ALGEBRA AND LOGIC SEMINAR
Centre for Algebra, Logic and Computation
Department of Mathematics and Statistics
University of Saskatchewan, Saskatoon, Canada
Coordinators:
M.R. Bremner, F.- V. Kuhlmann, S. Kuhlmann, M.A. Marshall (Emeritus)
TALKS 2011--2012
Thursday, October 27, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Professor Maria del Rosario Gonzalez-Dorrego, Universidad Autonoma de Madrid
Title: Smooth double subvarieties on singular varieties
Abstract
Tuesday, October 25, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Mark Spivakovsky, Toulouse CNRS
Title: On local uniformization in arbitrary characteristics
This is a continuation of the previous talk.
Thursday, October 20, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Franz-Viktor Kuhlmann
Title: Henselian rationality
This is a continuation of the previous talk.
Tuesday, October 18, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Mark Spivakovsky, Toulouse CNRS
Title: On local uniformization in arbitrary characteristics
The purpose of this series of lectures is to discuss some recent work on local uniformization in arbitrary characteristic. The first lecture is meant to be a leisurely introduction to the problem. In the subsequent lecture(s) we will discuss a selection of topics and results on the subject with emphasis on valuation-theoretic techniques such as key polynomials for simple extensions of valued fields.
Thursday, October 13, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Franz-Viktor Kuhlmann
Title: Henselian rationality
This is a continuation of the previous talk.
Thursday, October 6, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Franz-Viktor Kuhlmann
Title: Henselian rationality
This talk is about the structure of valued algebraic function fields, as they
come up in the problem of local uniformization and the model theory of valued
fields. I will sketch this background and describe the theorems that have been
proved and the theorems we would like to prove. A valued algebraic function
field is henselized rational if its henselization is the henselization of a
rational function field. The first talk will be an introduction, showing the
importance of henselian rationality. Later talks will give proofs for the
henselian rationality of certain function fields in detail and will discuss
possible improvements of these proofs, which would have important applications.
Thursday, September 29, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Josnei Antonio Novacoski
Title: The valuative tree
This is a continuation of the previous talk.
Thursday, September 22, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Josnei Antonio Novacoski
Title: The valuative tree
In this talk we present a few properties of the set of valuations on a two dimensional regular local ring. We present different ways to get a tree structure on such a set.
Thursday, September 15, 2011, 1:00-2:30 PM, McLean Hall 242.1
Speaker: Asim Naseem
Title: Subfields of a valued field
We assume the ramification index and inertia degree of (K'/K,v) to be finite
and give some conditions on K' such that the extension (K'/K,v) is finite
and defectless.
Moreover we give a 1-1 correspondence between the subfields K of k((X)) such
k((X))/K is Galois and the finite subgroups of Aut(k((X))/k), where k is a
field which is algebraically closed to radicals.
TALKS 2010--2011
Wednesday, March 30, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Murray Marshall
Title: Lower bounds for a polynomial in terms of its coefficients
We use an old result proved by A. Hurwitz in 1890 to establish a general
sufficient condition for a polynomial to be a sum of squares. Then it is
explained how this sufficient condition can be combined with Geometric
Programming to allow fast computation of global lower bounds for polynomials.
This is joint work with Mehdi Ghasemi. It is an extension of earlier work
Wednesday, March 23, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Professor Salma Kuhlmann, University of Konstanz
Title: Kaplansky revisited: Truncation closed embeddings of valued fields in
fields of generalized power series
This is a continuation of the previous talk.
Wednesday, March 16, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Professor Salma Kuhlmann, University of Konstanz
Title: Kaplansky revisited: Truncation closed embeddings of valued fields in
fields of generalized power series
We revisit Kaplansky's embedding theorem for valued fields by requiring
that the embedding satisfies the additional property of being "truncation
closed" (i.e. that the image of the embedding of the valued field in the
field of power series be closed under initial segments of series).
We start by motivating why such embeddings are interesting and relate them
to questions arising in the study of Models of Peano Arithmetic.
