**Seminar
on Algebra and Logic**

Seminar talks
are intended to be accessible to graduate students. If you would like to
receive e-mail announcements about the seminar, please contact Mai Gehrke (mgehrke@math.ru.nl)
or Wieb Bosma (bosma@math.ru.nl).

FALL 2010

**December 3, 10, and 17**: Sebastiaan Terwijn. *Connections between computability and lattice theory*. Alle voordrachten
zijn in Hg03.054, 13:30 - 14:30.

**Abstract: **De voordrachten zullen gaan over verbanden tussen berekenbaarheid en tralietheorie, in het bijzonder over het Medvedev lattice en het Muchnik
lattice. Het is de bedoeling om een aantal
dingen in detail

te behandelen. Onderwerpen die aan de orde kunnen
komen zijn: Turing reduceerbaarheid, continuous and computable functionals, Medvedev reducties factors van het Medvedev lattice, verbanden met constructieve logica, lattice embeddings and intervals, Pi-0-1 klassen.

The DIAMANT **Intercity
Number Theory Seminar** will be held in Nijmegen on **1 October** 2010. The first talk will be in room HG00.086 and
the last two talks in room HG00.071. This is also the day of the Wiskundetoernooi.

14:30-15:15 **Cecilia Salgado**,
*Zariski** density of
rational points on del Pezzo surfaces of low degree*

**Abstract.** Let *k* be a non-algebraically closed field and *X* be a
surface defined over *k*. An interesting problem is to know whether the set of *k*-rational
points *X*(*k*) is Zariski dense in *X*. A lot of
research is done in this field but, surprisingly, this problem is not
completely solved for the simplest class of surfaces, the rational, where one
expects a positive answer. In this lecture I will define del Pezzo surfaces, a important
subclass of rational surfaces. I will talk about the cases already treated
(mainly by Manin), as well as the two cases left
open, the del Pezzo surfaces of degrees one and two,
presenting recent results (in progress) in the field.

15:30-16:15 **Rajender**** Adibhatla** (Essen),
*Higher
congruence companion forms*

**Abstract.** This talk will discuss the local splitting behaviour of ordinary, modular Galois representations and
relate them to companion forms and complex multiplication. Two modular forms
(specifically *p*-ordinary,
normalised eigenforms) are
said to be "companions" if the Galois representations attached to
them satisfy a certain congruence property. Companion forms modulo *p* play a role
in the weight optimisation part of (the recently
established) Serre's Modularity Conjecture. Companion
formsmodulo *pn* can be used to reformulate a
question of Greenberg about when a normalised eigenform has CM.

16:30-17:15 **David Gruenewald **, *Explicit Complex Multiplication in Genus 2 *

**Abstract.** In this talk we make explicit the Galois
action on the CM moduli for genus 2 Jacobians. By using recently computed (3,3)-isogeny relations, we demonstrate how this can be used to
improve the CRT algorithm for computing Igusa class
polynomials, providing some examples. This is joint work with Reinier Bršker and Kristin Lauter.

**August 27**: Mehrnoosh
Sadrzadeh, University of Oxford, *Algebraic logic to reason about information
acquisition*

**Abstract: **The purpose of this talk is to
show how abstract algebraic structures such as modules and quantales
can be useful in modeling and reasoning about information acquisition in
concrete scenarios of computer science and AI, in particular in robot
navigation protocols. To do so, we add modalities to the setting to model
agents' uncertainties about states, but also crucial to modeling are axioms
that describe how these uncertainties change as a result of performing
state-changing actions. I present recent joint work with P. Panagaden
and discuss connections to the relational approaches such as Kripke update models.

SPRING 2010

**June 8:** Sep Thijssen, *Computing torsion in field extensions*

Iedereen
is van harte welkom!

**June 15:** Dion Coumans, *Duality for classical first order logic*

**Abstract: **A
logic consists of a collection of axioms and
reasoning rules. Many logics may be associated to a class of algebras in such a
way that we can obtain information about the logic by studying the class of
algebras. We call the class of algebras associated to a logic
the 'algebraic semantics' of the logic. The algebraic semantics for classical
propositional logic (CPL) are given by Boolean algebras. The class of Boolean
algebras is dually equivalent to the class of Stone spaces. This duality
enables us to use topological tools in our study of CPL. The aim of this talk
is to show how the duality for CPL may be extended to a duality for classical
first order logic (CFOL). I will first give a short introduction in logic and,
in particular, in the use of duality theory in logic. Thereafter we turn to
classical first order logic. The algebraic semantics for CFOL are given by
Boolean hyperdoctrines. I will explain what these are
by abstracting the essential properties of the collection of all first order
formulas over a given signature. Thereafter we may identify the dual notion of
a Boolean hyperdoctrine and consequently describe a
duality for CFOL.

