Professor Markus Szymik

School of Mathematics and Statistics

The University of Sheffield

Hicks Building

Hounsfield Road

Sheffield S3 7RH

United Kingdom

m.szymik@sheffield.ac.uk

In this section you can find out about the papers (and the book) that I have written (with the
indicated co-authors).

The publications may differ from the preprint versions; please refer to the former in the
interest of accuracy.

The manuscripts are often under revision; some links may sometimes be broken.

**Generalizations of Loday's assembly maps for Lawvere's algebraic
theories**

(with A.M. Bohmann)

J. Inst. Math. Jussieu (to appear)

Arxiv

**Groups, conjugation and powers**

(with T. Vik)

Arxiv

**Preludes to the Eilenberg-Moore and the Leray-Serre spectral
sequences**

(with F. Neumann)

Arxiv

**The homotopy types of free racks and quandles**

(with T. Lawson)

Arxiv

**Boolean algebras, Morita invariance, and the algebraic K-theory of
Lawvere theories**

(with A.M. Bohmann)

Arxiv

**The stable homotopy theory of vortices on Riemann surfaces**

Arxiv

**A new Kenzo module for computing the Eilenberg-Moore spectral
sequence**

(with A. Romero and J. Rubio and F. Sergeraert)

ACM Commun. Comput. Algebra 54 (2020) 57-60

Link

**Grundkurs Topologie** (with G. Laures)

Spektrum Akademischer Verlag, Heidelberg, 2009

2., überarbeitete Auflage, Springer Spektrum, Heidelberg, 2015

Link

**Equivariant Topology and Derived Algebra**

(edited with S. Balchin, D. Barnes, and M. Kędziorek)

Cambridge University Press, 2021

Link

**Topological Data Analysis**

(edited with N. Baas, G. Carlsson, G. Quick, and M. Thaule)

The Abel symposium 2018

Springer, 2020

Link

In this section you can learn about the presentations that I give and the conferences that I organize.

