Wednesday, November 29,
2006 (please note that the second talk will begin
at 5:15pm)
Speaker:
Dr. Amit Kulshrestha (Louvain-la-Neuve)
Title:
Similarity factors of hermitian forms over number fields
Abstract:
Undoubtedly, number fields exhibit many nice properties. Some of these
properties can be expressed in terms of Galois cohomology or quadratic
forms over fields (e.g. strong approximation property, torsion-free
third power of the fundamental ideal of Witt ring, etc.). In this talk
we discuss the group of similarity factors of hermitian forms over some
special fields (number fields always being typical examples of those
fields) and compare the group of similarities with "Hyp" group and
"certain norm" groups. The results mentioned in the talk are a part of
my PhD Thesis submitted last year.
Speaker: Ms.
Mélanie Raczek (Louvain-la-Neuve)
Title: Certain subspaces of order 3 matrices and their
automorphisms
Abstract: Let F be a separably closed field of
characteristic not equal to 2 or
3. We consider a 3-dimensional subspace V of the trace zero matrices of
M_3(F) which is totally isotropic for the trace form (x\mapsto
tr(x^2)). We compute the group of automorphisms of (M_3(F),V). Such a
computation is interesting with respect to the classification of degree
3 central simple algebras equipped with a subspace V as above.
Wednesday, November 15,
2006
Speaker:
Professor Werner Nahm (DIAS)
Title:
K_3(\bar Q) and modular forms
Abstract:
A conjectural relation between modular forms and torsion
elements of K_3(\bar Q) has been tested by D. Zagier and appears to
hold true. The relationship of the conjecture to the representation
theory of Yangians will be explained.
Wednesday, November 8,
2006
Speaker:
Professor Graham Everest (UEA)
Title: The Divisibility of Some Divisibility Sequences
Abstract: It is a pity
that the Mersenne Prime Conjecture seems to be out of reach at the
moment. However, in the late 19th century Zsigmondy and Bang found a
way to guarantee new prime factors of terms of this and other related
sequences. I will discuss this kind of result in the context of some
integer sequences which arise naturally from geometry using the theory
of elliptic curves.
Wednesday, October 18,
2006
Speaker: Professor Chris Smyth (Edinburgh)
Title:
Integer symmetric matrices with small eigenvalues
Abstract: In
joint work
with James McKee, we describe all integer symmetric matrices
whose eigenvalues have modulus at most 2. It turns out that they can be
essentially described by certain kinds of graphs, which we call charged
signed graphs. What is the significance of the bound 2? I'll explain
this in the talk. If time permits I'll also describe a potential
application for these matrices.
Wednesday, October 11,
2006
Speaker: Professor David Lewis (UCD)
Title: Similitudes
of algebras with involution under odd-degree extensions
Abstract: We
discuss the
following recent result of P. B. Barquero-Salavert: Let F be a field
and L/F be an odd-degree extension. Let (A_1, sigma_1) and (A_2,
sigma_2) be two central simple algebras with involution. We investigate
in what cases (including when char(F)=2), we have that (A_1, sigma_1)
and (A_2, sigma_2) are similar over L implies they are already similar
over F. This will have applications to the solution of injectivity
problems in nonabelian galois cohomology.
Wednesday, October 4, 2006
Speaker: Dr. Kevin Hutchinson (UCD)
Title: Galois
Groups, cup products and K-theory
Abstract: A much-studied paper of Sharifi and
McCallum
in 2003 relates the structure of the Galois group of the maximal
p-ramified (p prime) extension of a number field to a cup product in
cohomology and, via this, to K-theory and Iwasawa theory. We will give
a survey of these ideas and will provide a particularly simple
K-theoretic interpretation of part of the cup product which extends
some results of Sharifi and McCallum and gives information about the
Galois groups of certain number fields.
Wednesday, September 27,
2006
Speaker: Dr. Robert Osburn (UCD)
Title: Gaussian hypergeometric functions and
supercongruences
Abstract: In
1984, Greene
introduced the notion of hypergeometric functions over finite fields.
Special values of these functions have recently been of interest as
they are related to numbers of F_p points on algebraic varieties and to
Fourier coefficients of modular forms. In this talk, we discuss a
general modulo p^3 congruence for these functions which yields
extensions to recent supercongruences of Ono-Ahlgren, Loh-Rhodes, and
of Mortenson. This is joint work with Carsten Schneider (Linz).
Friday, June 9, 2006
Speaker: Dr. Mikael Lescop, Universite de Limoges
Title: Iwasawa Descent and Co-descent for units modulo
circular units
Abstract: pdf
Wednesday, May 17, 2006
Speaker: Dr. N. Semenov, Universitat Bielefeld
Title: Motivic decompositions of twisted flag varieties
Abstract:
The talk is devoted to motivic decompositions of projective homogeneous
varieties in the category of Chow motives. One of the motivations for
this problem is the recent progress achieved in proving celebrated
conjectures relating Galois cohomology and Milnor K-theory. In the talk
we describe a general decomposition method and provide a motivic
decomposition for varieties of type E_8.
