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Special day on Mathematics of Cryptology
January 21 2005
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On January 21 we organize a joint meeting with the
RISC
seminar.
The lectures will take place at Room 201 (Huygens Building) of the
Lorentz Center in Leiden.
Program
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10:30−11:00
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Coffee
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11:00−11:45
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Carles Padró (Barcelona)
Secret Sharing Schemes, Error Correcting Codes and Matroids
Abstract
Error correcting codes and matroids have been widely used in the study
of secret sharing schemes. This talk deals mainly with the connections
between codes, matroids and a special class of secret sharing schemes,
namely multiplicative linear secret sharing schemes (MLSSS). Such
schemes are known to enable multi-party computation protocols secure
against general (non-threshold) adversaries. Two open problems related
to the complexity of multiplicative linear secret sharing schemes will
be considered.
Hirt and Maurer proved that such a scheme can be costructed whenever
the set of players is not the union of two unqualified subsets.
Cramer, Damgard and Maurer proved that, in this case, a multiplicative
linear secret sharing scheme can be efficiently constructed from any
linear secret sharing scheme by increasing the complexity by a
constant factor of 2. The first open problem we consider is to
determine in which situations a multiplicative scheme can be obtained
without increasing the complexity. We study this problem in an
extremal case. Namely, to determine whether all self-dual vector space
access structures admit an ideal MLSSS. By the aforementioned
connection, this in fact constitutes an open problem about Matroid
Theory, since it can be re-stated in terms of representability of
identically self-dual matroids by self-dual codes. We introduce a new
concept, the flat-partition, that provides a useful classification of
identically self-dual matroids. Uniform identically self-dual
matroids, which are known to be representable by self-dual codes, form
one of the classes. We prove that this property also holds for the
family of matroids that, in a natural way, is the next class in the
above classification: the identically self-dual bipartite matroids.
The second open problem deals with strongly multiplicative linear
secret sharing schemes. As opposed to the case of multiplicative
LSSSs, it is not known whether there is an efficient method to
transform an LSSS into a strongly multiplicative LSSS for the same
access structure with a polynomial increase of the complexity. We
prove a property of strongly multiplicative LSSSs that could be useful
in solving this problem. Namely, using a suitable generalization of
the well-known Berlekamp-Welch decoder, we show that all strongly
multiplicative LSSSs enable efficient reconstruction of a shared
secret in the presence of malicious faults.
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11:45−12:00
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Coffee break
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12:00−12:45
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Salil Vadhan (Harvard)
Randomness Extractors and their Cryptographic Applications
Abstract:
Over the past two decades, a rich body of work has developed around
the problem of constructing randomness extractors --- algorithms that
extract high-quality randomness (i.e. nearly uniform and independent
bits) from low-quality randomness (i.e. sources of biased and
correlated bits). Although some of the early results on randomness
extraction came from the cryptography literature, most of the
subsequent theory has been developed in the setting of computational
complexity, where extractors have unified the study of a number of
fundamental objects (such as pseudorandom generators, expander graphs,
and list-decodable error-correcting codes). In the past few years, the
relevance of extractors to cryptography has been re-discovered, with
increasing variety of applications being found. In this talk, I will
survey the basic theory of randomness extractors, give a sense of the
current state-of-the-art, and describe some of their cryptographic
applications.
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12:45−13:45
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Lunch
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13:45−14:30
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Rafi Ostrovsky (UCLA)
Survey on Private Information Retrieval
Abstract:
Consider a setting where a user wishes to retrieve an item from a
database, without letting the database administrator know which item
is being retrieved. Of course, a trivial (but expensive) solution is
for the user to request contents of the entire database, thus
concealing from the database administrator which item is of interest
to the user. Can this be done with less communication? Perhaps
somewhat surprisingly, the answer is yes, under various assumptions
and settings. In this talk, I'll survey much progress that has been
achieved on this problem and point our some interesting connections to
other problems in coding theory and several hardness results in
cryptography.
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14:30−15:00
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Tea break
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15:00−15:45
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Phong Nguyen (ENS Parijs)
From Euclid to Lenstra-Lenstra-Lovasz: Revisiting Lattice Basis
Reduction
Abstract:
Lattices are simple yet fascinating mathematical objects. Roughly
speaking, they are linear deformations of the n-dimensional grid Z^n.
