Faculty Profile

Eivind Eriksen

Associate Professor - Department of Economics

Biography

I am a norwegian mathematician. My research is in algebra and algebraic geometry, and I am particularly interested in the following topics:

Noncommutative algebraic geometry.
Moduli problems and (noncommutative) deformation theory.
Differential structures in algebraic geometry (rings of differential operators, connections, D-modules).
Representation theory.

For further information, please see my personal homepage.

Publications

Eriksen, Eivind; Laudal, Olav Arnfinn & Siqveland, Arvid (2017)

Noncommutative Deformation Theory

CRC Press.

Eriksen, Eivind (2014)

Computing Noncommutative Deformations

Makhlouf, Abdenacer; Paal, Eugen, Silvestrov, Sergei & Stolin, Alexander (red.). Algebra, Geometry and Mathematical Physics

Eriksen, Eivind & Siqveland, Arvid (2011)

Geometry of Noncommutative Algebras

Banach Center Publications, 93, s. 69- 82. Doi: 10.4064/bc93-0-6

Eriksen, Eivind (2011)

The Generalized Burnside Theorem in Noncommutative Deformation Theory

Journal of Generalized Lie Theory and Applications, 5 Doi: 10.4303/jglta/G110109

Eriksen, Eivind & Gustavsen, Trond Stølen (2010)

Equivariant Lie-Rinehart cohomology

Proceedings of the Estonian Academy of Sciences : Physics, Mathematics, 59(4), s. 294- 300. Doi: 10.3176/proc.2010.4.07

In this paper we study Lie-Rinehart cohomology for quotients of singularities by finite groups, and interpret these cohomology groups in terms of integrable connection on modules.

Eriksen, Eivind (2010)

Computing Noncommutative Deformations of Presheaves and Sheaves of Modules

Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 62(3), s. 520- 542. Doi: 10.4153/CJM-2010-015-6

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases. In the last part of the paper, we describe a noncommutative deformation functor for quasi-coherent sheaves of modules on a ringed space (X,A) . We show that for any good A -affine open cover U of X , the forgetful functor QCohA→PreSh(U,A) induces an isomorphism of noncommutative deformation functors. Applications. We consider noncommutative deformations of quasi-coherent A -modules on X when (X,A)=(X,O X ) is a scheme or (X,A)=(X,D) is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of X closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left D X -module D X when X is an elliptic curve as an example.

Eriksen, Eivind & Gustavsen, Trond Stølen (2009)

Lie-Rinehart cohomology and integrable connections on modules of rank one

Journal of Algebra, 322(12), s. 4283- 4294.

Let k be an algebraically closed field of characteristic 0, let R be a commutative k-algebra, and let M be a torsion free R-module of rank one with a connection . We consider the Lie-Rinehart cohomology with values in EndR(M) with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on M. When R is an isolated singularity of dimension d2, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when R is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology.

Eriksen, Eivind & Gustavsen, Trond (2009)

Lie-Rinehart cohomology and integrable connections on modules of rank one

Journal of Algebra, 322(12), s. 4283- 4294. Doi: doi:10.1016/j.jalgebra.2009.09.015

Eriksen, Eivind & Gustavsen, Trond S. (2008)

Connections on modules over singularities of finite CM representation type

Journal of Pure and Applied Algebra, 212(7), s. 1561- 1574. Doi: 10.1016/j.jpaa.2007.10.008

Eriksen, Eivind (2008)

Connections on Modules Over Quasi-Homogeneous Plane Curves

Communications in Algebra, 36(8), s. 3032- 3041.

Eriksen, Eivind (2008)

An example of noncommutative deformations

Journal of Generalized Lie Theory and Applications, 2(3), s. 52- 56.

Eriksen, Eivind (2008)

Computing noncommutative global deformations of D-modules

Silvestrov, Sergei; Paal, Eugen, Abramov, Viktor, Stolin, Alexander, Eriksen, Eivind & Gustavsen, Trond S. (red.). Generalized Lie Theory in Mathematics, Physics and Beyond

Eriksen, Eivind & Gustavsen, Trond S. (2008)

Connections on modules over singularities of finite and tame CM representation type

Silvestrov, Sergei; Paal, Eugen, Abramov, Viktor, Stolin, Alexander, Eriksen, Eivind & Gustavsen, Trond S. (red.). Generalized Lie Theory in Mathematics, Physics and Beyond

Silvestrov, Sergei; Paal, Eugen, Abramov, Viktor, Stolin, Alexander, Eriksen, Eivind & Gustavsen, Trond S. (2008)

Generalized Lie Theory in Mathematics, Physics and Beyond

Springer.

Eriksen, Eivind & Gustavsen, Trond Stølen (2007)

Computing obstructions for existence of connections on modules

Journal of symbolic computation, 42, s. 313- 323.

Eriksen, Eivind & Gustavsen, TS (2007)

Computing obstructions for existence of connections on modules

Journal of symbolic computation, 42 Doi: 10.1016/j.jsc.2006.10.001

Sørhus, Vidar; Eriksen, Eivind, Grønningsæter, Nils, Halbwachs, Yvon, Hvidsten, Per Øyvind, Strøm, Kyrre, Westgaard, Geir & Røtnes, Jan Sigurd (2005)

A new platform for laparoscopic training and education

Studies in Health Technology and Informatics, 111, s. 502- 507.

