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 Bröker 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 hyperreële lichaam.

De hyperreële getallen zijn een uitbreiding van de reële getallen zodanig dat deze getallen een totaal geordend lichaam vormen waarin oneindig kleine en oneindig grote getallen bevinden. Dit hyperreële lichaam is elementair equivalent met het reële lichaam. Hierdoor is het mogelijk analyse te doen op de hyperreële getallen in plaats van de reële getallen. Ik zal de constructie van dit hyperreële lichaam behandelen. Deze constructie wordt ook wel de ultraproduct constructie van de reële 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 d’Algebra, 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 K₁ of a commutative domain R is the direct sum of its unit group and a group SK₁(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 SK₁ 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 K₂ 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 K₂ 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 ‘extracted’ 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 Lübeck, 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 q, as opposed to a field just defined up to isomorphism. The definition is complicated enough that it occupies most of the lecture. No easier definition is known that has the attractive algorithmic properties needed in the previous talk.

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 László Lovász 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, Lovász 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: Gyöngyvér 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. Gyöngyvér 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.

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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