Henrik Holm's homepage

University of Copenhagen

Teaching

Courses

Fall 2024		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2024		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2024		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2023		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2023		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2023		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2022		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2022		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2022		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2021		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2021		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2021		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2020		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2020		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2020		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2019		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2019		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2019		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2018		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2018		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2018		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2017		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2017		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2017		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2016		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2016		Algebra 2 (lecturer) (KU-SCIENCE)
Spring 2016		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2015		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2015		Algebra 2 (lecturer) (KU-SCIENCE)
Fall 2014		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Fall 2014		Mathematics and data processing (lecturer), calculus, and R (KU-SCIENCE)
Spring 2014		Algebra 2 (lecturer) (KU-SCIENCE)
Fall 2013		Linear algebra for computer scientists (lecturer) (KU-SCIENCE)
Spring 2013		Algebra 2 (lecturer) (KU-SCIENCE)

Fall 2011		Mathematics and data processing (lecturer), calculus, and R (KU-LIFE)
Spring 2011		Mathematics and optimization (lecturer), optimization theory (KU-LIFE)
Fall 2010		Mathematics and data processing (instructor), calculus, and R (KU-LIFE)
Spring 2009		Mathematics and planning (lecturer), optimization theory (KU-LIFE)
Fall 2008		Mathematics and data processing (instructor), calculus, and R (KU-LIFE)
Spring 2008		Mathematics and planning (lecturer), optimization theory (KU-LIFE)
Fall 2007		Mathematics and data processing (instructor), calculus, and R (KU-LIFE)

Spring 2007		Elementary commutative algebra (lecturer), graduate course in ring theory (AU)
Spring 2007		Linear transformations (lecturer), undergraduate course in linear algebra (AU)
Spring 2007		MatLab (chief consultant), mathematical laboratory in linear algebra (AU)
Fall 2006		MatLab (chief consultant), mathematical laboratory in calculus (AU)
May 2006		MAII (lecturer and examiner), undergraduate course in Fourier analysis (AU)
Spring 2006		MatLab (chief consultant), mathematical laboratory in linear algebra (AU)
Fall 2005		MatLab (chief consultant), mathematical laboratory in calculus (AU)
Spring 2005		MatLab (chief consultant), mathematical laboratory in linear algebra (AU)
Fall 2004		MatLab (chief consultant), mathematical laboratory in calculus (AU)

Oct. 2002		Mat4HA (lecturer), graduate course in homological algebra (KU-SCIENCE)
Fall 2002		Mat1GA (coordinator/instructor), course in linear algebra and calculus (KU-SCIENCE)
Fall 2002		MatLab (instructor), mathematical laboratory for all undergraduates (KU-SCIENCE)
Spring 2002		Mat1GB (instructor/examiner), course in linear algebra and calculus (KU-SCIENCE)
Spring 2002		MatLab (instructor), mathematical laboratory for all undergraduates (KU-SCIENCE)
Fall 2001		Mat1GA (instructor), course in linear algebra and calculus (KU-SCIENCE)
Fall 2001		MatLab (instructor), mathematical laboratory for all undergraduates (KU-SCIENCE)
Fall 2000		Mat3AL (instructor), undergraduate course in Galois theory (KU-SCIENCE)
Spring 2000		Mat3MI (instructor), undergraduate course in measure theory (KU-SCIENCE)
Fall 1999		Mat2AN (instructor), undergraduate course in mathematical analysis (KU-SCIENCE)
Spring 1999		Mat3MI (instructor), undergraduate course in measure theory (KU-SCIENCE)
Fall 1998		Mat2AN (instructor), undergraduate course in mathematical analysis (KU-SCIENCE)
Spring 1998		Mat3MI (instructor), undergraduate course in measure theory (KU-SCIENCE)
Fall 1997		Mat2AN (instructor), undergraduate course in mathematical analysis (KU-SCIENCE)
Fall 1996		Mat2AN (instructor), undergraduate course in mathematical analysis (KU-SCIENCE)