Eriksen, Eivind (2017)

Matematikk for økonomi og finans. Oppgaver og løsningsforslag.

[Textbook]. Cappelen Damm Akademisk.

Eriksen, Eivind (2016)

Matematikk for økonomi og finans

[Textbook]. Cappelen Damm Akademisk.

Eriksen, Eivind & Fausk, Halvard (2014)

Mattenøkkelen

[Textbook]. Gyldendal Akademisk.

Eriksen, Eivind (2013)

Simple modules over matric algebras and their geometry

[Academic lecture]. MODULI OPERADS DYNAMICS.

Eriksen, Eivind (2013)

Noncommutative deformations and geometry of simple modules

[Academic lecture]. Algebra, Combinatorics, Dynamics and Applications.

Eriksen, Eivind (2012)

A-infinity algebras and noncommutative deformations

[Academic lecture]. Conference AGMP-8 Brno'12.

Eriksen, Eivind (2009)

The Generalized Burnside Theorem in noncommutative deformation theory

[Academic lecture]. Algebra, Geometry, and Mathematical Physics, 5th Baltic-Nordic AGMP Workshop.

Eriksen, Eivind (2008)

Integrable connections on modules of rank one

[Academic lecture]. Algebra, Geometry, and Mathematical Physics. 4th Baltic-Nordic Workshop: Tartu, 09-11 October, 2008.

Eriksen, Eivind (2007)

Kryptografi og elliptiske kurver

[Academic lecture]. Principles of Computer Security.

Eriksen, Eivind (2007)

Examples in noncommutative deformation theory

[Academic lecture]. Algebra, Geometry, and Mathematical Physics, Baltic-Nordic Workshop.

Eriksen, Eivind & Gustavsen, Trond S. (2006)

Connections on modules over singularities of finite CM representation type

[Academic lecture]. Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop.

Eriksen, Eivind (2006)

Noncommutative deformations of differential structures

[Academic lecture]. Algebra-seminaret ved HiBu.

Eriksen, Eivind (2006)

Non-commutative deformations of differential structures

[Academic lecture]. Algebra, Geometry and Mathematical Physics Baltic-Nordic Network Workshop.

Eriksen, Eivind (2006)

Konneksjoner på moduler over singulariteter

[Academic lecture]. Algebra-seminaret.

Eriksen, Eivind (2004)

A_{\infty} algebras and A_{\infty} modules

[Academic lecture]. utenTitteltekst.

We introduced the notions of A_{\infty} algebras and A_{\infty} modules, and discussed some relations to Massey products and deformation theory.

Eriksen, Eivind (2004)

Noncommutative deformations of sheaves of modules

[Academic lecture]. utenTitteltekst.

Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def: a -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result: If X has a good A-affine open cover U, and F is a quasi-coherent left A-module for all i. Then Def has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism. In particular, if X is a scheme over k, and F is coherent for all i, then H^n(U,F_j,F) is isomorphic to Ext^n(F_j,F) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme.

Eriksen, Eivind (2004)

Noncommutative deformations of sheaves of modules

[Academic lecture]. utenTitteltekst.

Let k be an algebraically closed field, X a topological space, A a sheaf of associative k-algebras on X, and F_1, ... , F_p a finite family of sheaves of left A-modules on X. In this general situation, we define a noncommutative deformation functor Def_F: a_p -> Sets, generalizing the noncommutative deformations of modules (Laudal) and deformations of a sheaf of modules on a scheme (Siqveland). Moreover, we show the following result: If X has a good A-affine open cover U, and F_i is a quasi-coherent left A-module for all i. Then Def_F has a pro-representing hull. If the global Hochschild cohomology groups H^n(U,F_j,F_i) are finite dimensional vector spaces over k for n=1,2 and for all i,j, then this hull is determined by an obstruction morphism. In particular, if X is a scheme over k, and F_i is coherent for all i, then H^n(U,F_j,F_i) is isomorphic to Ext^n_A(F_j,F_i) and we obtain a generalization of the usual deformation theory of coherent modules on a scheme.

Sørhus, Vidar; Eriksen, Eivind, Grønningsæter, Nils, Halbwachs, Yvon, Hvidsten, Per Øyvind, Kaasa, Johannes, Strøm, Kyrre, Westgaard, Geir & Røtnes, Jan Sigurd (2004)

A comprehensive platform for laparoscopic education

[Academic lecture]. SMIT Conference.

Academic Degrees
Year Academic Department Degree
2000 University of Oslo Ph.D Dr. Scient.
1994 University of Oslo Master Cand. Scient.
1992 University of Oslo Cand.Mag
Work Experience
Year Employer Job Title
2010 - Present BI Norwegian Business School Associate Professor in Mathematics
2005 - 2010 Oslo University College Associate Professor
2004 - 2005 Buskerud University College Associate Professor
2003 - 2004 University of Oslo NRC Postdoctoral Fellow
2001 - 2003 University of Warwick Marie Curie Postdoctoral Fellow
1995 - 2000 University of Oslo Research Fellow