Lattices have many applications in mathematics and computer science.
Of particular importance is lattice basis reduction, which is the
problem of finding "nice" representations of lattices. For instance,
lattice basis reduction is the most popular tool to attack public-key
cryptosystems. In this talk, we will revisit lattice basis reduction,
from Euclid's gcd algorithm to the celebrated LLL algorithm. We will
also briefly discuss recent results. Curiously, it is possible to
obtain a Euclid-like complexity for lattice basis reduction: in some
sense, one can compute a reduced basis (without fast integer
arithmetic) in essentially the same time as the elementary method to
multiply integers.
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15:45−16:15
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Tea break
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16:15−17:00
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Steven Galbraith
(London)
The Eta Pairing
Abstract:
We introduce a new pairing on certain supersingular curves which is
very closely related to the Tate pairing, but which has some
implementation advantages. We interpret the results of Duursma and Lee
in terms of this pairing and we describe a fast pairing on genus 2
curves in characteristic 2.
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17:00−18:00
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Drinks
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February 4
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Special program in Utrecht on the occasion of Johnny Edwards' PhD defense
The afternoon talks are in room 018 at Kromme Nieuwegracht 66, number 11 on
this map
We can have lunch at "Trans 10", which is number 14 on the
map
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10:30
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PhD defense of Johnny Edwards in the Academiegebouw, number 1 on
this
map
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13:00−14:00
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Michael Stoll (Bremen), Proving non-existence of rational
points on curves
Abstract.
Let C be a curve of genus at least two over Q (or, more generally,
over a number field K).
An important problem is to decide whether C has any rational
points. The first step in trying to solve this problem is to check
if C has points everywhere locally (which can be done effectively).
If this is the case, but C
does not appear to have global points, there is the possibility that
the absence of global points can be explained by the Brauer-Manin
obstruction. It turns out that this is equivalent to the existence
of a descent obstruction coming from some abelian covering of C.
We will discuss how we can check for such an obstruction, focussing
on genus 2 curves, and we will give experimental results obtained
by Victor Flynn, Nils Bruin and myself.
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14:15−15:15
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Andreas Schweizer (KIAS Seoul), Construction of quadratic function
fields whose class groups have m-rank 4.
Abstract.
Fix a finite field Fq of odd characteristic and
an odd integer m that is not divisible by the characteristic
of Fq. We are interested in the m-rank of the
divisor class groups of quadratic extensions L of the rational
function field Fq(T).
First we show: If there is one L with m-rank at least r,
then there are infinitely many L with m-rank at least r.
Then for q = 1 mod 4 we explicitly construct an L with
m-rank at least 4. Similar constructions give slightly weaker
results if q = 3 mod 4 and also for the ideal class group
of the integral closure of Fq[T] in L.
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15:30−16:30
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Andreas Weiermann (Utrecht/Muenster),
Analytic combinatorics of the transfinite
Abstract:
We consider a natural well-ordered subclass H of Hardy's
1917 orders of infinity.
Let H be the least set of functions from the non negative
integers into the non negative integers such that:
1) the constant zero function belongs to H,
2) with f and g the function idf+g belongs to
H where
id denotes the identity function.
Let < be the ordering of eventual domination on H.
A norm function is a function from H into the non negative
integers such that for any non negative integer n there are
only finitely many elements in H having norm below n.
For a given norm N and a given f in H let
cfN(n)
be the number of elements g in H such that
g<f
and N(g)≤ n.
We are going to classify various of these count functions.
Partition functions appear as special cases of this construction.
We use tools from Tauberian theory and generating function
methodology. Moreover we indicate where the count functions
can be used in logic.
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February 18
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Leiden, room 312 (first talk), 405 (other talks).
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12:00−13:00
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Robin de Jong (Leiden),
Arakelov geometry and its applications to number
theory
Abstract.
The purpose of this talk is to make popular the idea that Arakelov
geometry can be used to obtain effective results in number theory. We
discuss in this respect the existence of an efficient algorithm to
compute the Ramanujan tau-function at a given prime number, as well as
the effective Shafarevich conjecture trying to make quantitative the
(proven) statement that given a number field K, a positive integer g
and a set S of primes of K, there exist only finitely many
K^alg-isomorphism classes of curves of genus g, defined over K and
having good reduction outside S. The former topic is work in progress
together with Edixhoven and Couveignes, the latter topic is probably
still far out of reach, although recently strong results have been
obtained by Heier in the function field case.