Students

PhD

2019.04.30		Georgios Dalezios, Relative homological algebra and exact model structures^†
2018.05.16		Rune Harder Bak, Tensor abelian categories in a noncommutative setting

		[^†double degree student cosupervised with Sergio Estrada]

MSc

2024.06.05		Natascha Schalburg (expected)⁴
2024.05.31		Peixiang Tan (expected)³
2024.05.31		Nicolai Sebastian Carstensen (expected)
2023.12.20		Magnus Rahbek Dalgaard Hansen, Invariant theory and graphs²
2023.05.31		Magnus Bendix Hansen, The undecidability of Whitehead's problem¹
2023.05.31		Peter Rützou Klinting, Ramsey theory: Milliken's tree theorems¹
2022.05.31		Marius Verner Bach Nielsen, The Q-shaped derived category^d
2021.12.20		Andreas Munch Mikkelsen, The Whitehead problem and Suslin's problem¹
2020.06.02		David Tintinago Pinzon, Model structures on chain complexes from cotorsion pairs^c
2019.03.04		Benjamin Pedersen, Flat objects in functor categories
2019.03.04		Anders Friis Frand-Madsen, Gorenstein representation type of local rings
2018.08.06		Michelle Mia Rasmussen, Properties of DG-injective complexes
2018.05.22		Jonas Thomas Seneberg Rudloff, The flat cover conjecture
2018.03.05		Kristian Peter Poulsen, Thick subcategories of the stable category of CM modules
2018.03.05		Astrid Høgenhaven, Abelian model category structures
2018.03.05		Anne H. K. Lützen, Gorenstein projective resolutions and Tate cohomology
2017.08.07		Kirsten Birkegaard Madsen, Matlis reflexive modules
2017.06.02		Anne Gregersen, Homological identities for differential graded algebras
2017.03.06		Ann-Sofie Grundahl, Flat quiver representations
2016.08.05		Beatrix Miranda Ginn Nielsen, Atom spectra of Grothendieck categories^b
2016.06.27		Rikke Stuart Christensen, The homotopy category of complexes of projective modules
2015.09.14		Tina Lysholm Hansen, Localizing subcategories of the derived category of a ring
2015.07.09		Georgios Dalezios, Local homomorphisms and dualizing complexes^a

		^aPhD at the Universities of Murcia and Copenhagen (double degree)
		^bPhD at Danmarks Tekniske Universitet (DTU)
		^cPhD at the University of Warwick
		^dPhD at the Norwegian University of Science and Technology (NTNU)
		¹cosupervised with Asger Törnquist
		²cosupervised with Søren Eilers
		³cosupervised with Benjamin Briggs
		⁴cosupervised with Peter Jørgensen

BSc

2024.06.07		Janus Jogvansson (expected)
2024.03.22		Peter Sæter Egeskov
2024.01.12		Sebastian Larsen
2024.01.12		Rannva Magnussen
2023.06.09		Sebastian Giovanni Murgia Kristensen^c
2023.06.09		Mikkel Dons Demsig
2023.06.09		Jeppe Bernhard Obel
2023.06.09		David Møller Andersen
2023.03.24		Veronika Mantzius Postgaard
2023.01.13		Rasmus Lykkeskov Underlien Jensen^c
2023.01.13		Emil Christopher Jennings
2022.01.14		Kasper Rask Jensen
2021.06.11		Mikkel Kristian Mølbak
2021.06.11		Magnus Rahbek Dalgaard Hansen
2021.04.06		Toke Ull Harremoës
2021.01.15		Mads Kofoed Madsen
2020.06.05		August Trier Nerenst Andersen
2020.01.10		Sebastian Kaels Vigh
2020.01.10		Linus Wegge Hjalsted
2020.01.10		Marius Enevold Gjesme
2020.01.10		Thomas Holme Surlykke
2019.06.07		Jacob Bast Hall
2019.06.07		Frederik Erslev Roesdahl
2019.06.07		Aske Lyngbak Hansen
2019.03.22		Jon Braarud Holm Jordal
2018.08.06		Alex Thomas Jørgensen
2018.01.12		Minna Olivia Christiansen
2017.06.09		Jannick Klitgaard Henriksen
2016.06.10		Morten Alkjær Lassen
2016.06.10		Jens Emil Østergaard Laursen
2016.04.01		Nikolaj Mathias Primault
2016.01.15		Anders Friis Frand-Madsen
2016.01.15		Jacob Harder and Nikolaj Theodor Thams
2015.06.04		Pernille Emma Hartung Hansen
2015.06.04		Lasse Damgaard Christensen
2015.06.04		Kasper Bro Müller
2015.06.04		Benjamin Pedersen
2015.06.04		Anne Hoff Kjeldsen Lützen
2015.03.20		Kristian Peter Poulsen
2014.06.06		Rolf Christian Jørgensen
2014.06.06		Lasse Bryder Nielsen
2014.06.06		Karen Elise Løvengreen
2014.06.06		Beatrix Miranda Ginn Nielsen
2014.01.10		Camille Maria Sieling Langdal Jensen
2013.06.07		Morten Elkjær Hansen
2013.06.07		Mikkel Bøhlers Nielsen