**June 8:** Sam van Gool, *Canonical extensions and Stone duality for strong proximity lattices*

**Abstract: **Strong
proximity lattices were introduced by Jung and Sunderhauf
[2] as the finitary algebraic structures dual to
stably compact spaces. A strong proximity lattice is a distributive lattice,
endowed with a binary relation, whose intended interpretation is `a is way below b'. Stably compact spaces are topological spaces which were proposed as the generalisation
of compact Hausdorff spaces to the T0 setting. Our
first result is that the duality of [2] can also be described algebraically and
in a point-free way, by de_ning the appropriate generalisation of canonical extensions of lattices to
strong proximity lattices.

Since distributive lattices are
in the famous Stone duality [3] with spectral spaces, it is natural to wonder
what additional structure on spectral spaces corresponds to the relation of a
strong proximity lattice. We show that, up to isomorphism, strong proximity
lattices correspond to spectral spaces with a retraction, and that the image of
this retraction is precisely the stably compact space which
the strong proximity lattice represents. In particular, we use this duality to
retrieve the result from Johnstone [1] that stably
compact spaces are precisely the retracts of spectral
spaces.

References:

[1] Peter T. Johnstone,
*Stone spaces*,
Cambridge studies in advanced mathematics, vol. 3, Cambridge University Press,
1982.

[2] Achim
Jung and Philipp Sunderhauf, On the duality of compact vs. open, *Papers on General Topology and Applications:
Eleventh Summer Conference at University of Southern Maine*, Proceedings (S.
Andima, R. C. Flagg, G. Itzkowitz,
P. Misra, Y. Kong, and R. Kopperman,* *eds.), *Annals of the New
York Academy of Sciences*, vol. 806, 1996, pp. 214-230.

[3] Marshall H. Stone,
Topological representation of distributive lattices and brouwerian
logics, Casopis pro pestovani
matematiky a fysiky 67
(1937), 1-25.

**April 29: **(HG01.058)
Tanneke Ouboter geeft een voordracht
geven (in het Nederlands)
over het onderwerp van haar
Master-scriptie, die zij heeft geschreven onder leiding van prof. Ronald Meester aan de Vrije Universiteit
Amsterdam.

**Abstract: **The
basic stochastic model for analyzing the spread of infection diseases is the
standard SIR (Susceptible →
Infectious →
Removed) model. One of the simplifying assumptions in this model is uniform
mixing between the individuals, which means that all individuals meet each
other at equal rate. We introduce two extended models, the Hierarchical and
Random model, where every individual (i.e. child) is part of precisely one
household and goes to precisely one school. So these social networks overlap
and the rate of disease transmission between two individuals depends on the
subgroup they both belong to. The two models are different in the way how the social levels of households and schools are
interlinked. In the Hierarchical model, all children in each household go to
the same school, while in the Random model every household member goes
independently of his or her sibling to a randomly chosen school. Additionally,
in both models we assume that all individuals are equally likely to meet each
other outside of their households and schools, called the community. To compute
the threshold parameter and expected final epidemic size of these models
heuristically, we have observed the spread of the disease in a slightly
different way (where the time dynamics are left out of consideration), such
that we could use branching approximations. These approximate results are shown
to be exact as the population size tends to infinity. We have compared both
models on their main characteristics heuristically by proving that in the start
of the epidemic, the Hierarchical model is stochastically dominated by the
Random model.

**January 13: **(HG01.058)

10.45-11.45 Noud
Aldenhoven presenteert zijn Bachelor scriptie:** ***Uniciteit** van het hyperre‘le
lichaam*.

De hyperre‘le
getallen zijn een uitbreiding van de re‘le getallen zodanig dat deze
getallen een totaal geordend lichaam vormen waarin oneindig kleine en oneindig grote getallen
bevinden. Dit hyperre‘le lichaam is elementair equivalent met het re‘le
lichaam. Hierdoor is het mogelijk analyse te doen
op de hyperre‘le getallen
in plaats van de re‘le getallen. Ik zal de constructie
van dit hyperre‘le lichaam behandelen. Deze constructie wordt ook wel de ultraproduct
constructie van de re‘le getallen genoemd. Voor het eerst goed beschreven door Robinson
(1966). Het zal blijken dat de constructie van dit lichaam niet
uniek is en verschillende lichamen oplevert. Echter, onder enkele
sterke aannames, kunnen we laten zien dat deze
lichamen isomorf met elkaar zijn. Dit
laatste kunnen we bewijzen door een resultaat te
gebruiken uit een artikel van Erdoes, Gillman en Henriksen
(1955).

