# Fundamental interactions between algebra and computation

The last decades have seen a development towards deeper understanding of the processes of computations. Fundamental algebraic structures governing these processes have been uncovered. The meeting informs on approaches to this from different stands.

## Hovedinnhold

- Talks Thursday in Auditorium 5, 3'rd floor, Realfagsbygget
- Talks Friday in Auditorium 4, 4'th floor, Realfagsbygget

## Speakers, double talk:

Kurusch Ebrahimi-Fard (NTNU Trondheim)

Hans Munthe-Kaas (Universitetet i Bergen)

Cordian Riener (UiT Norges arktiske universitet)

They give double lectures, one on Thursday morning, and one on Friday morning. The lecture on Thursday morning will be survey talks about their scientific work:

- What are the notable structures?
- What are the significant and interesting problems?
- What results/gains in insights have one reached in this area in the last one or two decades?
- What does one see as interesting further directions?

The lecture on Friday is about more recent research.

## Speakers, single talk:

Vladimir Dotsenko (Université de Strasbourg)

Morten Brun (UiB)

Gunnar Fløystad (UiB)

Trygve Johnsen (UiT, Norges arktiske universitet)

Markus Szymik (NTNU)

## Participants:

**Physical:**

Diego Caudillo(NTNU)

Kurusch Ebrahimi-Fard(NTNU)

Gunnar Fløystad(UiB)

Arne Lien(UiT)

Helger Lipmaa(Simula)

Hans Munthe-Kaas(UiB)

Kataryna Pavlyk(Simula)

Håvard Raddum(Simula)

Cordian Riener(UiB)

Belen Garcia Pascual(UiB)

Lars Salbu(UiB)

Jonathan Stava(UiB)

Hugues Verdure(UiT)

Øyvind Ytrehus(Simula)

**Likely non-physical:**

Morten Brun

Vladimir Dotsenko

Fedor Fomin

Lars Jaffke

Trygve Johnsen

Helge Maakestad

Rakhi Pratihar

Kristian Ranestad

Sondre Rønjom

Markus Szymik

Arvid Siqveland

Arne B. Sletsjøe

Yannic Vargas

Jon Eivind Vatne

Morten Øygarden

## Schedule Thursday 22'nd April:

**Overview talks 09.00-12.00:**

09.00-9.45 : Cordian Riener, Symmetries in algorithmic questions in real algebraic geometry

10.00-10.50: Kurusch Ebrahimi-Fard, The Magnus expansion, rooted trees and cumulants

11.15-12.00: Hans Munthe-Kaas, Structure preserving integration of differential equations; algebra, geometry and computation.

12-13.30: Lunch

13.30-14.15: Morten Brun, Clustering with a trans disciplinary perspective

14.30-15.15: Markus Szymik, Computing homology for algebraic theories

15.45-16.30: Vladimir Dotsenko, Functorial Poincaré-Birkhoff-Witt theorems

## Schedule Friday 23'rd April:

09.00-9.45 : Cordian Riener: Symmetries in algebraic varieties and semi-algebraic sets.

10.00-10.50: Kurusch Ebrahimi-Fard, The Magnus expansion, rooted trees and cumulants

11.15-12.00: Hans Munthe-Kaas, Structure preserving integration of differential equations: aromatic B-series.

12-13.15: Lunch

13.15-14.00: Trygve Johnsen, Codes from symmetric matrices

14.30-15.20: Gunnar Fløystad, Profunctors between posets: Applications in algebra and combinatorics

## Abstracts:

Morten Brun: Clustering with a trans disciplinary perspective

Abstract: From topological data analysis I give a motivation for cluster analysis and explain challenges, practical and theoretical. From this view I advocate breaking the rules and redefining the objective of this academic discipline. I tour from explicit examples to theoretical problems.

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Vladimir Dotsenko: Functorial Poincaré-Birkhoff-Witt theorems

Abstract: I explain how effective algorithmic methods for operads lead to proofs of functorial theorems of the Poincaré-Birkhoff-Witt type for many different types of universal enveloping algebras. This relies on a general result I proved in a joint work with Pedro Tamaroff.