		[^c Student from computer science]

Reading Courses / Projects Outside the Course Scope


2018		Benjamin Pedersen, Representations of quivers
2018		Krisitian Peter Poulsen, Triangulated categories
2018		Josefine Alberte Sjøstrøm, Triangulated categories
2017		Asbjørn Bækgaard Lauritsen, Adjoint functors
2017		Kristian Peter Poulsen, Triangulated categories
2017		Anne Hoff Kjeldsen Lützen, Triangulated categories
2016		Lasse Bryder Nielsen, Group (co)homology
2015		Georgios Dalezios, On Sharp/Foxby equivalence
2015		Adrien Vakili, Modules and vector spaces
2013		Sabine Trede, Groups and rings

Outside Examiner Jobs

Jan. 2024		B.Sc thesis, N. L. Falk, Infinity-category theory and the dg-nerve, 37 pp.
Jan. 2024		B.Sc thesis, S. C. Baatz, Galois theory, 34 pp.
June 2023		B.Sc thesis, P. L. Gammelgaard, Complex representations of GL(2,K), 25 pp.
June 2023		B.Sc thesis, S. Y. Gheith, Galois theory, 37 pp.
June 2023		M.Sc thesis, W. Overgaard-Larsen, Derived and Frobenius categories, 83 pp.
Jan. 2023		B.Sc thesis, L. N. Jæger, Galois theory, 31 pp.
June 2022		M.Sc thesis, M. M. Parylewicz and N. F. H. Larsen, Fantastic morphisms, 108 pp.
June 2022		Project, A. A. Christensen, The Riemann-Roch theorem, 27 pp.
Aug. 2021		B.Sc thesis, A. B. Poulsen, Reinforcement learning and Gröbner bases, 40 pp.
Aug. 2021		Algebra 2 (re-exam), University of Aarhus.
June 2021		Algebra 2, University of Aarhus.
June 2021		M.Sc thesis, P. Q. Nielsen, The Jacobian and the Dixmier conjectures, 43 pp.
Jan. 2021		B.Sc thesis, N. J. Dreijer, Galois theory, 33 pp.
Aug. 2020		B.Sc thesis, A. V. Sckerl, Continued fractions and factorization, 37 pp.
June 2020		B.Sc thesis, A. D. Christiansen, Bézout's theorem, 53 pp.
June 2020		B.Sc thesis, L. H. Brædder, The Weil pairing for elliptic curves, 42 pp.
June 2020		B.Sc thesis, K. Jensen, Elliptic curves and Hasse's theorem, 44 pp.
June 2020		M.Sc thesis, R. H.-Larsen, Gröbner basis computations in the Weyl algebra, 50 pp.
June 2020		M.Sc thesis, I. B. Christensen, Coding theory and Gröbner bases, 67 pp.
June 2020		B.Sc thesis, M. B. Pedersen, Galois theory, 36 pp.
Jan. 2020		B.Sc thesis, F. B. Hermansen, Grothendieck's Galois theory of split algebras, 30 pp.
Jan. 2020		B.Sc thesis, J. A. Færgeman, The punctual Hilbert scheme, 129 pp.
Jan. 2019		M.Sc thesis, E. B. Nielsen, Algebraic methods in integer programming, 82 pp.
June 2018		Ph.D thesis, M. Tarantino, Recollements, model structures, and hearts, 98 pp.
Oct. 2018		Part A exam, J. Christensen, Semi-regular bimodules over Lie algebras, 18 pp.
Aug. 2018		Part A exam, S. Ørsted, Homotopy descent via A-infinity comodules, 30 pp.