11.45 Jorik
Mandemaker talks about new work in the Algebra &
Logic seminar: *From
Boolean algebras to Effect algebras.*

Effect algebras are latest in a
line of structures aiming to model quantum logic. Effect algebras are algebraic
structures with a partially defined addition. The key example
are all positive self-adjoint linear maps on a
Hilbert space below the identity. We'll take a categorical look at effect
algebras. In particular we'll construct an adjunction between Boolean algebras
and effect algebras.

**January 7:** (HG.00.310) Extra seminars:

10.00-11.00 Samson Abramsky (Oxford): *Coalgebras**, Chu Spaces, and Representations of Physical Systems*

11.00-12.00 Peter Johnstone (Cambridge): *Remarks on Lawvere's Nullstellensatz*

followed
by: 13.30 Ph.D. defense
Chris Heunen!

FALL 2009

**December 11: **Michael Naehrig,** ***Pairings
for cryptography*

All are welcome!

**December 2: **Jonas Frey,
Universite Paris 7, *A universal characterization of the tripos-to-topos construction**
*

**Abstract:**
The concept of (elementary) topos was introduced by Lawvere and Tierney
around 1970 as a generalization of Grothendieck's
notion of topos (nowadays known as Grothendieck topos), motivated by
logical and foundational questions. Nevertheless, until 1980, all known toposes were in fact Grothendieck
toposes, so the greater generality of Lawvere/Tierney's definition was never really exploited.
Finally, in 1980, Hyland, Johnstone and Pitts
described a construction which gives rise to toposes
other than Grothendieck toposes.
This construction starts from "triposes",
which are certain fibrations used in categorical
logic. The most prominent topos that can be obtained
in this way is the "effective topos", which
was described in an article by Hyland in 1981.

I will begin my talk by giving a
detailed and elementary description of the effective topos,
using a decomposition of Hyland's original construction in two steps. Then I
will define the general concept of "tripos",
and explain how the decomposition of the construction which was demonstrated at
the example of the effective topos allows us to give
a characterization of the tripos-to-topos
construction as a kind of biadjunction between a
2-category of triposes and a 2-category of toposes.See abstract

**November 25:**
Agostinho Almeida, Centro dÕAlgebra,
University of Lisbon, *Toggling between expansions of RS-frames and Urquhart spaces*

**Abstract:**
RS-frames and Urquhart spaces are two approaches to provide dualities for (not
necessarily distributive) bounded lattices. The former uses the canonical
extension of the original lattice and it is in fact a skeleton of the canonical
extension. The later is based directly on the original lattice using topology,
but if we drop the topology, the resulting structure is equivalent to the
RS-frame of the canonical extension of the original lattice. The method of
toggling between these two frameworks is made explicit and then we proceed to
expand it to an additional operation (in this case, a negation), based on
earlier work on dualities for lattices with negation via Urquhart duality by W.
Dzik, E. Or\l owska
and C.van Alten and similar
work (for the same classes of algebras) by the present speaker using canonical
extensions and RS-frames. This correspondence enabled us to sort out a
problem posed by the former authors in their paper.

**September 16:**
Pierre Gillibert, Universite
de Caen, *From** lifting objects to lifting diagrams: CLL, a general categorical
tool for solving problems in universal algebra*

**Abstract:** See pdf

SPRING 2009

**February 17:** Ivano Ciardelli, ILLC, Amsterdam, *Inquisitive
Semantics and Logic*

**Abstract:**
Inquisitive semantics is a tool which allows us to
represent the inquisitive content of a formula as well as the informative one.
I will first introduce the system and explain some of its basic properties. I
will then move on to a discussion of the associated logic: the connections
with Intuitionistic Logic will be clarified and a
complete axiomatization established. Finally,
Inquisitive logic will be presented as the limit of a hierarchy of logics
arising by imposing restrictions on the semantics.