Finally, We give an intrinsic condition on the valued field (existence of
towers of complements to the fractional ideals of the valuation ring) to
admit a truncation closed embedding.
Joint with A. Fornasiero and F.-V. Kuhlmann
Wednesday, March 9, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Professor Victor Vinnikov, Ben-Gurion University
Title: On the Rank of Noncommutative Polynomials
Wednesday, March 2, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Mohammad Maghaddam
Title: The space of valuations in the case of rank 1 value groups.
This is a continuation of the previous talk.
Wednesday, February 16, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Mohammad Maghaddam
Title: The space of valuations in the case of rank 1 value groups.
This is a continuation of the previous talk.
Wednesday, February 9, 2011, 12:30-2:00 PM, McLean Hall 242.1
Speaker: Mohammad Maghaddam
Title: The space of valuations in the case of rank 1 value groups.
We give a brief introduction to the morphism of blowing ups with smooth centers of the smooth varieties. We introduce the dual graph of the configuration space of the exceptional divisors of a chain of blowing up morphisms. We show how this dual graph can be identified with the space of quasi-monomial valuations. This result shows that this subspace of the valuation space has a natural simplicial-complex structure. Using MacLane's idea of description of valuations in terms of key-polynomials, we give a more algebraic description of the space of valuations. We show how Favre and Jonsson used MacLane's idea to give an explicit description of the valuation space in the case of valuations centered over the ring $C[[x,y]]$; Using this method, one is able to give natural topological, metric, parametric and ultra-metric tree-structure on the valuation space in this case.
Friday, January 28, 2011, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Ron Brown, University of Hawaii
Title: The Krasner constant and main invariant for polynomials over Henselian fields.
Thursday, January 27, 2011, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Franz-Viktor Kuhlmann
Title: Spaces of R-places.
This is a continuation of the colloquium talk of January 21, 2011.
Wednesday, December 15, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Sven Wagner
Title: On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
This is a continuation of the previous talk.
Tuesday, November 30, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Sven Wagner
Title: On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
We will prove that the Pierce-Birkhoff Conjecture (PBC) holds for
non-singular two-dimensional affine real algebraic varieties over real
closed fields, i.e., if W is such a variety, then every piecewise
polynomial function on W can be written as suprema of infima of
polynomial functions on W. More precisely, we will give a proof of the
so-called Connectedness Conjecture for the coordinate rings of such
varieties, which implies the Pierce-Birkhoff Conjecture.
In the first part of the talk, we will recall Madden's abstract
reformulation of the PBC using separating ideals. We then consider the
Connectedness Conjecture (CC) of Lucas, Madden Schaub and Spivakovsky,
and we will prove this conjecture for the one-dimensional case.
In the second part, we will give the definition of a quadratic
transformation of a ring along a valuation. Then we study the effect of
quadratic transformations on separating ideals. At last, we outline the
proof of the CC in the two-dimensional case.
Tuesday, November 16, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Katarzyna Osiak, Silesian University, and Franz-Viktor Kuhlmann
Title: Introduction to spaces of real places
Katarzyna Osiak will continue with her talk. Franz-Viktor Kuhlmann will talk on Brown's Theorem and related open problems.
Tuesday, November 2, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Katarzyna Osiak, Silesian University
Title: Introduction to spaces of real places
This will be a continuation of the talk of the previous week.
Tuesday, October 26, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Katarzyna Osiak, Silesian University
Title: Introduction to spaces of real places
At the last seminar we have shown that the spaces of R-places are
closed under the operation of taking closed subspaces. We will show,
that they are also closed under two other topological constructions:
1) finite disjoint unions
2) direct product with a Boolean space.
In the second part of the talk we will deal with spaces of R-places of
function fields and the problem of metrizability of these spaces.
Tuesday, October 19, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Katarzyna Osiak, Silesian University
Title: Introduction to spaces of real places
We will outline the definition of R-places (real places with
archimedean residue fields) and their connection with orderings. We will
describe the topology on the space of all R-places of a field and its
connection with the Harrison topology of the space of its orderings. We will
then give a survey on the basic properties of spaces of R-places and mention
some open problems.
