Dedicated to

Murray Marshall

**Organized by:**

Franz-Viktor Kuhlmann
(Institute of Mathematics, University of Silesia at Katowice, Poland),

Katarzyna Kuhlmann
(Institute of Mathematics, University of Silesia at Katowice, Poland),

Salma Kuhlmann
(University of Konstanz, Germany),

Victor Vinnikov
(Ben Gurion University of the Negev, Israel)

**Local organizer:**

Mehdi Ghasemi (University of Saskatchewan)

**This event was generously supported by the**

Department of Mathematics and Statistics

University of Saskatchewan

106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

Phone: (306) 966-6081 - Fax: (306) 966-6086

**Dates:**

*Tuesday and Wednesday, August 30/31, 2016*

The Nineteenth Colloquiumfest was devoted to the memory of Murray Marshall, who died on May 1, 2015. Most of the talks were in some way connected to the work of Murray. But we saw this meeting also as a gathering of people who were connected to Murray as friends, collaborators or participants of the algebra and the FPT seminar in Saskatoon.

9:45-10:20 am:

TEA and COFFEE

10:30-11:20 am:

**Danielle Gondard**
(Paris, France):

*Marshall's Problem & Some Further Developments
*

11:30 am - 12:20 pm:

**Vicki Powers**
(Emory University, USA):

*The Moment Problem and Representations of Positive Polynomials on Noncompact Semialgebraic Sets
*

12:30-2:00 pm:

**LUNCH**

2:00 - 2:50 pm:

**Mihai Popa**
(San Antonio, USA):

*On the transpose of certain classes of random matrices
*

2:55 - 3:45 pm:

**Douglas Farenick**
(University of Regina):

*Operator systems from discrete groups
*

3:50-4:20 pm:

TEA and COFFEE

4:30 - 5:20 pm:

**Mehdi Ghasemi**
(University of Saskatchewan):

*Truncated moment problem from a topological point of view
*

5:25 - 5:55 pm:

**Ray Spiteri**
(Department of Computer Science, University of Saskatchewan):

*Designing error control circuits for 4-qubit quantum computing
*

7:00 pm:

**SOCIAL DINNER**

9:45 - 10:35 am:

**Daniel Plaumann**
(Dortmund, Germany):

*Positive Polynomials on Real Projective Varieties
*

10:45-11:20 am:

TEA and COFFEE

11:30 am - 12:20 pm:

**Tim Netzer**
(Innsbruck, Austria):

*Free semialgebraic geometry and convexity
*

12:30-2:00 pm:

**LUNCH**

2:00 - 2:50 pm:

**Maria Infusino**
(Konstanz, Germany):

*Moment problems on symmetric algebras of locally convex real spaces
*

2:55 - 3:45 pm:

**Pawel Gladki**
(University of Silesia at Katowice, Poland):

*Homological aspects of multivalued addition
*

3:50-4:20 pm:

TEA and COFFEE

4:30 - 5:20 pm:

**Christopher Dutchyn**
(Department of Computer Science, University of Saskatchewan):

*Modern Software-supported Mathematics
*

5:25 - 5:55 pm:

**Victor Vinnikov**
(Beer Sheva, Israel):

*Sums of hermitian squares
*

6:00 - 6:30 pm:

**Franz-Viktor Kuhlmann**
(University of Silesia at Katowice, Poland):

*Coincidence theorems for ultrametric and ball spaces
*

**Christopher Dutchyn**: *Modern Software-supported Mathematics*

Abstract: Computer-aided mathematics is becoming a reality: not just computational statistics or other applied areas, but highly abstract areas are becoming accessible. Computerized proof assistants can help organize the bureaucracy, verify completeness and correctness of mathematical proofs, and generally reduce the ancillary burdens of writing proofs. Using Samer Assaf's recent PhD dissertation as a source of examples, we demonstrate how proof assistants can aid in the practice of mathematics. We conclude with a discussion of deficiencies and possible improvements for proof-assistants.

**Douglas Farenick**: *Operator systems from discrete groups*

Abstract:
In this lecture I will discuss some results obtained in collaboration with A. Kavruk, V. Paulsen, and I.G. Todorov on certain operator subsystems of C*-algebras of discrete groups. Of special interest and importance are operator systems on free groups with finitely many generators, and I will discuss this case in particular, especially with regards to their tensor products and their use in quantum information theory.

**Mehdi Ghasemi**: *Truncated moment problem from a topological point of view*

Abstract:
Let *R[X] _{m}* be the vector space of polynomials of degree at most m in n variables,

**Pawel Gladki**: *Homological aspects of multivalued addition*

Abstract:
In this talk we will introduce the notion of the category of accessible posets, as well as provide numerous examples of accessible posets. In particular, we will show how algebras with multivalued addition such as hypergroups, hyperrings or hyperfields can be regarded as accessible posets. Next, by means of localization and order collapse, we will demonstrate a "canonical" way of assigning to each hypergroup (hyperring, hyperfield) a group (ring, field, respectively) with a certain ordering relation. As (abstract) Witt rings are an important class of examples of hyperfields, these methods provide new ideas to study some classical problems in the algebraic theory of quadratic forms. Some applications, in particular to Witt equivalence, will be also discussed. This is joint work with Krzysztof Worytkiewicz.