**March 10:** Jean-Eric
Pin, LIAFA CNRS, Paris, *Equational** theory of regular languages*

**Abstract:**
(co-authors: Mai Gehrke, Serge Grigorieff) I will
present a survey of the equational theory of regular
languages. This theory makes use of a metric on words that depends on the
minimal size of an automaton separating two given words. The completion of the
set of words for this metric is the set of profinite
words. Using Stone duality, one can show that any lattice of regular languages
can be defined by a set of "profinite"
equations. This result applies in particular to classes of regular languages
defined by fragments of first order logic and can be extended to infinite words
and even to trees.

**March 17:** Isar Stubbe, Department
of Maths and CS, University of Antwerp, *Suprema** versus enriched colimits*

**Abstract: **A
quantale Q is a monoid in
the category of sup-lattices. Since a quantale is in
particular a monoidal category, we can
straightforwardly define Q-enriched categories, which can** **be thought of as "Q-valued
ordered sets". Every such Q-enriched** **category has an
underlying (ordinary) ordered set, and in this talk** **I shall relate enriched colimits in a Q-category with suprema
in its** **underlying
ordered set.

**March 24:** Frans Keune, IMAPP,
Radboud University Nijmegen,** ***Higher
algebraic K-theory*

**Abstract:** In
classical algebraic K-theory functors K_{n} are defined (say from rings to Abelian groups) for n=0,1,2. These groups contain a lot of information, including classical
mathematical results such as for example the quadratic reciprocity law in
number theory. The way they are defined is typically algebraic. Around 1970
constructions have been made to extend the theory to functors
K_{n} for all non-negative integers n. The resulting theory
is known as higher algebraic K-theory. What is the idea behind, what do these
constructions look like and is such an approach also useful for the classical
algebraic K-theory?

**March 31: Frans Keune**,
IMAPP, Radboud University Nijmegen, *Reciprocities on Dedekind domains*

**Abstract:**
The K_{1} of a commutative domain R is the direct sum of
its unit group and a group SK_{1}(R). For Dedekind domains the latter
can be described by means of "Mennicke
symbols" and there is an equivalent description by
"reciprocities". There also is a relative version describing the
SK<sub>1</sub> of an ideal of a Dedekind domain. A complete account
of this can be found in the book "Algebraic K-Theory" by Hyman Bass,
published in 1969. That was classical algebraic K-theory. There was no K_{2}
in Bass' book. I will show how higher algebraic K-theory can be used to obtain
these results. In 1974 I used reciprocities for a computation of the K_{2}
of a field that was completely different of the original computation by
Matsumoto in 1969.

**April 21: **Sam
van Gool, ILLC,Universiteit
van Amsterdam, *Canonical
extensions of partially ordered sets: reconstructing the lost duality*

**Abstract:**
Dualities between algebras and spaces are used to obtain representation
theorems, which yield completeness results in logic. The theory of canonical
extensions was developed in order to give a purely algebraic account of
representation theorems. Using canonical extensions, representation theorems
can be obtained, even in cases where a duality is not available. The process
can be reversed: previously unavailable dualities can be ÔextractedÕ from
canonical extensions. These extracted dualities can then provide explicit frame
semantics, both discrete and topological, for a logic.
We try to reconstruct (work in progress) the lost topological duality for
partially ordered sets, which are the algebras for substructural
logics.

**May 19: ** Raul
Leal Rodriguez, ILLC,Universiteit
van Amsterdam, *Of the Hennessy-Milner Property and other Demons: An elementary
construction of final coalgebras *

**Abstract: **Coalgebras are the dual of
algebras; however, a better intuition is to see** **coalgebras
as a generalisation of transition systems. In** **coalgebra a crucial issue is the description of the behaviour of a state in a** **coalgebraic
system; this can be done using e.g. final** **coalgebras. In this
talk we will give a basic introduction to** **coalgebra and coalgebraic logic. We will illustrate how logic help us to understand** **coalgebraic systems
presenting an elementary** **construction of final coalgebras.

** **

**May 26: **Dennis
Meffert, IMAPP/ICCIS, Radboud University Nijmegen, *Pairing**-Based
Cryptography*

**Abstract:**
Interest in bilinear pairings in the cryptographic research community has been
steadily rising since the publication of the paper 'Identity-Based Encryption
from the Weil Pairing' by Boneh and Franklin in 2001.
In this talk I will give an overview of the mathematics of
bilinear pairings on elliptic curves, as well as some cryptographic
applications.