Category Theory at Work in Computational Mathematics and Theoretical Informatics

Bergen, June 26-30, 2023

Equivariant Topology and Derived Algebra

NTNU Trondheim, Jul 29-Aug 2, 2019

Knots and Braids in Norway

NTNU Trondheim, May 13-17, 2019

Abelsymposium 2018: Topological Data Analysis

Geiranger, Jun 4-8, 2018

Topology and Applications

NTNU Trondheim, Nov 30-Dec 2, 2016

Aberdeen, Nov 30, 2022

Algebra Seminar

Belfast, Nov 7, 2022

MSRC Colloquium

Tromsø, Sep 2, 2022

Nasjonalt Matematikermøte

Tromsø, Aug 30, 2022

Lie-Størmer Days

Warwick, May 10, 2022

Warwick Algebraic Topology Seminar

Ankara (via Zoom), Nov 15, 2021

Bilkent University Topology Seminar

Leicester (via Zoom), Jun 3, 2021

LAGOON - Leicester Algebra & Geometry Open Online Seminar

Bergen (via Zoom), Apr 22, 2021

Fundamental Interactions Between Algebra and Computation

Berlin (via Zoom), Apr 20, 2021

Forschungsseminar Geometrie und Topologie

Regensburg (via Zoom), Apr 15, 2021

Seminar in Homotopy Theory and Related Areas

Oberwolfach (via Zoom), Apr 7, 2021

Arbeitsgemeinschaft

Marburg (via Zoom), Mar 24, 2021

Algebraic Topology for Amateurs

University of Tromsø (via Zoom), Jan 15, 2021

Mørketidens Mattemøte

Bochum (via Zoom), Jul 7, 2020

Oberseminar Topologie

Queen's University Belfast (via Zoom), June 15, 2020

Online Algebraic Topology Seminar

Leicester, Nov 19-22, 2019

Topological Methods in Group Representation Theory

Caen, Oct 21-23, 2019

Journée Normandes de Topologie

Regensburg, Jul 19, 2019

SFB-Seminar

The Fields Institute, Toronto, Jul 8-12, 2019

Four Manifolds: Confluence of High and Low Dimensions

Leeds, Jul 1-4, 2019

Loops in Leeds

Copenhagen, Jun 24-28, 2019

10 years with SYM in Copenhagen

Oxford, Jun 20, 2019

Topology Seminar

Bielefeld, May 23, 2019

Mathematisches Kolloquium

Bochum, May 10-11, 2019

NRW Topology Meeting

Stockholm, Mar 19, 2019

Topology Seminar

Leeds, Dec 12, 2018

Algebra Seminar

Aberdeen, Nov 21, 2018

Topology Seminar

Cambridge, Nov 20, 2018

HHH Seminar

Sheffield, Nov 1, 2018

Topology Seminar

Montpellier, Oct 23-26, 2018

Rencontre 2018 du GdR de Topologie Algébrique

Lille, Oct 5, 2018

Topology Seminar

Strasbourg, Apr 17, 2018

Séminaire Algèbre et Topologie

Durham, Dec 14, 2017

Geometry and Topology Seminar

Cambridge, Nov 22, 2017

Differential Geometry and Topology Seminar

Grenoble, Oct 9, 2017

Séminaire Algèbre et Géométries

Leicester, Sep 8, 2017

British Topology Meeting

Leicester, Sep 5, 2017

Pure Maths Colloquium

National University of Ireland, Galway, May 18-20, 2017

Groups in Galway 2017

Københavns Universitet, March 20, 2017

Topology Seimnar

Johns Hopkins University, Nov 7, 2016

Topology Seminar

University of Rochester, Nov 4, 2016

Topology Seminar

University of British Columbia, Oct 26, 2016

Topology Seminar

UCLA, Oct 12, 2016

Algebraic Topology Seminar

Indiana University-Purdue University Indianapolis, Sep 20, 2016

Modern Analysis and Geometry Seminar

Wayne State University, Sep 14, 2016

Topology Seminar

Fields Institute, Jun 13-17, 2016

Group Actions Workshop

Trondheim, Sep 3, 2019

Birralee International School Trondheim
*On Learning Mathematics*

Trondheim, Mar 18, 2018

Study Group at Galleri KiT
*Badiou - Inaesthetics - Topology*

Here is an account of my main teaching. I often teach an additional reading course each term. Please feel free to contact me when you are interested in taking such a course.

MAS400 Project Presentation in Mathematics and Statistics

MAS435 Algebraic Topology

Spring 2021: Calculus 3 (TMA4115)

Autumn 2020: Introduction to Lie Theory (MA3407)

Spring 2020: Algebraic Topology 2 (MA3408)

Autumn 2019: Advanced Topics in Topology (MA8403)

Spring 2019: Introduction to Lie Theory (MA3407)

Spring 2018: Calculus 3 (TMA4115)

Autumn 2017: Algebraic Topology (MA3403)

Spring 2017: Introduction to Topology (TMA4190)

Spring 2016: General Topology (MA3002)

Autumn 2015: Calculus 3 (TMA4110)

Spring 2015: General Topology (MA3002)

Autumn 2014: Calculus 3 (TMA4110)

In this section you can learn about the people who I currently mentor.

Abigail Linton

postdoc, since 2020

Erlend Bergtun

PhD, since 2020

William Hornslien

PhD, since 2020

Tam-lin Moonstone

PhD, since 2022

Jake Saunders

PhD, since 2022

Taizhen Shi

Master, Sheffield, 2023 (expected)

Luca dal Molin

Master, visiting from the University of Pavia, 2023 (expected)

Rachael Boyd

ERCIM postdoctoral fellow

Paul André Dillon Trygsland

PhD, Trondheim, 2022

Truls Bakkejord Ræder

PhD, Trondheim, 2017

Jinsong Wang

Master, Sheffield, 2022

Gustav Kjærbye Bagger

Master, Trondheim, 2022

Emil August Hovd Olaisen

Master, Trondheim, 2022

Odin Hoff Gardå

Master, Trondheim, 2022

Sverre Myhre Lien

Master, Trondheim, 2021

Tobias Grøsfjeld

Master, Trondheim, 2017

Reidun Persdatter Ødegaard

Master, Trondheim, 2015

Jens Jakob Kjær

Master Thesis, København, 2013

Rasmus Nørtoft Johansen

Master Project, København, 2013

Robert Wilms

Master Thesis, Bochum, 2011

Norman Schumann

Diplomarbeit, Bochum, 2009

Anders Krøger Evensen

Bachelor, Trondheim, 2021

Tallak Manum

Bachelor, Trondheim, 2021

Torstein Vik

Bachelor, Trondheim, 2021

Kamilla Weka

Bachelor, Trondheim, 2019

Christopher Schwartz Kvarme

Bachelor, Trondheim, 2017

Adrián Javaloy Bornás

Bachelor, Trondheim, 2017

Peter Marius Flydal

Bachelor, Trondheim, 2017

Eivind Otto Hjelle

Bachelor, Trondheim, 2016

Andreas Hamre

Bachelor, Trondheim, 2015

Bjørnar Gullikstad Hem

StudForsk, Trondheim, 202?

Lise Millerjord

StudForsk, Trondheim, 201?

Erlend Børve

StudForsk, Trondheim, 2016

Johanne Haugland

StudForsk, Trondheim, 2016

If you might be interested in working with me, why not drop by my office or email me? Topics can range from topology, geometry, and algebra to applications in biology and network science. Or, you can find ways to experiment with homology on your computer. Below you will find samples of ideas for future projects that I am willing to supervise, and the teaching section above contains a list of some of my previous projects. I am happy if you contact me to see if I can offer something else that matches your interests.