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Kurusch Ebrahimi-Fard: The Magnus expansion, rooted trees and cumulants

Abstract: It is common to write the solution of a matrix-valued linear initial value problem in terms of the time-ordered exponential (aka Dyson series). In his seminal 1954 paper, Wilhelm Magnus proposed an intricate differential equation for the computation of the logarithm of the time-ordered exponential. Since then, the Magnus expansion has become a household name in applied mathematics, control theory, physics and chemistry.

It is closely related to the Baker-Campbell-Hausdorff expansion. In fact, the Chen-Mielnik-Plebanski-Strichartz formula, a variant of the Magnus expansion, is also known as the continuous Baker-Campbell-Hausdorff formula. More recently, the mathematical fine structure of the Magnus expansion has been unfolded using more advanced algebraic tools such as pre- and post-Lie products on non-planar, respectively planar rooted trees.

Part I: Basic aspects of the Magnus expansion and its algebraic properties.

Part II: We show the relations between the Magnus expansion and cumulant expansions in commutative as well as non-commutative probability theory.

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Gunnar Fløystad: Profunctors between posets: Applications in combinatorics and algebra

Abstract: We give an introduction to profunctors between posets P +>Q, a more general notion than order preserving maps. Defining the graph and ascent of such profunctors, we use these to construct large classes of simplicial complexes, in particular lots of triangulated spheres. We also exemplify with profunctors between natural numbers N +> N.

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Trygve Johnsen: Codes from symmetric matrices

Abstract: We consider a vector space of linear homogeneous polynomials in m^{2} variables with coefficients in the finite field F_{q}. We evaluate these polynomials at fixed sets S of symmetric matrices [a_{i,j}]. The image consists of (code) words of length |S|.

We look in particular at the codes C_{t} obtained by letting: S=S_{t }be all symmetric matrices of rank at most t, for all 1 <=t<=m. Alternatively, we pick only one non-zero such matrix for each multiplicative equivalence class giving rise to a point in P(S_{t}), and take the code (vector space) generated by the words thus obtained. In this way we obtain a smaller “sister code” C_{t}’ of C_{t}.

We want to say as much as possible about the weight distribution of such codes. In particular we want to determine their minimum distances. This is on-going joint work with Prasant Singh, UiT, and is a natural continuation, or analogue of work by Singh and Peter Beelen, DTU, on corresponding codes from skew-symmetric matrices.

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Hans Munthe-Kaas: Structure preserving integration of differential equations; algebra, geometry and computation.

Abstract: In numerical discretisation of dynamical systems (time evolution of differential equations), the problem of structure preservation has been central over the last decades. How construct discrete approximations of flow maps, which preserve conservation laws and geometric constraints? Are there ’no-go’ theorems? Basic tools are B-series and related algebraic structures: Formal power series in special algebras such as preLie and postLie algebras.

Part I: A general overview of the field, status and open questions.

Part II: Aromatic B-series arise in the study of volume preserving discretisations, and in the characterisation of equivariance properties of approximate flow maps. Joint works with Olivier Verdier, Robert McLachlan, Klas Modinand with Gunnar Fløystad and Dominique Manchon.

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Coridan Riener: I. Symmetries in algorithmic questions in real algebraic geometry

Abstract: Symmetry is a form of algebraic structure, omnipresent in algebra and geometry. In algorithmic tasks the presence of symmetry may reduce algorithmic complexity of problems. We highlight several instances of such, in particular from optimization and algorithmic real algebraic geometry.

II. Symmetries in algebraic varieties and semi-algebraic sets.

Abstract: We focus on the symmetric group $S_n$ acting on algebraic varieties and semi-algebraic sets. We present generalisations of works of Arnold on so called Vandermonde varieties and their implications for the cohomology of symmetric semi-algebraic sets. We consider the infinite symmetric group acting on infinite polynomial rings. We present results and several conjectures and open problems, which our group in Tromsø is working on.

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Markus Szymik: Computing homology for algebraic theories

Abstract: We discuss problems that we encounter when trying to compute the homology of algebraic structures. Crucial examples are the homology of groups, the homology of Lie algebras, and the homology of associative algebras. Unfortunately, these examples are misleading, in general, as the homology of commutative algebras shows. I give a gentle introduction, using bits and pieces from recent work as illustrations.