**June 2:** Joost Berson, IMAPP, Radboud
University Nijmegen, *Tameness of automorphisms over Artinian rings*

**Abstract:** One
of the open problems about polynomial automorphisms
over a field, is the question whether they are tame, i.e. a composition of
linear and elementary automorphisms. The famous
Jung - Van der Kulk Theorem
(1942/1953) states, that this is true in the case of 2 variables. The
case of 3 variables was not solved until 2004 (negative answer). However,
a non-tame automorphism in 3 variables could still be
tame when seen as an automorphism in 4 or more
variables (it would then be called stably tame), so the general problem
remains open. In fact, D. Wright, A. van den Essen and the speaker recently
showed that all known automorphisms in 3
variables over a field are stably tame. For this result, 2-variable
special automorphisms over Artinian rings
proved to be crucial. After a survey of the tameness problem, we will
examine them. It will turn out that they are tame in case of an Artinian Q-algebra. In characteristic p, the problem can be
reformulated in very basic algebra.

The DIAMANT **Intercity Number Theory Seminar** will be
held in Nijmegen on 12 june.

13:00-14:00 Sander Zwegers, Mock modular forms: an introduction

**Abstract.** The
main motivation for the theory of mock modular forms comes from the desire to
provide a framework to understand the mysterious and intriguing mock theta
functions, defined by Ramanujan in 1920, as well as
related functions. In this talk, we will describe the nature of the modularity
of the original mock theta functions, formulate a general definition of mock
modular forms, and consider some further examples. Time permitting,
we will also consider a generalization to higher depth mock modular forms.

14:30-15:30 Oliver Lorscheid, Toroidal Eisenstein
series and double Dirichlet series

**Abstract.** A
formula of Erich Hecke in an article from 1917 laid a
connection between a sum of values of an Eisenstein series E(-,s)
with the value ζ(s) of the zeta function ζ. We call an automorphic form toroidal if the
corresponding sum (or integral in its adelic
formulation) vanishes for all right translates. The importance of this
definition lies in a reformulation of the Riemann hypothesis in terms of the space
of toroidal automorphic
forms as observed by Don Zagier.

Namely, the Eisenstein series E(-,s) lies in a tempered representation if and only if s
has real part 1/2, and by Hecke's formula, E(-,s) is toroidal if s is a zero of the zeta functions. In order to
reverse the latter statement, non-vanishing results has to be shown for the
factors occuring in Hecke's
formula. In a joint work with Gunther Cornelissen, double Dirichlet
series are used for this purpose. In this talk, we will introduce into the theory
of (toroidal) automorphic
forms and give an overview over results in this direction. Then we will explain
how to use double Dirichlet series to show
non-vanishing results.

16:00-17:00 Dimitar
Jetchev, Global divisibility of Heegner
points and Tamagawa numbers

**Abstract.** We
improve Kolyvagin's upper bound on the order of the
p-primary part of the Shafarevich-Tate group of an
elliptic curve of rank one over a quadratic imaginary field. In many cases, our
bound is precisely the one predicted by the Birch and Swinnerton-Dyer
conjectural formula.

**Donderdag**** 2 juli:**
Janneke van den Boomen
van 14.30 tot 15.30 in zaal HG00.303,** ***Grafen** en hun niet-isomorfe opspannende bomen*

**Abstracts:** Iedere samenhangende graaf bevat een
of meerdere opspannende bomen. Deze bomen zijn
op te delen in isomorfieklassen. Op deze manier kunnen we dus bepalen welke niet-isomorfe opspannende bomen een graaf
bevat. We zullen de algoritmes die hiervoor worden gebruikt gaan bekijken. Als je dat eenmaal kunt
bepalen komen er vele nieuwe,
interessante vragen naar
boven. Een aantal daarvan zullen we proberen te beantwoorden.
We zullen gaan bekijken hoe je voor bipartiete grafen een formule
kunt bepalen voor het aantal niet-isomorfe opspannende bomen, wat de kleinste
graaf is die alle niet-isomorfe bomen bevat en wat de grootste graaf is die er maar een
bevat.

FALL 2008

**Friday, September 26:**
A day on **Standard
models of finite fields **addressing the theoretical and practical aspects of
defining finite fields algorithmically. The lectures will take place in room HG
00.071 (See also http://www.math.leidenuniv.nl/~desmit/ic/2008.html)

11:30-12:15 **Frank LŸbeck**,
*Conway
polynomials*

**Abstract:** I
will give the definition of the Conway polynomials that define finite fields,
mention some cases where they are used, and explain how they can be computed.
Then I will address the problem that the Conway polynomials that are not yet
known are very difficult to compute. On the other hand one would like to know
them for any field GF(*q*) for which the factorization of (*q*-1) is known
(these are the fields in which elements can be tested for primitivity).
I will propose a modification of the definition such that the modified
polynomials can be computed in reasonable time.