Topological structures like knots, braids, or Möbius strips help engineers to construct more efficient conveyor belts, computer scientists to plan the motion of robots and to construct quantum computers, and chemists and biologists to understand the structure of large molecules and genes.

Lorenz has described a three-dimensional system of non-linear ordinary differential equations that arises in many contexts such as a simplified mathematical model for atmospheric convection, lasers, dynamos, electric circuits, chemical reactions, and forward osmosis. And for some choices of parameters, it has knotted periodic orbits.

The mathematical theory of braids, knots, and links is attractive because the objects appeal to geometric intuition and are easily visualized. But, it turns out that there are deep connections to algebra: Braids form groups, and they describe features of the representation theory of quantum groups and Hopf algebras. The lesser known racks and quandles are algebraic structures that are easy to describe, but strong enough to classify all knots. There is still much to be explored in this area.

As the saying goes: Nothing in biology makes sense except in the light of evolution. The foundations of evolutionary biology were laid by Darwin (1809-1882). He also sketched the first evolutionary tree.

Ever since then, the ability to think in terms of trees has become a central competence for evolutionary biologists, and the study of patterns of descent in the form of trees has developed into an important branch of life sciences: phylogenetics.

Phylogenetics has spawned new research in mathematics, involving finite metric spaces, spaces (in the sense of geometry/topology) of phylogenetic trees, affine buildings (in the sense of Lie theory), tropical geometry... leading to both practical as well as theoretical results.

Network science is an interdisciplinary academic field which studies, among other things, computer networks, telecommunication networks, sensor networks, and biological networks. As such, it links the mathematical sciences to other core research areas, such as computer and information science, electronics and telecommunications, and telematics, of course.

Arguably one of the earliest results in network theory dates back to Euler (1707-1783): the problem to find a walk through the city of Königsberg that would cross each of the seven bridges once and only once, or show that such a path does not exist. His solution was the beginning of the development of graph theory and lead to central ideas of modern topology and homotopy theory.

While there is no doubt that numbers and functions have proven to be very useful concepts to model certain quantitative aspects of everyday life, topologists have developed ideas that can also be used to study qualitative features in networks and other general systems of interrelations.

The theory of the equation *f(x)=x* has produced some of the most generally useful results in
mathematics. Banach's fixed point theorem and Brouwer's fixed point theorem are two pillars of the
theory that every student will learn about, and they are in turn the main ingredients for
fundamental applications ranging from biology, numerical computations, economics, hydrodynamics,
differential equations to game theory.

From a topologist's perspective, one may wonder: What can be said about the space of fixed points of
a given function? (Is it non-empty?) What can be said about the space of all functions that have a
fixed point? (Is it the space of *all* continuous functions?) There are various variants of
these questions that can be starting points for projects in this area.

As elementary as it might seem at a first glance, fixed point theory interacts with many seemingly more elaborate mathematical theories, and can, therefore, be an easy way into other subjects of interest, such as Burnside rings, Hochschild homology, Hopf-Conley index theory, bordism theory, and even stable homotopy theory.

Every vector space has a basis. This is useful to know because each basis gives rise to a coordinate system, which in turn can be used to do calculations with vectors and matrices. However, there are modules over other rings that look like they should have a basis, but in fact, they have not. This failure is measured by lower K-theory. Even when bases and coordinates exist, there is usually no preferred choice, and the coordinate changes have to be kept under control, so as to make sure that observed features do not depend on arbitrary choices. This is codified in higher K-theory. As such it is one foundation for the mathematical study of symmetries.

K-theory is concerned with universal (read: best) invariants of mathematical structures. For that reason, explicit evaluations of these invariants are difficult, and new calculations are always welcome with open arms. On the other hand, because of its universal approach, the areas of applications range throughout mathematics, and it is possible (but not necessary) to learn a lot of mathematics when working on examples with K-theory in mind. Projects in this area typically focus on examples that motivate and illustrate some of the general K-theory machinery.

Symmetries are everywhere, and groups are the mathematical structure to describe symmetries. Some of the most prominent Norwegian mathematicians devoted large parts of their lives to the theory of groups: Niels Henrik Abel (1802-1829), Peter Ludwig Mejdell Sylow (1832-1918), and Marius Sophus Lie (1842-1899).

Topology has many methods to offer to study symmetries and groups, homology for instance. This is based on the idea that more complicated groups can be approximated by means of more elementary ones, the abelian groups, named in honor of Abel. But, homology of groups is typically not easy to compute.

It is worth the while to spend some time with the calculation of the homology of the full permutation groups, or the full matrix groups over finite fields, for example. There are computer programs that allow for sample experiments, but the computational complexity of the problem offers some severe challenges. The subject allows for ample exploration in many directions.