Wednesday, April 19, 2006
Speaker: Dr. Nicholas Grenier-Boley, University of
Nottingham
Title: Criteria
for certain Witt rings of real fields to be isomorphic
Abstract: Two fields K and L of characteristic different
from 2
are said to be Witt equivalent if their Witt rings W(K), W(L) are
isomorphic as rings. In 1970, Harrison has found an interesting
criterion giving necessary and sufficient conditions for two fields to
be Witt equivalent: this result is known as Harrison's criterion.
In this talk, we will explain how to obtain such criteria (in the case
where the considered fields are supposed to be real) in three
situations: the Witt ring of a field whose u-invariant is smaller than
2, the Witt ring of a quadratic field extension L/K endowed with its
nontrivial automorphism in the case where the u-invariant of K is
smaller than 4, and the reduced Witt ring at a preordering. This is a
joint work with Detlev Hoffmann.
Wednesday, April 12, 2006
Speaker: Professor Burt Totaro, DPMMS, University of
Cambridge
Title: Splitting
Fields for Quadratic Forms
Abstract: There
is a natural
"simplest" quadratic form over any field, the hyperbolic form x_1x_2+
...+ x_{2n-1}x_{2n}. Any (nondegenerate) quadratic form over a field
becomes hyperbolic after some finite extension of the field. We give
some estimates for the smallest possible degree needed to make a
quadratic from hyperbolic; the focus is on bounds which work for all
fields, no matter how complicated. The known results use the topology
of the spin groups and the binary expansion of the square root of 2.
Wednesday, March 8, 2006
Speaker: Professor Detlev Hoffman, University of
Nottingham
Title: Levels
of Quaternion Algebras
Abstract: Let R
be a
unitary ring. If -1 can be written as a sum of squares
of R,
then the smallest positive integer n such that -1 is a sum
of n squares is called the level of R,
otherwise R is
said to be of infinite level. A famous result of Pfister
asserts
that the level of a field, if finite, is always a power of 2,
and
that every power of 2 can be realized as level of some field.
In
the case of noncommutative division rings, David Lewis constructed
quaternion division algebras of level 2^k and 2^k+1
for every
nonnegative integer k, and he asked whether other values can
occur
as the level of a quaternion division algebra. We
will show
that infinitely many other values can be realized as such levels.
Wednesday, March 1, 2006
Speaker: Dr
Jeremy Lovejoy, LIAFA, Universite Paris VII
Title:
Partitions, overpartitions, and indefinite quadratic forms
Abstract: In
1988, Andrews, Dyson, and Hickerson described a simple partition
function whose values are multiplicative and determined by an
indefinite quadratic form. Today there are still only a handful of
examples like theirs that have been discovered. I will describe some of
these, the related combinatorial objects, and the basic hypergeometric
series machinery required to establish the link to indefinite quadratic
forms. At the end of the talk I will briefly discuss connections with
mock theta functions and Maass waveforms.
Wednesday, February 15,
2006
Speaker: Dr
Robert Osburn,
UCD School of Mathematical Sciences
Title: Two
dimensional lattices with few distances
Abstract: It
is an old problem in combinatorial geometry
how to place a given number of distinct points in
n-dimensional Euclidean space in order to minimize
the total number of associated distances. In 1991,
Conway and Sloane conjectured that the best lattices
for this purpose are those with minimal Erdos number.
They proved that for n \geq 3 the lattices with minimal Erdos
number are precisely the even lattices of minimal determinant.
In this talk, we prove an analogous result for n=2. This is joint
work with Pieter Moree (MPIM).
Wednesday, January 25th,
2006
Speaker: Dr
Philippe Elbaz-Vincent, Universite de Montpellier
Title:
Cohomology of Modular Groups, Perfect Forms and k-theory of
Integers
Abstract:
For $N=5,6,7$, we compute the cellular complex as defined by Voronoi
associated to the real quadratic forms of rank $N$.
We deduce from this geometric complex the cohomologies of
$GL_N(\mathbb{Z})$ and $SL_N(\mathbb{Z})$ "modulo small torsion".
We will explain how we can deduce, with further computations, the
cohomology of more general modular groups.
We will show that $K_5(\mathbb{Z})$ is isomorphic to $\mathbb{Z}$ and
$K_6(\mathbb{Z})=0$.