13:30-14:15 **Wieb**** Bosma**, *Dealing with finite fields in Magma*

**Abstract:** In
computations with finite fields it is essential to maintain subfield relations
in a consistent way. In this talk I will describe the different representations
for finite fields in the computer algebra system Magma, and the mechanism used
for ensuring that subfield diagrams commute.

14:45-15:30 **Bart de Smit**,
*Consistent isomorphisms between finite fields*

**Abstract:** We
give a deterministic polynomial time algorithm that on input two finite fields
of the same cardinality produces an isomorphism between the two. Moreover, if
for three finite fields of the same cardinality one applies the algorithm to
the three pairs of fields then one obtains a commutative triangle. The
algorithm depends on the definition given in the next talk.

16:00-16:45 **Hendrik**** Lenstra**, *Defining** Fq*

**Abstract:** The
lecture provides a definition of **F*** _{q}* as an actual field of
cardinality

**October 21:**
Alessandra Palmigiano, ILLC, Universiteit van
Amsterdam, *Topological** groupoid quantales:
an algebraic study of noncommutative topologies*

**Abstract:** See Palmigiano&Re

**November 11:**
Bas Spitters, Radboud Universiteit/TU Eindhoven, *A
computer-verified implementation of Riemann integration – an introduction
to computer mathematics* (jww Russell O'Connor)

**Abstract:**
The use of floating point real numbers is fast, but may cause incorrect answers
due to overflows. These errors can be avoided by hand. Better, exact real
arithmetic allows one to move this bookkeeping process entirely to the computer
allowing one to focus on the algorithms instead. For maximal certainty, one
uses a computer to check the proof of correctness of the implementation of this
algorithm. We illustrate this process by implementing the Riemann integral in
constructive mathematics based on type theory. The implementation and its
correctness proof were driven by an algebraic/categorical treatment of the
Riemann integral that is of independent interest.
This work builds on
O'Connor's implementation of exact real arithmetic. A demo session will be
included.

**November 18:**
Roel Willems, IMAPP,
Radboud Universiteit, *Polynomial automorphisms over finite fields*

**Abstract:**
Let k be a field. We want to find a good description of the group of polynomial
automorphisms Aut_n(k) of k^n.
If n=1, this is trivial, because the only polynomial automorphisms
are the affine maps. For n=2, Jung in 1942 showed that if char(k)=0,
then the automorphism group is generated by the tame
maps. In 1953 Van der Kulk
generalized this to any field. For n>2 it is still open, but in 2004 Shestakov and Umirbaev showed
that in case n=3, N = ( x-2(xz+y2)y-(xz+y2)^2z,
y+(xz+y2)z, z ) Nagata's map (1972) in char(k)=0 is not tame. If char(k) > 0 the problem is still open for n>2. This
talk will be about some results and some open problems in the case where k=Fq, a finite field.

**November 25:**
Ionica Smeets, Mathematisch Instituut,
Universiteit Leiden, *The** LLL-algorithm: how it originated and how we can
use it as a multidimensional continued fraction algorithm
*

**Abstract:**
Hendrik Lenstra, Arjen Lenstra and L‡szl— Lov‡sz published their
famous LLL-algorithm for basis reduction in 1982. Last year the 25th birthday
of this algorithm was celebrated in Caen, France with a three-day conference. Lenstra, Lenstra, Lov‡sz and close bystander Peter van Emde
Boas started the conference by telling how the algorithm emerged from
misunderstandings, errors and coincidences. Ionica Smeets wrote down these memories for an upcoming book from
Springer about the LLL-algorithm. The first part of her talk will be this nice
historic story. In the second part she will talk about continued fractions and
explain how you can iterate the LLL-algorithm to find a series of multidimensional
continued fractions.

**December 2:** GyšngyvŽr Kiss, IMAPP, Radboud Universiteit (Budapest), *Analysis and
implementation of a deterministic primality test *

**Abstract:**
Elliptic curves have been applied very successfully to the problem of proving primality in practice for large prime numbers. GyšngyvŽr will describe the primality
test, and point out how heuristics on the behaviour
may improve the practical performance.