**Danielle Gondard**: *Marshall's Problem & Some Further Developments*

Abstract:
The talk deals with Murray Marshall's work on abstract spaces of orderings, and especially on the still open problem of the realizability of abstract spaces of orderings as spaces of orderings of fields.
After the needed background, we list known results and present some theorems obtained in two papers written with Murray.
We also make some steps towards other possible approaches.

**Maria Infusino**:
*Moment problems on symmetric algebras of locally convex real spaces*

Abstract:
This talk aims to introduce some infinite dimensional versions of the classical full moment problem which naturally arise in several applied fields. The general theoretical question addressed is whether a linear functional L on the symmetric algebra S(V) of a locally convex real vector space V (possibly infinite dimensional) can be represented as an integral w.r.t. a non-negative Radon measure supported on a fixed subset of the algebraic dual V* of V. I present a recent joint work with M. Ghasemi, S. Kuhlmann and M. Marshall, where we get representations of continuous positive linear functionals L on S(V) as integrals w.r.t. uniquely determined Radon measures supported in special sorts of closed balls in the topological dual space V' of V. A better characterization of the support is obtained when L is positive on a 2d-power module of S(V). I compare these results with the corresponding ones for the full moment problem on locally convex nuclear spaces, pointing out the crucial roles played by the continuity and the quasi-analyticity assumptions on L in localizing the support of the representing measure and in the question of determinacy.

**Franz-Viktor Kuhlmann**: *Coincidence theorems for ultrametric and ball spaces*

Abstract:
We improve a coincidence theorem for ultrametric spaces due to Sibylla Priess-Crampe and Paulo Ribenboim. Then we derive a very general and flexible coincidence theorem for ball spaces. Ball spaces are a very simple structure that still allows to encode spherical completeness, the completeness property that makes coincidence theorems work in various settings. We discuss possible applications of this general theorem.

This is joint work with Katarzyna Kuhlmann and Fatma Sonaallah.

**Tim Netzer**: *Free semialgebraic geometry and convexity*

Abstract: Free semialgebraic geometry investigates sets of matrices of all sizes, defined by non-commutative polynomial inequalities. A suitable notion of convexity also exists in this context. Compared to the commutative setup, some results and methods are much stronger in the free case, but some crucial results seem to fail completely. I will report on recent progress in this area, in particular on free quantifier elimination and properties of free convex hulls.

**Daniel Plaumann**: *Positive Polynomials on Real Projective Varieties*

Abstract:
Hilbert classified all pairs (n,d) such that every nonnegative homogeneous polynomial in n variables of degree 2d is a sum of squares. More recently, this classification has been extended by Blekherman, Smith, and Velasco to nonnegative polynomials on projective varieties, leading to a generalisation, a unified proof and deeper geometric understanding of Hilbert's result.

In this talk, a different approach to these results will be presented, originating from the case of ternary quartics (n,2d)=(3,4). This allows for new results concerning the number of squares and the number of different representations. (Based on joint work with Greg Blekherman, Lynn Chua, Rainer Sinn, and Cynthia Vinzant).

**Mihai Popa**: *On the transpose of certain classes of random matrices*

Abstract: The transpose is one of the most natural operation with matrices. Yet, the relation between a given collection of matrices (with entries in some algebra) and the collection of its transposes is not trivial nor straightforward. The lecture will present some recent results concerning asymptotic non-commutative independence between certain classes of random matrices and their transposes.

**Vicki Powers**: *The Moment Problem and Representations of Positive Polynomials on Noncompact
Semialgebraic Sets*

Abstract: The multi-dimensional moment problem asks for a characterization of the moment multi-sequences of positive Borel measures with support in a given closed subset K of R^n. This problem is directly related to questions about representations of polynomials nonnegative on K. In this expository talk, we discuss results in the case where K is a non-compact semialgebraic set with particular emphasis on the work of Murray Marshall.

**Ray Spiteri**: *Designing error control circuits for 4-qubit quantum computing*

Abstract:
Quantum computing offers the promise of a spectral leap in performance compared to transistor-based computers. However, with this promise comes the obligation of increased error checking and correction of quantum bits (qubits) in order to guarantee sufficient robustness to fault tolerance. In this talk, I describe the journey toward producing a design for a circuit that can guarantee a sufficiently high fidelity of the error correction component for four-qubit computations. This will allow experimentalists to build such a device and tackle even harder problems than the current three-qubit systems.

**Vicctor Vinnikov**: *Sums of hermitian squares*

Abstract

*Last update: September 7, 2016
--------- created and maintained by Franz-Viktor Kuhlmann*