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Abstracts for Number theory lunch seminar

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On Drinfeld type automorphic forms and Rankin triple product L-functions over function fields

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Speaker: 
Fu Tsun-Fei
Affiliation: 
National Tsing Hua U/MPI
Date: 
Thu, 2013-05-02 14:00 - 15:00
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Automorphic forms of Drinfeld type can be viewed as function field analogue of weight 2 modular forms. After a brief review of basic facts about Drinfeld type cusp forms, I will discuss the Rankin triple product L-functions associated to these forms. From the Garrett-type integral representation, we obtain the functional equation of these L-functions. When the "root number" is positive, I present an analogue of the Gross-Kudla formula for the central critical values. Two examples will be shown at the end.

Threshold functions for systems of equations on random sets

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Speaker: 
Ana Zumalacárregui
Affiliation: 
U. Autónoma de Madrid
Date: 
Wed, 2013-05-08 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

I will present a unified framework to deal with threshold
functions for the existence of certain combinatorial structures in random
sets. More precisely, let M·x=0 be a linear system defining our structure
(k-arithmetic progressions, k-sums, B_h[g] sets or Hilbert cubes, for
example), and A be a random set on {1,...,n} where each element is chosen
independently with the same probability.

I will show that, under certain natural conditions, there exists a
threshold function for the property "A^m contains a non-trivial solution of

Toward multiple zeta values cycles

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Speaker: 
Ismaël Soudères
Affiliation: 
Duisburg-Essen/MPI
Date: 
Wed, 2013-05-15 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

We will review the setting of Bloch-Kriz cycle complex over the
projective line minus three points.
We will then show how to recover in this context the algebraic cycles
associated to the classical polylogarithms
using a pull-back by the multiplication on the affine line.

For a low weight example, we will explain how a twisted multiplication
map allows to build more general cycles.
We will conclude by showing how  a multiple zeta value arises from
the Gangl-Goncharov-Levin seesaw process.

Representation equivalence, characteristic equivalence and commensurability of arithmetic lattices

Posted in
Speaker: 
C. Rajan
Affiliation: 
TIFR/MPI
Date: 
Tue, 2013-05-21 15:10 - 16:00
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Gopal Prasad and A. S. Rapinchuk defined a notion of weakly
commensurable lattices in a semisimple group, and gave a
classification of weakly commensurable Zariski dense subgroups. A
motivation was to classify pairs of locally symmetric spaces
isospectral with respect to the Laplacian on functions. For this, in
higher ranks, they assume the validity of Schanuel's conjecture.
In this talk, we observe that if we use the stronger notion of
representation equivalence of lattices, then Schanuel's conjecture can

Kudla's modularity conjecture for cycles of codimension 2

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Speaker: 
Martin Raum
Affiliation: 
ETH
Date: 
Wed, 2013-05-29 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar 
We establish Kudla's modularity conjecture for the generating function 
of special cycles of codimension 2.  The proof is based on a 
convergence result for formal Fourier Jacobi expansions.  We define 
such expansions in some simple cases, and outline the approach taken to 
show that they always converge.  Finally, we showcase the connection to 
previous results by Zhang, which yields the desired proof of Kudla's 
conjecture.

The 3x+1 Problem: Status and Recent Work

Posted in
Speaker: 
Marc Chamberland
Affiliation: 
Grinnell College
Date: 
Wed, 2013-06-19 14:15 - 15:15
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar 
The 3x+1 Problem is a long-standing conjecture. Let T be 
a map from the positive integers into itself, where T(x)=x/2 if x is even 
and T(x) = (3x+1)/2 if x is odd. The conjecture asks whether, under 
iteration of the map T, any positive integer eventually reaches the value 
one. This talk gives a survey of the various approaches and results, 
intersecting areas such as number theory, dynamical systems, and 
functional equations. The speaker's approach involving generating
functions is also presented.

 

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