**December 4:** Mirte Dekkers, IMAPP, Radboud
Universiteit (14.00 tot 15.00 in HG 00.071), *Stone** dualiteit: een toepassing in de theorie van formele talen*

**Abstract:** Afstudeervoordracht. Na afloop is
er koffie en thee met
(home-made) brownies in de 37-gang.

SPRING 2008

**January 7:**
Mai Gehrke, IMAPP, Radboud Universiteit Nijmegen,** ***The p-adic numbers from a
duality perspective*

**Abstract: **One
may think of the p-adic numbers as an
algebra, but from the perspective of topological methods in algebra,
this structure is actually a topo-relational space on
which the relations happen to be functions. In this talk I expose this point of
view.

**January 14:**
Ruben van den Brink, IMAPP, Radboud Universiteit Nijmegen, *Intuitionistic** Independence Results*

**Abstract: **Reverse
mathematics is concerned with calibrating theorems with axioms: given a basic
formal system and an axiom, which theorems are equivalent to the axiom and
which theorems can not be proved? For instance, the
fan theorem is equivalent to a number of theorems (e.g. Heine-Borel and an approximating version of Brouwers
Fixed Point Theorem, cf. [Veldman, report 0509, July
2007]), provably in a weak formal system of analysis using constructive logic.
For showing certain formulas are not provable from a given set of axioms,
various techniques have been developed (Kripke
models, sheaf models, realizability, just to name a
few). Unfortunately, most of these shamelessly use the apparatus of classical
mathematics. One can seriously question the significance of these
independence results from an intuitionistic point of
view. Also, because some of them are stated somewhat loosely and apparently
contradict established intuitionistic results.
However, not all might be lost. Taking Scott's topological interpretation as a
starting point, we show how to adapt his method to obtain some intuitionistically meaningful independence results.

.

**January 21:**
Roel Willems, IMAPP,
Radboud Universiteit Nijmegen, *A vanishing conjecture on differential operators*

**Abstract: **
In
this talk, I will discuss the connection between the Jacobian
Conjecture and the aforementioned vanishing conjecture. Furthermore I will give
a proof of this conjecture for a special class of differential operators.

**January 28:** Michiel de Bondt, IMAPP, Radboud
Universiteit Nijmegen, *Homogeneous Keller maps*

**Abstract: **In
this talk, I will discuss what I have done the past years and will lead to the
"cancelation of the s". Reductions of the Jacobian
Conjecture to special classes of polynomial mapping are discussed. For these
special classes of polynomial mappings, the Jacobian
conjecture is proved for small dimensions.

**March 13:** Vicenzo Marra, Dipartimento di Informatica e Comunicazione, Universita' degli Studi di Milano, *An introduction
to lattice-ordered groups*

**Abstract: **We
give an overview of some parts of the theory of lattice-ordered groups, aimed
at an audience of non-specialists. While we do address general lattice-groups,
we mainly concentrate on the Abelian case. Indeed, we
devote a substantial part of the talk to a discussion of the Baker-Beynon representation theory for finitely generated
projective lattice-ordered Abelian groups and real
vector spaces, and its connections with piecewise linear topology.

**March 20:** Klaas Landsman, IMAPP, Radboud Universiteit Nijmegen, *Algebraic and
logical notions of space*

**Abstract: The relationship between classical logic and space
was discovered by George Boole in his 'Laws of Thought' from 1854**.
In the 1930s, John von Neumann attempted to adapt Boole's work to the quantum
setting (whatever that may mean), replacing general sets as in Boole by Hilbert
spaces (and subsets by closed linear subspaces). Although still
"spatial" in the sense of Boole, the ensuing "quantum
logic" of Birkhoff and von Neumann is
suspicious, however. A better logical structure relevant to quantum theory,
developed in collaboration with Chris Heunen and Bas Spitters, emerges when Boole's sets are replaced by locales
in a topos. Such objects are simply called
"spaces" by topos theorists, and, being
complete Heyting algebras, they automatically carry
an intuitionistic logical structure. The connection
with quantum theory is given by our basis construction, which associates a
locale to a noncommutative C*-algebra (relying on the
constructive Gelfand duality of Banaschewski
and Mulvey). The goal of the seminar is to
motivate and explain this construction, including an introduction to all
concepts used in this abstract. Remarkably, the full title of Boole's
book is: 'An investigation of the laws of thought on which are founded the
mathematical theories of logic and probabilities' - as it is ultimately the
connection between the logical and the probabilistic structure of quantum
mechanics that we are trying to unravel.

**June 10**: Joris Sprunken, IMAPP, Radboud
Universiteit (15:45-17:30 in HG 00.633), Ultrafilters:
wat kun je ermee doen en wat
moet je daarvan vinden?