Tuesday, October 12, 2010, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Sven Wagner
Title: The Elementary Type Conjecture
This will be a continuation of the talk of the previous week.
Tuesday, October 5, 2010, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Sven Wagner
Title: The Elementary Type Conjecture
The Elementary Type Conjecture states that every finite
quadratic form scheme emerges from a finite iteration of direct product
and group extension constructions starting with so-called local types.
In the first part of the talk, we will provide the background in the
theory of quadratic forms in order to talk about the Elementary Type
Conjecture. Then we will present some of the partial answers that are known.
In the second part, we will discuss a special case of this conjecture
and indicate how new (partial) answers can be found.
This is joint work with Karim J. Becher (Konstanz).
Tuesday, September 21, 2010, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Salma Kuhlmann,
Schwerpunkt Reelle Geometrie und Algebra, Universitaet Konstanz Germany.
Title: The General Moment Problem.
Let $\tau$ be a locally convex topology on V, the countable
dimensional polynomial $\mathbb{R}$-Algebra. Let $K$ be a closed subset of
$\mathbb{R}^n$, and $M:=QM(g_1, \cdots g_s)$ a finitely generated module in V.
We investigate the following question: When is the cone $Pos(K)$ (of polynomials
nonnegative on K) included in the closure of M? We give an interpretation of
this inclusion w.r.t. representing continuous linear functionals by measures.
We discuss several examples and application to Berg's et al "exponentially
bounded moment sequences".
This is joint work with M. Ghasemi and E. Samei.
Tuesday, September 14, 2010, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Murray Marshall
Title: Positivstellensate for real function algebras.
This will be a continuation of the talk of the previous week.
Tuesday, September 7, 2010, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Murray Marshall
Title: Positivstellensate for real function algebras.
Suppose $A$ is some algebra of real-valued functions on some set $X$.
Typically $X$ is a topological space of some kind. Often the elements of $A$
are continuous. A quadratic module of $A$ is a subset $Q$ of $A$ satisfying
$Q+Q \subseteq Q$, $f^2Q \subseteq Q$ for all $f \in A$, and $1 \in Q$. We look
at the set $K_{Q,X} = \{ x \in X \mid g(x) \ge 0 for all g\in Q\}$ (the
non-negativity set of $Q$ in $X$). The question is: When can elements of $A$
which are positive or non-negative on $K_{Q,X}$ be described in some simple way
in terms of $Q$? We are mainly interested in the case where $A$ is finitely
generated as an algebra and $Q$ is finitely generated as a quadratic module,
i.e., there exist $g_1,\dots, g_m$ such that every element of $Q$ is
expressible in the form $g = s_0+s_1g_1+\dots +s_mg_m$ where each $s_i$ is a
sum of squares of elements of $A$. We give an answer involving the idea of
'hidden positivity', a term which will be defined in the course of the talk.
This is joint work with Tim Netzer (Leipzig). It is a follow up on recent work by Jean-Bernard Lasserre (Toulouse) and Mihai Putinar (Santa Barbara).
Tuesday, August 31, 2010, 11:00-12:30 AM, McLean Hall 242.1
Speaker: Pawel Gladki, Silesian University
Title: Symbol length and stability index.
Let $F$ be a field of characteristic different from $2$, let $WF$
denote the Witt ring of quadratic forms over $F$ and $IF$ its
fundamental ideal. The maximal number of additive generators of
$I^nF/I^{n+1}F$ (or $\infty$, if no such maximum exists), is called the
{\em $n$-symbol length of $F$} and denoted by $\lambda_n(F)$. While
$\lambda_0(F)=\lambda_1(F)=1$, the $2$-symbol length $\lambda_2(F)$ is
of particular interest and was already studied by Bruno Kahn. On the
other hand, if $F$ is a Pythagorean field, its {\em stability index} is
an invariant that measures the complexity of the space of orderings of
the field $F$. We show that a Pythagorean field (more generally, a
reduced abstract Witt ring) has finite stability index if and only if it
has finite 2-symbol length. We give explicit bounds for the two
invariants in terms of one another. To approach the question whether
those bounds are optimal we consider some examples of Pythagorean fields.