In order to bring Arakelov theory into action, it is essential to
obtain explicit representations of the various objects that occur in
this theory. As an example we discuss an explicit formula for the
Arakelov-Green function of a compact and connected Riemann surface of
positive genus.
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14:00−14:45
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Eleonora Pellegrini,
Power bases for pure cubic fields
Abstract.
In this talk we will consider the problem of the monogenicity
of a number field, with a particular attention at cubic fields of the
form Q(3√m), where
m is a cubefree positive
integer. After giving a criterion of monogenicity, we will discuss
the
distribution of those m that give rise to monogenic pure cubic
fields.
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15:00−15:45
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Clemens Fuchs (Leiden),
Polynomial generalisations of a problem of
Diophantus
Abstract.
A Diophantine m-tuple is a set of m positive integers with the
property that theproduct of any two distinct elements plus one is a
square of an integer. In my
talk I will start with a survey on Diophantine m-tuples and possible
generalisations with an emphasis on variants of the problem for
polynomials withinteger coefficients. E.g. I will discuss a recent
joint result with A. Dujella
and G. Walsh: we proved that there does not exist a set of more than
12
polynomials with integer coefficients, not all constant, and with the
property
that the product of any two of them plus a linear polynomial is a
square of a
polynomial with integer coefficients.
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16:00−16:45
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Capi Corrales Rodrigañez (Madrid),
On the unit group of an order in a non-split quaternion algebra
Abstract.
I will speak on some joint work with E. Jespers, G. Leal and
A. del Río, in which we give an algorithm to determine a finite set
of generators
of the unit group of an order in a non-split classical quaternion
algebra H(K)
over an imaginary quadratic extension K of the rationals.
We then apply this method to obtain a presentation for the unit group
of H(Z[(1+√-7)/2]). As a consequence, we get a
presentation for
the orthogonal group
SO3(Z[(1+√-7)/2]).
These results provide the first examples of a characterization of the
unit group of some group rings that have an epimorphic image that is
an order in a non-commutative division algebra that is not a totally
definite quaternion algebra.
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March 4
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Universtiteit Gent,
building S22 at Galglaan 2, lokaal 14.
On the
map
of Gent
this is between numbers 14 and 18, and on the
Campus
map it is building 40-22
[further info].
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13:30-14:30
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Joost van Hamel (Leuven),
Abelianised Galois cohomology of reductive groups
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14:45-15:45
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Lenny Taelman (Groningen),
On analytic unformization of abelian varieties and Anderson
modules
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16:15-17:15
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Jan Schepers (Leuven),
Introduction to motivic integration
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March 18
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Groningen, room A901 at Broerstraat 9,
next to the Academiegebouw
(see the map)
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11:15−12:00
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Matthias Schuett (Hannover),
Extremal elliptic K3 fibrations
Abstract.
In this talk we will consider elliptic K3 surfaces. After a brief
introduction we will especially emphasize the extremal fibrations.
With view
to their classification and Frits Beukers' talk, we would like to
discuss how
many of them arise as pull-back from rational elliptic surfaces by a
base
change of low degree.
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12:15−13:00
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Ronald van Luijk
(Berkeley),
K3 surfaces with Picard number one and infinitely many rational
points
Abstract.
Not much is known about the arithmetic of K3 surfaces in
general. Once the Picard number, which is the rank of the Neron-Severi
group, is high enough, more structure is known and more can be said.
But still we don't know of a single K3 surface whose set of rational
points has been proved to be neither empty, nor Zariski dense.
Also, until recently, not even a single K3 surface was known with
Neron-Severi rank 1 and infinitely many rational points. We will give
an explicit example of such a surface over Q, where even the Picard
number over the algebraic closure is equal to 1. This solves an old
problem, that has been attributed to Mumford. The method used has been
extended by Remke Kloosterman to find elliptic K3 surfaces of rank 15.
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13:45−14:30
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Frits Beukers (Utrecht),
Computation of extremal elliptic K3 fibrations
Abstract.