June 2018		B.Sc thesis, B. S. Nielsen, Galois theory, 33 pp.
Jan. 2018		B.Sc thesis, M. B. Villadsen, Koszul duality, 38 pp.
Jan. 2018		M.Sc. thesis, C. S. Kjer, Hilbert’s irreducibility theorem, 47 pp. (+ web)
June 2016		Ph.D thesis, M. Akhavin, G-Gorenstein complexes, 82 pp.
June 2016		B.Sc thesis, S. E. Mikkelsen, Galois theory, 32 pp.
June 2016		B.Sc thesis, M. Frey, Galois theory, 43 pp.
Dec. 2015		M.Sc thesis, K. Chu, A membership algorithm for state polytopes, 65 pp.
Jan. 2015		M.Sc thesis, A. R. Abolfazlian, Koszul duality and matrix factorizations, 107 pp.
July. 2013		Part A exam, T. Kanstrup, Demazure descent theory, 33 pp.
Jan. 2012		Homological algebra, University of Copenhagen
Sept. 2011		Elementary number theory (re-exam), University of Copenhagen
Sept. 2011		Algebra 2 (re-examination), University of Copenhagen
Aug. 2011		M.Sc thesis, P. K. Jørgensen, Clifford theory, 56 pp.
June 2011		Algebra 2, University of Copenhagen
April 2011		M.Sc thesis, I. Bonnen, Serre's problem on projective modules, 70 pp.
Feb. 2011		Grad. project, K. R. O. Kjær, Triangulated and derived functors, 44 pp.
Nov. 2010		Grad. project, J. Gabe, The Dold-Kan correspondance, 37 pp.
June 2010		Grad. project, M. B. Willumsen, Prüfer rings, 51 pp.
Jan. 2010		Algebra 2 (re-examination), University of Copenhagen
June 2009		B.Sc thesis, M. B. H. Gath, Frobenius groups, 40 pp.
June 2009		M.Sc thesis, L. Sprogø, Serre's conjecture, 73 pp.
May 2009		M.Sc thesis, T. Tofteby, Rings of finite representation type, 85 pp.
Dec. 2008		M.Sc thesis, K. G. Jensen, PI-algebras, 101 pp.
Dec. 2008		Grad. project, T. Nørgård-Sørensen, Categories, nerves, and (co)homology, 21 pp.
Aug. 2008		Grad. project, C. Søndergaard, Regular rings are UFD, 15 pp.
Aug. 2008		Linear Transformations (re-examination), University of Aarhus
Aug. 2008		M.Sc thesis, S. G.Thomsen, The Krull dimension of a commutative ring, 92 pp.
May 2008		Grad. project, T. Tofteby, Commutative rings of finite representation type, 37 pp.
April 2008		B.Sc thesis, M. S. Christensen, The inverse Galois problem, 18 pp.
Aug. 2007		M.Sc thesis, M. T. P. Noack, Semi-dualizing modules and G-dimensions, 88 pp.
July 2007		B.Sc thesis, U. H. Antonsen, Continued fractions, 26 pp.
July 2007		Grad. project, T. Bargheer and M. L. Christophersen, A days work in D(R), 69 pp.
June 2007		Grad. project, S. Arklint, On derived projective limits, 37 pp.
Jan. 2007		Grad. project, M. T. N. Jensen, Auslander's G-dimension, 31 pp.
July 2006		M.Sc thesis, R. Bruhn, Gorenstein dimensions over CM rings, 80 pp.
June 2006		Algebra 2, University of Copenhagen