**Abstract:**
Report on Master Thesis work

**Monday June 23 15:00 HG 00.308 :**
Jacob Vosmaer, ILLC, Universiteit van Amsterdam, *Representation of
topological modal algebras*

**Abstract: ** A
recurring question in topological algebra is: when is every Stone (compact, Hausdorff, zero-dimensional) topological algebra A of
a
certain type representable as a limit of its
finite continuous quotients (i.e. when is A
profinite)? In this talk we show that if A is a modal algebra, the answer to this question
is negative in general. The technical result behind this example is a duality
theorem characterizing compact Hausdorff modal
algebras as the duals of image-finite Kripke frames.

**June 26**: Mirte Dekkers, IMAPP, Radboud
Universiteit (13:30 HG 00.304), *Languages, lattices and duality: an algebraic
perspective on languages*

**Abstract:**
Report on Master Thesis work

FALL 2007

**October 15:**
Frans Keune, IMAPP, Radboud
Universiteit Nijmegen, *The K_2 of a field*

**Abstract: **In
algebraic K-theory one studies the functors K_n from rings to Abelian groups.
In this talk I will, after an introduction to algebraic K-theory, focus on the
K_2 of a field. The main theorem is Matsumoto's theorem which
describes this group in terms of generators and relations. There
are now several proofs of this theorem, some of them
are mine. Finding generators and relations is not the hard problem.
Showing that relations are sufficient is difficult. Some months ago I started
yet another approach which I will describe briefly. There are deep theorems
describing the behaviour of the K_2 under field
extensions. I have some hope that this other approach is of interest in
connection with these theorems. Number fields are of course interesting fields.
I will conclude with some remarks on the K_2 of a number field.

**October 22:**
Stefan Maubach, IMAPP, Radboud Universiteit Nijmegen
(currently visiting Oberwolfach).

**November 12:**
Bernd Souvignier, IMAPP, Radboud Universiteit
Nijmegen, *Some
issues in representation theory*

**Abstract:** In
this talk I will discuss some topics arising from dealing with representations
of finite groups in characteristic 0, in particular over the rational field or
algebraic number fields. The main theme is the decomposition of representations
into their irreducible constituents. Powerful computational methods for this
task are available over finite fields (the MeatAxe),
but they do not carry over easily due to the infiniteness of the rational
field. However, in many cases surprisingly simple
methods turn out to be surprisingly efficient. On the other hand, it is
easy to find examples where basically everythings
fails. These give rise to some interesting open questions. If time permits,
some remarks on the application of rational representations to mathematical
crystallography will be made.

**PS:** There will
be 'beschuit en muisjes'
served

**November 19****:**
Wim Veldman, IMAPP, Radboud
Universiteit Nijmegen, *How to formulate and prove Ramsey's Theorem and Kruskal's
Theorem?*

**Abstract:** The
subject of this talk is intuitionistic infinite combinatorics.

**November 26****:**
Arno van den Essen, IMAPP, Radboud Universiteit Nijmegen, *The power of nilpotent elements.*

**Abstract:** Nilpotent
elements in commutative and non-commutative rings contain the key to the
understanding of several problems. In this talk I discuss several examples from
affine algebraic geometry and dynamical systems to justify this statement.

**December 3: ** Chris
Mulvey, University of Cambridge and University of
Sussex, *Quantales** and Spectra*

**Abstract: ** This
talk is the first in a series of three informal talks. Please read the combined
abstract here.

**December 10:**
Eric Antokoletz (UC Berkeley) *** Cancelled (will be
rescheduled in February),** ***Higher
Semidirect Products of Groups and Algebraic Models of
Homotopy Types*

**Abstract: ** I
will begin with a brief introduction to the role of simplicial
sets and simplicial groups in classical homotopy theory (namely as combinatorial versions of spaces
and their loop spaces). Then I will explain a notion of higher semidirect product of groups that was shown by Carrasco and
Cegarra to play a fundamental role in the structure
of simplicial groups. In particular, I will sketch
their nonabelian version of the well-known Dold-Kan correspondence, giving an equivalence of
categories between simplicial groups and certain
algebraic structures called hypercrossed complexes (HXCs). These HXCs are complicated
to the extent that the enumeration of their axioms is a problem in itself. I
will describe some tools enabling the algorithmic enumeration of these axioms,
thus providing complete descriptions of HXCs. Finally, I will explain how a similar program may be carried
out for symmetric-simplicial groups, such that the
corresponding HXCs are significantly simpler, both in
terms of their data and their axioms.

**December 17:**
No seminar due to event honouring Henk
Barendregt