In a joint effort with H. Montanus we computed all extremal
semistable elliptic K3 fibrations over P1. Although the subject
matter
is geometrical, this talk will be computational. We focus
our attention on the determination of 188 plane 'dessins d'enfant'
and their associated Belyi maps.
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14:45−15:30
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Jozef Steenbrink (Nijmegen),
The billiard problem on an ellipse
Abstract.
I will show how one may visualize the solution to the billiard
problem on an ellipse and how this gives rise to a nice deformation of
a double hyperbola. It is a demonstration of the use of the
CABRI-geometry program on academic level (I hope).
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16:15
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PhD defense of
Remke Kloosterman in the Academiegebouw
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April 1
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Utrecht, room K11
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11:00−12:00
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Roelof Bruggeman (Utrecht),
Period functions and Maass cusp forms
Abstract.
For holomorphic modular cusp forms there is a well established
relation with period polynomials. In the case of modular Maass cusp
forms, Lewis and Zagier have studied the associated period functions.
I'll speak about ongoing work of Lewis, Zagier and me on these period
functions and their cohomological interpretation.
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12:15−13:15
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Yiannis Petridis (Bonn),
Modular symbols and spectral theory
Abstract.
A conjecture of Goldfeld about periods of elliptic curves implies a
weak version of ABC. On the other hand this conjecture is equivalent
with the modular symbol conjecture about their relative growth. The
modular symbols encode geometric information about the cohomology and
the fundamental group of the modular curves. Through trace
type-identities we can relate modular symbols and the spectral theory
of the Laplace operator on the modular curve. The objects to study
are: families of automorphic forms, Eisenstein series twisted by
modular symbols and Selberg-zeta functions. The technique is
perturbation theory of operators. The result is the distribution of
the (normalized) modular symbols: a Gaussian law.
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14:15−15:15
| Byoung Ki Seo (Utrecht),
Asymptotic behaviors of the first return time of
translations on a torus
Abstract.
We investigate asymptotic behaviors of the first return time of
translations on a torus using Diophantine approximations. It is known
that the first return time to an element of the equipartition equals
to the 1 over the size of the element asymptotically for almost but
not all irrational translation on a 1 dimensional unit interval. We
expand the results to the case for an irrational translation on a
multidimensional torus.
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15:30−16:30
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Gunther Cornelissen (Utrecht),
Complexity of the rational numbers and conjectures about elliptic
curves
Abstract.
We don't know whether or not there is an algorithm to decide whether
a diophantine equation has a rational solution or not ("Hilbert's
tenth problem for Q", HTP(Q)), but Julia Robinson has proven in 1949
that
one cannot decide the truth of more general statements about the
rationals.
I will discuss conjectures about elliptic curves that allow us to
improve
upon Robinson's result and - in a sense that I will make precise -
closer
to a solution of HTP(Q). The keyword is "Zsigmondy type theorems for
elliptic divisibility sequences with extra conditions". I will present
some heuristics to support the conjectures.
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April 15
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Special day on Lambda rings
Nijmegen, room CZ N7 (=N1004 on the first floor of building N1)
On the
map look for the red dot with green arrow "oudbouw". Enter there
and go up two floors. For parking you might get lucky at the bottom of
the map near "INGANG VANAF BRAKKESTEIN" on the Driehuizerweg, which
is easier to spot on this
map
Speakers: Jim Borger (Max Planck), Frans Clauwens (Nijmegen), Bart de
Smit (Leiden).
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12:00−13:00
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Jim Borger (Max Planck),
Lambda rings for beginners part I
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14:00−15:00
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Jim Borger (Max Planck),
Lambda rings for beginners part II
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15:15−16:15
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Frans Clauwens (Nijmegen),
Natural operations on lambda rings
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16:30−17:30
| Bart de Smit (Leiden),
Integral lambda ring structure on finite étale Q-agebras
Abstract.
For a lambda ring K which is
finite étale as a Q-algebra we determine whether
K has a lambda subring R
of finite rank over Z so that
K=R⊗Q.
We also determine the maximal such R if it exists.
This is joint work with Jim Borger.
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May 13
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Leiden, room 412
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13:30−14:15
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Joost Batenburg (Leiden/CWI),
Small, smaller, smallest. Steps toward the atomic resolution
microscope.