Bachelor Projects (suggestions)

Completion of rings (show)

Group (co)homology (show)

Gröbner bases (show)

Injective modules (show)

The lower K-groups of a ring (show)

Last updated: 15 April 2024

Description	Given an ideal \(I\) in a commutative ring \(R\) one can construct the \(I\)-adic completion \(\widehat{R}_I\). For example, \(\widehat{k[x]}_{(x)}\) is the formal power series ring and \(\widehat{\mathbb{Z}}_{(p)}\) is the ring of \(p\)-adic integers. The aim of this project is to define adic completions and to investigate their basic properties.
Keywords	Inverse limit; adic completion; Hensel's lemma; the Artin–Rees lemma; Krull's intersection theorem; Noetherian-ness.
Literature	H. Matsumura, Commutative ring theory, §8 (+ background).

Description	To a group \(G\) one can associate a sequence of (abelian) homology groups \( H_n(G)\) and cohomology groups \( H^n(G)\) that contain information about \( G\). For example, \( H_1(G) = G_\mathrm{ab}\) is the abelianization of \( G\). The aim of this project is to define group (co)homology and to give group theoretical descriptions of the lower (co)homology groups.
Keywords	Group ring; augmentation ideal; derivations; semi-direct products; (co)homology of finite cyclic groups; Hopf's formula; lower central series; extensions.
Literature	P. J. Hilton and U. Stammbach, A course in homological algebra, VI.1-VI.10.

Description	Given an ideal \( I\) in the polynomial ring \( k[x_1,\ldots,x_n]\) and a term ordering \( \leqslant\) one can always find a so-called Gröbner basis \( g_1,\ldots,g_m\) of \( I\) with respect to \( \leqslant\). For example, a Gröbner basis for the ideal \( I = (y^2-x^3+x,y^3-x^2)\) with respect to the lexicographic ordering (where \( x \geqslant y\)) consists of \( g_1 = y^9-2y^6-y^4+y^3\) (which depends only on \( y\)) and \( g_2 = x-y^7+y^4+y^2\). Gröbner bases are powerful tools to solve e.g. polynomial equations and the ideal membership problem. The aim of this project is to define, and to show the existence of, Gröbner bases, and to demonstrate some applications.
Keywords	Term orderings; multivariate division algorithm; Gröbner bases; Hilbert's basis theorem; Buchberger's algorithm; applications.
Literature	N. Lauritzen, Concrete abstract algebra, Chap. 5 D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms, 2.1-2.8.
Software	E.g. Singular.

Description	An object in a category is called injective if it has a certain lifting property. For example, the injective objects in the category of abelian groups are precisely the divisible abelian groups (such as the group of rational numbers \( \mathbb{Q}\) and the Prüfer groups \( \mathbb{Z}(p^\infty)\) where \( p\) is a prime). The aim of this project is to develop the theory of injective modules over an arbitrary ring.
Keywords	Divisible abelian groups; existence of enough injective modules; injective envelopes; direct sum of injective modules; injective cogenerators; the Faith-Walker theorem; structure of injective modules over commutative noetherian rings.
Literature	F. W. Anderson and K. R. Fuller, Rings and categories of modules, §18, §25. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, Chap. 3.3.

Description	The algebraic K-theory of a ring \( R\) is a certain sequence \( K_n(R)\) of abelian groups that contains information about \( R\). For example, if \( R\) is a field, then \( K_0(R) = \mathbb{Z}\) is the additive group of integers and \( K_1(R) = R^\times\) is the multiplicative group of units in \( R\). The aim of this project is to define and investigate the lower K-groups for certain classes of rings.
Keywords	Finitely generated projective modules; Grothendieck group; idempotents; elementary matrices; PIDs; local rings; Dedekind domains.
Literature	J. Rosenberg, Algebraic K-theory and its applications, 1.1-1.3 and 2.1-2.2. There is an erratum on the author's homepage.