This is a joint work with Karim Becher.
Thursday, August 5, 2010, 10:00-11:30 AM, McLean Hall 242.1
Speaker: Pawel Gladki, Silesian University
Title: Orderings of higher level on multifields and multirings.
Fourth talk in the series.
Tuesday, August 3, 2010, 10:00-11:30 AM, McLean Hall 242.1
Speaker: Pawel Gladki, Silesian University
Title: Orderings of higher level on multifields and multirings.
Third talk in the series.
Thursday, July 29, 2010, 10:00-11:30 AM, McLean Hall 242.1
Speaker: Pawel Gladki, Silesian University
Title: Orderings of higher level on multifields and multirings.
Second talk in the series.
Tuesday, July 27, 2010, 10:00-11:30 AM, McLean Hall 242.1
Speaker: Pawel Gladki, Silesian University
Title: Orderings of higher level on multifields and multirings.
Abstract: In this talk we will outline the theory of orderings of higher level on multirings and multifields. Basic notions in the theory of algebraic structures with multivalued addition will be also discussed, as well as necessary background in the algebraic theory of quadratic forms, spaces of orderings, and abstract real spectra.
This will be the first in a series of talks on this subject.
TALKS 2009--2010
Thursday, March 25, 2010, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Mehdi Ghasemi
Title: Closure of SOS cone in norm-p topologies
Abstract: First I will introduce the norm-p topology and then the weighted norm-p topology on the ring of real polynomials of n variables. For the norm-1 topology, the closure of SOS cone is determined by Lasserre and Netzer, I will use their result to find the closure of SOS cone in weighted norm-p topologies for p >= 1. Finally, by applying these results, I will determine the closure of SOS cone in the coefficient-wise convergent topology.
Tuesday, November 24, 2009, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Katarzyna Osiak, Silesian University
Title: On the realization of topological spaces as spaces of R-places.
This will be a continuation of the seminar presentation begun on November 19.
Thursday, November 19, 2009, 2:30-4:00 PM, McLean Hall 201
Speaker: Katarzyna Osiak, Silesian University
Title: On the realization of topological spaces as spaces of R-places.
Abstract: We prove that the class of topological spaces which are
realizable as spaces of R-places of fields is closed under finite disjoint
unions, closed subsets, and products with Boolean spaces.
This is joint work with Ido Efrat.
Thursday, November 12, 2009, 2:30-4:00 PM, McLean Hall 201
Speaker: Franz-Viktor Kuhlmann
Title: Metrizability of the space of $\bb R$-places of function fields of transcendence degree 1 over real closed fields
Abstract: We deal with the problem of metrizability of the space of $\bb R$-places of a
function field $K$ of transcendence degree 1 over a real closed field $R$. For
the rational function field $K = R(X)$ we give a necessary and sufficient
condition for the metrizability of $M(K)$, namely, that $R$ contains a
countable dense subfield. Moreover, this condition is necessary for the
metrizability of the space of $\bb R$-places of any function field over $R$.
This is joint work with Michal Machura and Kasia Osiak.
Thursday, October 29, 2009, 2:30-4:00 PM, McLean Hall 201
Speaker: Franz-Viktor Kuhlmann
Title: the defect
This will be a continuation of the seminar presentation begun on October 27.
Tuesday, October 27, 2009, 2:30-4:00 PM, McLean Hall 242.2
Speaker: Franz-Viktor Kuhlmann
Title: The defect
Abstract: I will report on a survey paper which I wrote for a Springer volume on recent developments in commutative algebra. In that paper, I give an introduction to the valuation theoretical phenomenon of ``defect''. I describe the role it plays in deep open problems in positive characteristic: local uniformization, the local form of resolution of singularities, and the model theory of valued fields. I give several examples of extensions with non-trivial defect and describe a way to classify them. Further, I give an overview of various results about the defect that help to tame it, their applications, and open problems.