Abstract.
Building an electron microscope that can view samples at atomic
resolution is a longstanding goal in the microscopy community. It has
recently become possible to acquire images of projected crystalline
structures in which separate atom columns can be resolved. By
themselves, these images do not provide sufficient information to
determine the positions and types of individual atoms inside the
crystal. However, when projections from several viewpoints are
combined, it is possible to make a full 3D reconstruction of the
crystal by performing a tomographic inversion procedure.
Because the number of measured projections is typically very small,
methods from continuous tomography, which are used in medical imaging,
cannot be used for this application. The young field of discrete
tomography is suited particularly well for computing reconstructions
from few projections. Discrete tomography is an interdisciplinary
field, with links to number theory, combinatorics, operations research
and coding theory.
In this talk I will describe how discrete tomography can be used to
compute atomic resolution 3D reconstructions of crystals and discuss a
variety of problems that still need to be solved.
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14:30−15:15
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Oliver Lorscheid (Utrecht),
Completeness and compactness for varieties over local fields
[math.AG/0410346]
Abstract. For a complex variety it is well known that it is
complete if and only if the rational points form a compact space in
the complex topology. The only essential property of the complex
numbers for this statement to hold is their locally compactness.
One finds the following generalisation: let K be a local field
and X a variety over K, then X is complete if and
only if for every finite field extension L of K,
X(L) is compact in its strong topology.
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15:30−17:00
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Pre-Journées program
Three PhD students from Leiden present their
proposed talks for the
Journées Arithmétiques
Willem Jan Palenstijn,
Computing near-primitive root densities
Abstract.
Artin's primitive root conjecture gives, for an integer x, an
expression for the density of primes q for which x is a
primitive root modulo q. In this talk, we consider the
following generalization: given a number field K, a non-zero
element x of K, and a positive
integer d, what is the density of primes q of K
for which the subgroup of the multiplicative group of the residue
class field of q generated by x has index dividing
d?
Bas Jansen,
Mersenne primes and Lehmer's observation
Abstract.
The Lucas-Lehmer test is an algorithm to check whether a number of the
form M=2p-1, with
p an odd positive integer, is prime.
The algorithm produces a sequence of
p-1 numbers modulo M,
starting with the number 4 and each time squaring
the previous number and subtracting 2.
Then the last number is zero modulo M if and only if
M is prime. Lehmer observed that if the last number is zero
then the penultimate number can be either PLUS or MINUS 2(p+1)/2 modulo M.
Gebre-Egziabher showed that if you start your sequence with 2/3
instead of 4, then the test also works, and the sign will be plus if
and only if p is 1 modulo 4 (p>5). In this talk we
generalize this result.
Reinier Bröker,
Class invariants in a non-archimedean setting
Abstract.
One of the goals of explicit class field theory is to compute a
generating polynomial for the Hilbert class field of a given number
field. For imaginary quadratic fields, the theory of complex
multiplication provides us with an elegant solution. The classical
approach has two improvements. Firstly, one can use `smaller'
functions than the classical j-function, leading to smaller
polynomials. This leads to the theory of class invariants, which was
initiated by Weber. Secondly, one can work in a non-archimedean
setting, avoiding the problem of rounding errors in the classical
approach. In this talk we will combine both improvements, i.e., we
will show how to use class invariants in a non-archimedean setting.
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September 2 |
Special program around PhD defense of
Gabor Wiese.
Location of the talks: Spectrumzaal of Studentencentrum Plexus,
Kaiserstraat 25, Leiden (
directions)
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10:00−10:45
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Gabor Wiese (Leiden),
Modular Forms of Weight One Over Finite Fields.
Abstract.
The talk having the same title as my thesis to be defended
in the afternoon will start by presenting the main
motivation for my research, namely (some of) the number
theoretic information really and conjecturally provided by
modular forms. Next, there will be a short overview over the
thesis. Finally, I will sketch the proof of one of the
results.
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11:00−11:45
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Loïc Merel (Paris),
The formula relating modular symbols to L-values.
Abstract.
