres T. Xylouris)

Posted in
Speaker: 
B.Z. Moroz
Affiliation: 
Bonn
Date: 
Wed, 2011-12-14 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Let a and q be two natural numbers with (a, q) = (1). By a classical theorem of J.P.G.L.Dirichlet,
there is a rational prime p and an integer l such that p = ql + a. No upper estimate for l in terms
of q was known, however, until the celebrated work of Yu.V.Linnik, who proved in 1944 that
$l < C q^L$ for some effectively computable constants C and L. In 1957  Ch.D.Pan proved
that one can take L=10000. The admissible value of L was improved in the works of several
authors. In 1992 D.R.Heath-Brown proved that the value L = 5.5 is admissable; in

Higher Chow cycles on Abelian Surfaces

Posted in
Speaker: 
Ramesh Sreekantan
Date: 
Wed, 2012-01-18 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

In this talk we discuss the construction of  new
indecomposable higher Chow cycles on a principally polarized  Abelian
surface over a non- Archimedean local field, which generalize a
construction due to Collino. The construction uses a generalization -
due to Birkenhake and Wilhelm - of some classical work of Humbert and
can be used to prove a  non-Archimedean analogue of the
Hodge-D-conjecture in the case when the Abelian surface has good and
ordinary reduction.
 

A weighted Brun-Titchmarsh inequality

Posted in
Speaker: 
Jan Büthe
Affiliation: 
Uni Bonn
Date: 
Wed, 2012-01-25 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The Brun-Titchmarsh inequality gives an upper bound for
the number of primes in finite arithmetic progressions. In this talk a
generalization of this inequality to some weighted sums over primes
is discussed.

The Brauer-Manin obstruction to the local-global principle for the embedding problem

Posted in
Speaker: 
P. Ambrus
Date: 
Wed, 2012-02-01 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We study an analogue of the Brauer-Manin obstruction to the
local-global principle for embedding problems over global fields. We will
prove the analogues of several fundamental structural results. In
particular we show that the Brauer-Manin obstruction is the only one to
strong approximation when the embedding problem has abelian kernel and show
that the analogue of the algebraic Brauer-Manin obstruction is equivalent
to the analogue of the abelian descent obstruction. In the course of our
investigations we give a new, elegant description of the Tate duality

Hopf algebra characters and multiple zeta values

Posted in
Speaker: 
M. Hoffman
Affiliation: 
US Naval Academy/MPI
Date: 
Wed, 2012-02-15 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Multiple zeta values can be thought of as images of a
homomorphism Z from the algebra QSym of quasi-symmetric
functions (an extension of the algebra of symmetric functions)
to the reals.  If this homomorphism is suitably defined, one
has the pleasing formula Z(H(t)) = Gamma(1-t), where H(t)
is the generating function of the complete symmetric
functions.  Aguilar, Bergeron and Sotille introduced the
idea of splitting a character on a Hopf algebra (which is
what Z: QSym -> R is) into odd and even factors.  We will

Heisenberg harmonic weak Maass-Jacobi forms

Posted in
Speaker: 
Martin Raum
Date: 
Wed, 2012-02-22 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The space of harmonic weak Maass-Jacobi forms has originally been
defined by Bringmann and Richter. We consider a subspace, that we call
the subspace of Heisenberg harmonic Maass-Jacobi forms. This space, as
opposed to the space of all Maass-Jacobi forms, allows for a detailed
analysis. We will decompose the space with respect to an analog of the
xi-operator for harmonic weak Maass forms, and we will demonstrate how
a theta-like decomposition comes up in the theory. We also discuss
singularities of Heisenberg harmonic Maass-Jacobi forms, which play a

Homotopy theory and rational points

Posted in
Speaker: 
Ambrus Pal
Date: 
Wed, 2012-03-07 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

It is possible to formulate very general local-global principles using homotopy theories of
algebraic varieties. In many cases these principles specialise to well-known obstructions
or conjectures, in other cases they tell something new. In my talk I will try to give an
overview of this subject.
 

Density of rational points on del Pezzo surfaces of degree one

Posted in
Speaker: 
Cecilia Salgado
Affiliation: 
Leiden/MPI
Date: 
Thu, 2012-03-15 11:15 - 12:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

The Segre-Manin Theorem implies that if a del Pezzo surface  S of degree at least three,
defined over the rational numbers, has a  rational point, then the rational points are Zariski
dense in S. A  result of Manin yields the same for degree two, as long as the initial  point
does not lie on a certain divisor. Similar results for del Pezzo  surfaces of degree one are
meager: they either depend on conjectures,  or they are restricted to small families of surfaces.

The surface of cuboids and Siegel modular threefolds

Posted in
Speaker: 
Damiano Testa
Affiliation: 
Warwick
Date: 
Wed, 2012-03-21 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

A perfect cuboid is a parallelepiped with rectangular faces all of whose edges, face diagonals and long diagonal have integer length.  A question going back to Euler asks for the existence of a perfect cuboid. 
No perfect cuboid has been found, nor it is known that they do not exist.

 

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