We will also give a bound on the torsion $K_7(\mathbb{Z})$ and explain
what kind of informations we can get on $K_8(\mathbb{Z})$ and its
arithmetic consequences. Finally we will discuss the generalisation of
the
result to the case of imaginary quadratic rings (work in progress with
R.
Coulangeon, U. Bordeaux 1).
For $N=5,6,7$, it is a joint work with H. Gangl (U. Durham, UK) and C.
Soulé (CNRS et IHÉS).
Wednesday, November 23rd,
2005
Speaker: Dr
Robert Osburn,
UCD School of Mathematical Sciences
Title: 4-core
partitions and class numbers.
Abstract:
We discuss a proof (and possible generalization) of an elegant result
of Ono and Sze which relates a combinatorial object,
4-core partitions of a positive integer n, to
a number theoretic object, the form class
group of discriminant -32n-20.
Wednesday,
November 16th, 2005
Speaker:
Dr Ramesh Sreekantan (Tata Institute of Fundamental Research, Max
Planck
Institute)
Title: Drinfeld
Modular Curves and special values of L-functions
Abstract: Beilinson,
building on the work of others, formulated some remarkable
conjectures relating special values of L-functions and
K-groups of smooth projective varieties over number fields. He proved
them in few special
cases as well - in particular, for $K_1$ of the self product
of a modular curve. In this talk we will discuss these conjectures and
their analogues in the case of function fields and outline the proof of
an analogue of his theorem in the case of a self product of Drinfeld
modular curves. This is joint work with C. Consani.
Wednesday, November 2nd,
2005
Speaker:
Dr Pieter Moree (Max-Planck-Institute for Mathematics, Bonn, Germany)
Title: The
hexagonal versus the square lattice
Abstract:
Schmutz-Schaller formulated in 1995 a conjecture concerning lattices of
dimensions 2 to 8 and proved its analogue in hyperbolic geometry. As a
particular case he mentioned that the hexagonal lattice ought to be
`better' than the square lattice. This statement is equivalent with the
statement that for every x the number of integers
n<=x that can be written as a sum of two squares is not less
than the number of integers m<=x that can be written as a sum of
square and three times a square.
Together with Herman te Riele (CWI, Amsterdam) I recently proved this
by methods from computational number theory and the asymptotic theory
of arithmetic functions.
As a byproduct I disproved some claims on the divisibility of the
tau-function Ramanujan made in his unpublished intriguing manuscript on
the partition and tau function (two famous functions in number theory).
Wednesday, October 26th,
2005
Speaker:
Prof Stefan Muller-Stach (Johannes Gutenberg-Universität Mainz)
Title:
Cohomology of some arithmetic groups
Abstract: We
present a new method in order to
compute the cohomology of certain arithmetic subgroups
of semisimple Lie groups. We are mainly interested in
applications to Shimura type groups, e.g. SL(2) and
SU(2,1). The talk combines Algebraic geometry and cohomology of
arithmetic groups but uses only a fairly modest amount of either one.
We give one application to one of Grothendiecks standard conjectures.
Wednesday, October 19th,
2005
Speaker: Mr
James O' Shea (UCD School of Mathematical Sciences)
Title:
Levels and sublevels of composition algebras
Abstract: We
discuss the level problem for composition algebras, generalising
results regarding the level of quaternion algebras. In doing
so, we offer some refinements and simplifications of existing
statements and proofs. In addition, we present a simple
method of extending known
level and sublevel values up the chain of composition algebras.
Wednesday, October 12th,
2005
Speaker: Dr
Thomas Unger (UCD School of Mathematical Sciences)
Title: Hasse
principles for hermitian forms
Abstract: I
will discuss a number of important Hasse principles in the
theory of quadratic forms and how they can be generalized to
hermitianforms. I will also mention some work in progress with V.
Astier.
Wednesday, October 5th,
2005
Speaker: Dr
Herbert Gangl (Max Planck Institute and Durham)
Title:
Multiple logarithms, polygons, trees and algebraic cycles
Abstract:
(joint work with A. Levin and A.B. Goncharov)
We construct cycles in Bloch's algebraic cycle complexes which play the
role of multiple logarithms. The combinatorial encoding of the cycles
leads to Hopf algebras on plane trees and on polygons. We relate the
construction to Goncharov's Hopf algebra on iterated integrals and to
the one of Connes-Kreimer in renormalization theory.
Wednesday, September
28th, 2005
Speaker: Dr
Robert Osburn (UCD School of Mathematical Sciences)
Title:
Vanishing of eigenspaces and cyclotomic fields
Abstract: We
discuss a result of Thaine and how it can be used to give a proof, via
quadratic forms, of the following: for a prime $p>3$ congruent
to $3$ modulo $4$, the component $e_{(p+1)/2}$ of the $p$-Sylow
subgroup of the ideal class group of $\mathbb Q(\zeta_{p})$ is trivial.