There are well known formulas relating values of L-functions of
modular forms to modular symbols. These formulas enable to construct
p-adic L-functions etc. Modular symbols contain a finite
generating set consisting of the so-called Manin symbols. I will
describe how one can express a Manin symbol at level N in terms
of L-values obtained by twisting modular forms of level
N by characters of level dividing N and of a few local
invariants. Here is an elementary corollary of this formula : the
regular representation of
Gal(Q(μN)/Q), where
Q(μN) is the field generated by a
primitive N-th root of unity, does not occur in the group of
Q(μN)-rational points of an elliptic curve
E over Q of conductor N.
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14:15−15:00
| Thesis defense of Gabor Wiese at the Academiegebouw,
Rapenburg 73 in Leiden.
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Special day on the ABC-conjecture, September 9 2005
This is the kick-off meeting of an NWO sponsored "Leraar in Onderzoek"
project that will help Kennislink
to take ABC to the masses.
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September 9 |
Leiden, room 412.
| | 11:15-12:00 | Frits Beukers (Utrecht), Introduction to the
ABC conjecture
[PDF]
| | 12:15-12:45 | Jaap Top (Groningen), Finding good ABC triples,
part I; notes in Dutch (PDF)
| | 12:45-14:00 | Lunch
| 14:00-14:30 | Johan Bosman (Leiden), Finding good ABC
triples, part II;
notes in Dutch (PDF)
| | 14:45-15:30 | Hendrik Lenstra (Leiden), Granville's
theorem;
notes in Dutch (PDF)
Abstract.
Barry Mazur defined the `power' of a number to be the logarithm
of the number to the base its radical. For example, every perfect
square has power at least 2. How many integers up to a large
bound have power at least a given number? This question is
answered by Granville's theorem. It is of importance both in
understanding why the ABC-conjecture has a chance of being true,
and in analyzing an algorithm for enumerating ABC-triples.
| | 15:45-16:15 | Willem Jan Palenstijn (Leiden), Enumerating
ABC triples;
notes in Dutch (PDF)
Abstract.
An ABC triple is a triple of coprime positive integers a,
b, c with a + b = c and c
larger than the radical of abc. In this talk we present an
algorithm that enumerates all ABC triples with c smaller than a
given upper bound N with a runtime essentially linear in
N.
| | 16:30-17:00 | Carl Koppeschaar (Kennislink), Reken mee met
ABC
[PPT]
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September 23
|
Utrecht room K11
Joint session with
Intercity Arithmetic Geometry
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11:00-11:45 |
Bas Edixhoven,
Introduction to Serre's conjecture [PDF]
Abstract.
The conjecture will be stated, and put in its historical context and
in the wider context of the Langlands program. Serre's level and
weight of a 2-dimensional mod p Galois representation will be defined.
Khare's result will be stated. An overview will be given of what will
be treated in the seminar.
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12:00-12:45 |
Johan Bosman,
Galois representations associated to modular forms [PDF]
Abstract.
Modular forms, Hecke operators and eigenforms will be defined. The
existence of Galois representations associated to eigenforms will be
stated. The construction of these representations will be postponed
until later, in the more general case of Hilbert modular forms.
Something about weight 2 being easier and about weight 1 being special
can be said.
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13:30-15:15 |
Gunther Cornelissen,
Remarks on a conjecture of Fontaine and Mazur [PS]
Abstract.
The following statement will be explained and proved:
Let K be a
number field, and S a finite set of places of K. Let
KS be a maximal algebraic extension of K in which all
places in S are completely split. Let X be a smooth irreducible
quasi-projective scheme over K, such that for every v in
S the
set X(Kv) is non-empty. Then
X(KS) is
Zariski dense in X.
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October 7
|
Leiden, room 312.
Joint session with
Intercity Arithmetic Geometry
| |
12:15-13:00 |
Sander Dahmen,
Lower bounds for discriminants,
Abstract.
Lower bounds for discriminants of number fields will be
roved, in terms of their degree only; see
Odlyzko
| |
14:00-14:45 |
Frits Beukers,
Upper bounds for discriminants,
Abstract.
Let p be prime. Suppose that r is an irreducible
representation of the absolute Galois group of Q on a
2-dimensional vector space over an algebraic closure of
Fp, with p in {2,3}. The image of such
a representation is finite, hence determines a number field K
that is Galois over Q. An upper bound for the discriminant of
K will be proved that contradicts the lower bound of the
previous lecture in the case that p is in {2,3} and r is
unramified outside p. The audience will draw the logical
conclusion. For p=2 this result is due to Tate, and for
p=3 it is due to Serre.
