About Me

I am a mathematician.
My research focuses on problems at different interfaces between topology, geometry, and algebra. Everyone believes to know what algebra (numbers!?) and geometry (triangles!?) are about. Topology is certainly different, and no less exciting and useful.
One of my main objectives is to understand the symmetries and connectedness in various structures. This involves, but is not restricted to: group theory and generalizations, algebraic topology, manifolds and cell complexes, K-theory, homological algebra, and category theory (in some more or less random order).
In the end, I also like to compute homology.

Markus Szymik
Department of Mathematical Sciences
NTNU Norwegian University of Science and Technology
7491 Trondheim


1252, Sentralbygg II


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.

Published or Accepted Papers

String bordism and chromatic characteristics
Homotopy Theory: Tools and Applications. Contemp. Math. (to appear)
Preprint - Arxiv

Alexander–Beck modules detect the unknot
Fund. Math. (to appear)
Preprint - Arxiv

Quandle cohomology is a Quillen cohomology
Trans. Amer. Math. Soc. (to appear)
Preprint - Arxiv

The rational stable homology of mapping class groups of universal nil-manifolds
Ann. Inst. Fourier (to appear)
Preprint - Arxiv

Homotopy coherent centers versus centers of homotopy categories
New Directions in Homotopy Theory. Contemp. Math. 707 (2018) 121-142
Link - Preprint - Arxiv

Permutations, power operations, and the center of the category of racks
Comm. Algebra 46 (2018) 230-240
Link - Preprint - Arxiv

Spectral sequences for Hochschild cohomology and graded centers of derived categories
(with F. Neumann)
Selecta Math. 23 (2017) 1997-2018
Link - Preprint - Arxiv

Frobenius and the derived centers of algebraic theories
(with W.G. Dwyer)
Math. Z. 285 (2017) 1181-1203
Link - Preprint - Arxiv

Brauer spaces for commutative rings and structured ring spectra
Manifolds and K-theory. Contemp. Math. 682 (2017) 189-208
Link - Preprint - Arxiv

A non-trivial ghost kernel for complex projective spaces with symmetries
Publ. Res. Inst. Math. Sci. 52 (2016) 249-270
Link - Preprint - Arxiv

Drinfeld centers for bicategories
(with E. Meir)
Doc. Math. 20 (2015) 707–735
Link - Preprint - Arxiv

Homotopies and the universal fixed point property
Order 32 (2015) 301-311
Link - Preprint - Arxiv

Commutative S-algebras of prime characteristics and applications to unoriented bordism
Algebr. Geom. Topol. 14 (2014) 3717-3743
Link - Preprint - Arxiv

Twisted homological stability for extensions and automorphism groups of free nilpotent groups
J. K-Theory 14 (2014) 185-201
Link - Preprint - Arxiv

The chromatic filtration of the Burnside category
Math. Proc. Cambridge Philos. Soc. (2013) 287-302
Link - Preprint

Bauer-Furuta invariants and Galois symmetries
Q. J. Math. 63 (2012) 1033-1054
Link - Preprint

Brauer groups for commutative S-algebras
(with A. Baker and B. Richter)
J. Pure Appl. Algebra 216 (2012) 2361-2376
Link - Arxiv

Crystals and derived local moduli for ordinary K3 surfaces
Adv. Math. 228 (2011) 1-21
Link - Preprint - Arxiv

The Brauer group of Burnside rings
J. Algebra 324 (2010) 2589-2593
Link - Preprint

Characteristic cohomotopy classes for families of 4-manifolds
Forum Math. 22 (2010) 509-523
Link - Preprint

K3 spectra
Bull. Lond. Math. Soc. 42 (2010) 137-148
Link - Preprint

Stable diffeomorphism groups of 4-manifolds
Math. Res. Lett. 15 (2008) 1003-1016
Link - Preprint

Equivariant stable stems for prime order groups
J. Homotopy Relat. Struct. 2 (2007) 141-162
Link - Preprint - Arxiv

A stable approach to the equivariant Hopf theorem
Topology Appl. 154 (2007) 2323-2332
Link - Preprint


Adams operations and symmetries of representation categories
(with E. Meir)
Preprint - Arxiv

The homology of the Higman-Thompson groups
(with N. Wahl)
Preprint - Arxiv

The stable homotopy theory of vortices on Riemann surfaces
Preprint - Arxiv


Grundkurs Topologie
(with G. Laures)
Spektrum Akademischer Verlag, Heidelberg, 2009
2., überarbeitete Auflage, Springer Spektrum, Heidelberg, 2015

Seminars and Conferences

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


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


Copenhagen, Jun 24–28, 2019
10 years with SYM in Copenhagen

Aberdeen, Nov 21, 2018
Topology Seminar
Symmetry groups of algebraic structures and their homology

Cambridge, Nov 20, 2018
HHH Seminar
Symmetry groups of algebraic structures and their homology

