Research of Tomaž Košir

Research interests

My interests are in applications of algebra (commutative and associative) and algebraic geometry to the problems in linear algebra and partial differential equations. Major part of my published work is in multiparameter spectral theory, but I am also interested in a number of related problems in the areas of commuting matrices, commuting varieties, representations of commutative zero-dimensional algebras, matrix groups and semigroups, and others. My interests were greatly influenced by concrete applied problems. I also help in developing numerical algorithms that can be applied to such problems.

Recently, I joint also a group that is interested in modelling dependencies via copula theory. One of our interests is in modelling dependencies in shock models.

Editorial work:

In the period 2008-2014, I was a member of the editorial board of the Rendiconti dell’Istituto di Matematica dell’Università di Trieste: An International Journal of Mathematics that is published since 1969.

Selected Publications:

On commuting matrices:

• with J. Irving and M. Mastnak. A proof of the Box Conjecture for commuting pairs of matrices. arXiv
• with P. Oblak: On pairs of commuting nilpotent matrices. Transform. Groups 14 (2009), no. 1, 175-182. pdf
• with B. Plestenjak: On stability of invariant subspaces of commuting matrices, Linear Algebra and its Applications 342 (2002), 133-147. pdf

Multiparameter eigenvalue problems:

• with M.E. Hochstenbach and B. Plestenjak: Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models. Num. Lin. Alg. Appl. 31 (2024), Issue 2, Paper No. e2540. open access
• with M.E. Hochstenbach and B. Plestenjak: A Jacobi-Davidson Type Method for the Two-Parameter Eigenvalue Problem, SIAM J. Matrix Anal. Appl. 26 (2004/05), 477-497. pdf
• with P. Binding: Root Vectors for Geometrically Simple Two-Parameter Eigenvalues, Transactions Amer. Math. Soc. 356 (2004), 1705-1726. pdf

• Root Vectors for Geometrically Simple Multiparameter Eigenvalues, Integr. Eq. Op. Theory 48 (2004), 365-396. pdf
• The Cayley-Hamilton Theorem and Inverse Problems for Multiparameter Systems, Linear Algebra and its Applications 367 (2003), 155-163. pdf
• with L. Grunenfelder: Geometric Aspects of Multiparameter Spectral Theory, Transactions Amer. Math. Soc. 350 (1998), 2525-2546. pdf

• with L. Grunenfelder : An Algebraic Approach to Multiparameter Spectral Theory, Transactions Amer. Math. Soc. 348 (1996), 2983-2998. pdf

• with P. Binding : Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type, Transactions Amer. Math. Soc. 348 (1996), 229-249. pdf

On copula theory:

• with E. Peronne: Discrete imprecise copulas. Fuzzy Sets and Systems 504 (2025), Paper No. 109251, 20 pp. open access
• with M. Omladič: Singular components of shock model copulas. J. Comp. Appl. Math. 400 (2022) Paper No. 113749. paper
• with D. Kokol Bukovšek, B. Mojškerc, and M. Omladič: Extreme generators of shock induced copulas. Appl. Math. Comput. 429 (2022), Paper No. 127214, 16 pp. paper
• with D. Kokol Bukovšek, M. Omladič and N. Stopar: Extending sub-quasi-copulas. J. Math. Anal. Appl. 500 (2021), no. 1, Paper No. 125099, 16 pp. paper
• with M. Omladič: Reflected maxmin copulas and modeling quadrant subindependence. Fuzzy Sets and Systems 378 (2020), 125–143. arxiv
• with D. Kokol Bukovšek, B. Mojškerc, and M. Omladič: Relation between non-exchangeability and measures of concordance of copulas. J. Math. Anal. Appl. 487 (2020), no. 1, 123951, 26 pp. arxiv

Determinantal varieties and determinantal representations:

• with A. Buckley: Plane curves as Pfaffians. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 10 (2011), no. 2, 363-388. pdf
• with A. Buckley: Determinantal representations of smooth cubic surfaces, Geometria Dedicata 125 (2007), 115-140. pdf
• with A. Buckley: Construction of self-adjoint determinantal representations of smooth cubic surfaces, Proceedings of EUROSIM 2007, Ljubljana, Slovenia, pdf.
• with B.A. Sethuraman: A Groebner basis for the 2 x 2 determinantal ideal mod t, Journal of Algebra 292 (2005), 138-153, pdf.
• with B.A. Sethuraman: Determinantal Varieties Over Truncated Polynomial Rings, Journal of Pure and Applied Algebra 195 (2005), 75-95. pdf

Other topics:

• with L. Grunenfelder, M. Omladič, and H. Radjavi: Finite groups with submultiplicative spectra. J. Pure Appl. Algebra 216 (2012), no. 5, 1196-1206. arxiv
• with J. Bernik, R. Drnovšek, D. Kokol-Bukovšek, M. Omladič, and H. Radjavi: On semitransitive Jordan algebras of matrices. J. Algebra Appl. 10 (2011), no. 2, 319-333.
• with K. Cvetko-Vah, D. Kokol-Bukovšek, G. Kudryavtseva, Y. Lavrenyuk, and A. Oliynyk: Semitransitive and transitive subsemigroups of the inverse symmetric semigroups, Semigroup Forum 78 (2009), no. 1, 138-147.
• with B.A. Sethuraman we wrote an appendix to J. Holbrook and J. P. Schoch: Moving zeros among matrices, Linear Algebra and its Applications 424 (2007), 83-95, pdf
• with J. Bernik, R. Drnovšek, T.J. Laffey, G. MacDonald, R. Meshulam, M. Omladič, and H. Radjavi: Common fixed points and common eigenvectors for sets of matrices, Linear and Multilinear Algebra 53 (2005), 137-146. pdf
• with J. Bernik, R. Drnovšek, M. Omladič, and H. Radjavi: Irreducible semigroups of matrices with eigenvalue one, Semigroup Forum 67 (2003), 271-287.

Slides of some recent talks

Link to the complete bibliography

Help in organizing meetings

IN ALGEBRA:

I helped in organizing a series of workshops Linear Algebra Workshop (LAW) in Slovenia from 1996 until 2017.

The 7^th LAW was organized in conjuction with the 23^rd International Workshop on Matrices and Statistics (IWMS), Ljubljana, June 8-12, 2014.

The 4^th LAW was held in conjunction with the Conference in Honor of Heydar Radjavi’s 70^th Birthday in Bled, Slovenia, in May 2005.

The 9^th LAW will take place in Portorož, Slovenia, June 2-6, 2025.

IN FINANCIAL MATHEMATICS:

We also organized Summer schools in financial mathematics in Ljubljana in September 2009 and in August/September 2011, and 5^th General Conference in Advanced Mathematical Methods in Finance in Bled, Slovenia, May 4 – 8, 2010. The latter was organized as a part of the AMaMeF scientific program sponsored by the European Science Foundation (ESF).

IN ALGEBRAIC GEOMETRY:

With researchers from the Universities of Trieste and Udine, and SISSA with ICTP we organized one day meetings in Algebraic Geometry. TULS I in 2003 and TULS IV in 2006 were held in Ljubljana. In 2010 also the University of Ferrara joined the group. TULSF VIII was held in September 2013 in Ljubljana.