Thursday, October 22, 2009, 2:30-4:00 PM, McLean Hall 201
Speaker: Josnei Antonio Novacoski, Sao Paulo
Title: Classification of algebraic function fields with divisor class number two
Abstract
Tuesday September 15, 2009, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Franz-Viktor Kuhlmann
Title: The story of extremality
Abstract: When does a valued field (K,v) have the property that for every n and every
polynomial f in n variables the set of all values vf(a_1,...,a_n) has a maximum
(where infinity is allowed), when the a_i run through... yes, through what? I
will tell the story of how the right definition for what we call an extremal
valued field was found, after a mathematical error at first led us the wrong
way.
The perfect extremal fields of equal characteristic are now fully characterized.
For mixed characteristic and non-perfect fields no full characterization is
known yet, but necessary conditions are known. For the case of non-perfect
fields I will give an example to show what can go wrong there.
Interestingly, the theory of extremal fields has turned out to have connections
with the theory of large fields. Model theory is used in some of the proofs.
This is joint work with Salih Azgin and Florian Pop.
Thursday September 10, 2009, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Salma Kuhlmann, University of Konstanz
Title: Hardy type derivations on generalized series fields
Abstract: We consider the valued field $\mathds{K}:=\mathbb{R}((\Gamma))$ of
generalized series (with real coefficients and monomials in a totally ordered
multiplicative group $\Gamma\>$). We investigate how to endow $\mathds{K}$ with
a series derivation, that is a derivation that satisfies some natural properties
such as commuting with infinite sums (strong linearity) and (an infinite version
of) Leibniz rule. We characterize when such a derivation is of Hardy type, that
is, when it behaves like differentiation of germs of real valued functions in a
Hardy field. We provide a necessary and sufficient condition for a series
derivation of Hardy type to be surjective.
Joint work with Micka\"el Matusinski.
Tuesday September 8, 2009, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Mehdi Ghasemi
Title: Lower bound for a polynomial in terms of its coefficients
This will be a continuation of the seminar presentation begun on September 3.
Thursday September 3, 2009, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Mehdi Ghasemi
Title: Lower bound for a polynomial in terms of its coefficients
Abstract: I will start with some recently found sufficient conditions by Lasserre and also by Fidalgo and Kovacec on the coefficients of a real polynomial of even degree and n variables to be a sum of squares of polynomials. Then I will use these results to determine a lower bound for a given polynomial. Finally, I will make a comparison among three different resulting lower bounds in terms of coefficients and number of monomials appearing on a polynomial.
Thursday August 27, 2009, 2:30-4:00 PM, McLean Hall 242.1
Speaker: Victor Vinnikov, Ben Gurion University
Title: Noncommutative positive kernels and their matrix evaluations
Abstract: I will start by stating and proving a version of noncommutative positivstellensatz for so called hereditary noncommutative polynomials and nilpotent matrix evaluations. Nilpotent substitutions allow us to consider formal power series as well as polynomials, and the result fits naturally into the framework of positive noncommutative kernels. If time permits I will also introduce, and speculate about, the associated reproducing kernel Hilbert spaces.
This is a joint work with Dmitry Kalyuzhnyi-Verbovetskii.
Thursday June 4, 2009, 2:30-3:30 PM, McLean Hall 242.1
Speaker: Andreas Fischer, Fields Institute
Title: Extension theorems of Kirszbraun and Tietze in definably
complete structures.
Abstract: Let R denote the reals and let A be a closed subset of R^n. The seminal
Kirszbraun Theorem states that an L-Lipschitz function f:A-->R^k (that is,
\norm{f(x)-f(y)}\leq L\norm{x-y} for all x,y\in A) can be exended to an
L-Lipschitz function on R^n. The classical proofs use Zorn's lemma. However,
using monotone set-valued operators, one obtains a constructive approach to the
Kirszbraun theorem, which also works over definably complete expansions of real
closed fields. In these structures, also the Tietze extension theorem and its
variants hold true.
PAST TALKS