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15:00-15:45 |
Bas Edixhoven,
Overview of Khare's proof
Abstract.
An overview will be given of Khare's proof of Serre's conjecture
in level one. Those who will not attend the rest of the seminar
will have an idea of Khare's proof, and it is hoped that those
who will attend the rest of the seminar will now be sufficiently
motivated to digest the more technical parts that are to come.
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16:00-16:45 |
Johan de Jong,
Kloosterman lecture:
Rational points and rational connectivity
Abstract.
Recently, Graber, Harris and Starr proved that any family of
rationally connected projective varieties over a smooth curve has a
section. A complex projective variety (or manifold) M is rationally
connected when every two points in M lie on a rational curve in
M.
In this lecture we will explain a generalization of this
result, joint with Jason Starr, to the case when the base of the
family is a surface.
In the other three lectures (October 14, November 11 and
November 25) we will explain the idea behind the proof of the theorem,
the analogy with Tsen's theorem, the connection with weak
approximation and the connection with the period-index problem for
Brauer groups.
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October 21
|
Delft, Snijderszaal (1e verdieping laagbouw EWI), Mekelweg 4.
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11:00-11:45 |
Klaas Pieter Hart (Delft),
Ultrafilters and combinatorics
Abstract.
In this talk I describe how ultrafilters may be used to prove
combinatorial
theorems about subsets of the set of natural numbers.
For example, an ultrafilter helps in making various choices in a
standard
proof of Ramsey's theorem.
For more intricate applications one needs to bring in some algebra.
One can extend ordinary addition to the whole set of ultrafilters and
thus obtain a compact semi-topological semigroup.
An idempotent in this semigroup is instrumental in proving Hindman's
finite-sum theorem and a special idempotent can be used to prove
Van der Waerden's theorem on arithmetic progressions.
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12:00-13:00 |
Valerie Berthe (Montpellier),
Some applications of numeration systems
Abstract.
The aim of this talk is to survey several applications of numeration
systems, first in discrete geometry, and second, in cryptology. We
first detail some natural connections between the study of discrete
lines (and more generally discrete hyperplanes), and beta-expansions
and tiling theory. For the latter application to cryptology, we then
focus
on some redundant numeration systems in mixed bases.
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14:00-15:00 |
Tom Ward (East Anglia), Heights and highly effective
Zsigmondy theorems
Abstract.
Elliptic divisibility sequences are known to eventually have
primitive divisors (indeed, the primitive part is very large).
This talk will describe some families of elliptic divisibility
sequences for which effective bounds can be given. For example,
every term beyond the fourth term in the Somos-4 sequence
1,1,1,1,2,3,7,23,... is shown to have a primitive divisor.
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15:15-16:15 |
Jan-Hendrik Evertse (Leiden),
Linear equations with unknowns from a multiplicative group
Abstract.
Let G be a finitely generated multiplicative group contained in
the complex numbers, and let a, b be non-zero complex numbers.
In 1960, Lang (building further on work of Siegel and Mahler),
proved that the equation
has at most finitely many solutions x, y in G.
Two equations ax+by=1, a'x+b'y=1 may be
called isomorphic if a'=au, b'=bv for
certain u, v in G. Clearly, two isomorphic
equations (1) have the same number of solutions. It is not difficult
to construct equations (1) having two solutions. In 1988, Györy,
Stewart, Tijdeman and the speaker proved that for given G, up
to isomorphism there are only finitely many equations of type (1)
having more than two solutions.
In my lecture I will discuss a generalization to linear equations
in more than two variables with unknowns from G. At the end
of my lecture I will explain how this fits into a more general
geometrical setting, and discuss a recent result by Gael Rémond
on semi-abelian varieties, of which the results on linear equations
mentioned above are special cases.
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16:30-17:15 |
Robbert Fokkink (Delft),
Minimal sets and Euclidean rings
Abstract.
Recently J.P. Cerri settled some conjectures on the euclidean
minimum of a number field, using results from topological dynamics.
This may seem like a remarkable link between two unrelated fields,
but it really is not.