Sheffield, Nov 1, 2018
Topology Seminar
Quandles, knots, and homotopical algebra

Montpellier, Oct 23–26, 2018
Rencontre 2018 du GdR de Topologie Algébrique
Symmetry groups of algebraic structures and their homology

Lille, Oct 5, 2018
Topology Seminar
Knots and homotopy theory

Strasbourg, Apr 17, 2018
Séminaire Algèbre et Topologie
Homotopical ideas in the theory of knots

Durham, Dec 14, 2017
Geometry and Topology Seminar
Homotopical ideas in the theory of knots

Cambridge, Nov 22, 2017
Differential Geometry and Topology Seminar
Homotopical ideas in the theory of knots

Grenoble, Oct 9, 2017
Séminaire Algèbre et Géométries
Symmetry groups of algebraic structures and their homology

Leicester, Sep 8, 2017
British Topology Meeting
Homotopical ideas in the theory of knots

Leicester, Sep 5, 2017
Pure Maths Colloquium
Symmetry groups of algebraic structures and their homology

National University of Ireland, Galway, May 18-20, 2017
Groups in Galway 2017
Homology of the automorphism groups of free nilpotent groups

Københavns Universitet, March 20, 2017
Topology Seimnar
Homotopical knot theory

Johns Hopkins University, Nov 7, 2016
Topology Seminar
The homology of the Higman-Thompson groups

University of Rochester, Nov 4, 2016
Topology Seminar
The homology of the Higman-Thompson groups

University of British Columbia, Oct 26, 2016
Topology Seminar
Freaks of algebra - a topologist's exhibition

UCLA, Oct 12, 2016
Algebraic Topology Seminar
The homology of the Higman-Thompson groups

Indiana University-Purdue University Indianapolis, Sep 20, 2016
Modern Analysis and Geometry Seminar
Freaks of algebra - a topologist's exhibition

Wayne State University, Sep 14, 2016
Topology Seminar
Symmetry groups of algebraic structures and their homology

Fields Institute, Jun 13-17, 2016
Group Actions Workshop
Homology of automorphism groups in algebra and topology


Trondheim, Mar 18, 2018
Study Group at Galleri KiT
Badiou - Inaesthetics - Topology


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

Current Course

Spring 2019: Introduction to Lie Theory Fall 2018: Reseach Leave at Cambridge University

Past Courses

Spring 2018: Calculus 3 (TMA4115)
Fall 2017: Algebraic Topology (MA3403)
Spring 2017: Introduction to Topology (TMA4190)
Fall 2016: Reseach Leave at Stanford University
Spring 2016: General Topology (MA3002)
Fall 2015: Calculus 3 (TMA4110)
Spring 2015: General Topology (MA3002)
Fall 2014: Calculus 3 (TMA4110)


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

Current Members

Rachael Boyd
ERCIM postdoctoral fellow

Paul André Dillon Trygsland
PhD, current
works on Morse theory

Christopher Schwartz Kvarme
Master, current
works on curves on surfaces

Kamilla Weka
Bachelor, current
works on wallpaper patterns

Lise Millerjord
StudForsk, current
works on symmetries of trees

Past Members

Truls Bakkejord Ræder
PhD, Trondheim, 2017
Rational Tambara functors

Christopher Schwartz Kvarme
Bachelor, Trondheim, 2017
Algebraic aspects of origami

Adrián Javaloy Bornás
Bachelor, Trondheim, 2017
A topological approach to clustering

Peter Marius Flydal
Bachelor, Trondheim, 2017
The model category of simplicial groups is proper

Tobias Grøsfjeld
Master, Trondheim, 2017
Thesaurus racks

Eivind Otto Hjelle
Bachelor, Trondheim, 2016
Delta-generated spaces

Erlend Børve
StudForsk, Trondheim, 2016
The homology of general linear groups over finite fields

Johanne Haugland
StudForsk, Trondheim, 2016
United representation rings of finite groups

Reidun Persdatter Ødegaard
Master, Trondheim, 2015
Algebraic invariants of links and 3-manifolds

Andreas Hamre
Bachelor, Trondheim, 2015
Manes' theorem

Jens Jakob Kjær
Master Thesis, Copenhagen, 2013
The Kahn-Priddy theorem

Rasmus Nørtoft Johansen
Master Project, Copenhagen, 2013
Thom spectra and bordism theories

Robert Wilms
Master Thesis, Bochum, 2011

Norman Schumann
Diplomarbeit, Bochum, 2009
Die chromatische Filtrierung des Burnside-Ringes

Projects for students

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.

Graphs and Networks

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 within the IME faculty at NTNU: 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.

the seven bridges of Königsberg

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.

Trees and Evolution

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.

Darwin's 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.

Fixed Point Theory

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.

Algebraic K-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.

Homology of Groups

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).

Abel Sylow Lie

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.

Braids, knots, and links

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.

a knot a braid

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.