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November 4
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Groningen, Room WSN 31, WSN-building
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11:30-12:30 |
Lenny Taelman,
An introduction to Drinfeld's shortest paper
Abstract.
The paper referred to in the title is called
"Commutative subrings of certain non-commutative rings"
(Funkcional. Anal. i Prilozen. 11 (1977), no. 1, 11-14).
We discuss its results and applications to number theory
and differential equations.
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13:15-14:15 |
Marius van der Put,
A local-global problem for differential equations
Abstract.
Consider a linear differential operator
L:=an(d/dz)n + ... +
a1(d/dz) + a0. The question
studied in the lecture is: if the inhomogeneous
equation L(y)=f has a formal solution everywhere locally,
does it follow that a global solution exists?
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14:30--15:00 |
Khuong An Nguyen,
d-solvable linear differential equations
Abstract.
A linear differential equation over a differential
field K is called
d-solvable if all its solutions are in a tower
K⊂K1⊂Kn of
differential fields, in which
each Ki+1/Ki is
either a finite extension or an
extension obtained by adjoining to Ki all
solutions
and their (higher) derivatives of some linear diff. eq.
over Ki of degree at most d. We
discuss this notion
and give examples.
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15:00--15:45
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Steve Meagher,
Forms of algebraic curves
Abstract.
Consider the Fermat curve
X4 + Y4 + Z4 =
0; this
curve has 96 automorphisms, and thus many `forms' (i.e.,
curves defined over the same ground field, which are not
isomorphic to the given one but which become isomorphic
over an extension field).
We compute these forms and give a formula for their number of points
over a finite field.
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16:00--17:00 |
Heide Gluesing-Luerssen,
A survey on convolution codes
Abstract.
Convolutional codes are (free) submodules
of F[z]n, for F a
finite field.
They can be interpreted as encoding arbitrarily long sequences
of message blocks into sequences of codeword blocks.
As is the case for classical block codes, a certain weight
function is needed for describing and investigating the
error-correcting properties of the code.
In this talk I will explain the basic notions for convolutional
codes.
Thereafter, I will sketch some current research projects.
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December 16
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Leiden, Room 412
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11:30-12:30
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Sven Verdoolaege
(LIACS, Leiden),
Enumerating parametric convex integer sets and their projections
Abstract.
Given a convex set of integer tuples defined by linear
inequalities over a fixed number of variables, where some of
the variables are considered as parameters, we consider two
different ways of representing the number of elements in the set
in terms of the parameters. The first is an explicit function
which generalizes Ehrhart quasi-polynomials. The second is its
corresponding generating function and generalizes the classical
Ehrhart series. Both can be computed in polynomial time based
on Barvinok's unimodular decomposition of cones. Furthermore, we
can convert between the two representations in polynomial time.
Finally, we present some ideas on how to handle projections of
convex integer sets.
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13:30-14:15
| Willemien Ekkelkamp (CWI/Leiden),
A variation of the MPQS factoring algorithm: analysis and
experiments
Abstract.
One of the methods used for factoring is the multiple polynomial
quadratic sieve factoring algorithm. In this talk I will present a
variation of this algorithm and some experimental results. Further,
we will take a look at the (theoretical) ratio of the different types
of relations coming out of the sieve. This ratio will be used in a
simulation program for estimating the time needed for factoring a
number.
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14:30-15:15
| Jeanine Daems (Leiden),
A historical introduction to mathematical crystallography
Abstract.
What is a crystallographic group? How many crystallographic groups are
there? The first part of Hilbert's 18th problem deals with
crystallographic groups. In 1910 Bieberbach solved this problem by
proving
that the number of crystallographic groups in each dimension is only
finite. In this talk I will discuss Bieberbach's proof and a modern
proof
of the same theorem. In dimensions 2, 3 and 4 the exact number of
crystallographic groups is known, I will talk about the methods used
to find them.
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15:30-16:30
| Peter Stevenhagen (Leiden),
Algorithmic class field theory
Abstract.
Class field theory `describes' the abelian extensions of a number
field, but does not readily provide generators for these extensions.
Finding a `natural' set of generators is a Hilbert problem that
is still largely unsolved.
We will discuss to which extent the theory furnishes algorithms to
generate class fields, and how this can be applied in algorithmic
practice.
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