Contact

University of Ljubljana
Faculty for Computer and Information Science
Personal website
Večna pot 113, 1000 Ljubljana, Slovenia

e-mail: aljaz.zalar@fri.uni-lj.si

Research interests

Real algebraic geometry, truncated moment problems, moment problems, matrix polynomials, completely positive maps.

Preprints

26. with I. Zobovič: Matricial Gaussian quadrature rules: nonsingular case.
25. with M. Kummer: Positive polynomials and the truncated moment problem on plane cubics.
24. with L. Fialkow: The pure Y=X^{d} truncated moment problem.
23. with S. Sun: Matrix Fejér-Riesz type theorem for a union of an interval and a point.
22. with S. Yoo: Bivariate Truncated Moment Sequences with the Column Relation XY=X^m + q(X), with q of degree m-1.
21. with R. Nailwal: Moment theory approach to Gaussian quadratures with prescribed nodes
20. with I. Klep, K. Šivic: Cross-positive linear maps, sums of squares and positive polynomials
19. with M. Omladič, M. Vuk: Extreme mass distributions for quasi-copulas.

Publications

18. with T. Štrekelj: Construction of exceptional copositive matrices, Linear Algebra Appl. 727 (2025) 368-384. (pdf)
17. with M. Belšak, M. Omladič, M. Vuk: Extreme values of the mass distribution associated with d-quasi-copulas via linear programming, Fuzzy Sets and Systems 517 (2025) 17 pp. (pdf)
16. with R. Nailwal: The truncated univariate rational moment problem, Linear Algebra Appl. 708 (2025) 280-301. (pdf)
15. with I. Klep, T. Štrekelj: A random copositive matrix is completely positive with positive probability, SIAM Journal on Applied Algebra and Geometry 8, 3 (2024) 583-611. (pdf)
14. with S. Yoo: The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations, Complex Anal. Oper. Theory. 18, 111 (2024) 54 pp. (pdf)
13. The truncated moment problem on curves y=q(x) and yx^ℓ=1, Linear and Multilinear Algebra 72, 12 (2024) 1922-1966. (pdf)
12. The strong truncated Hamburger moment problem with and without gaps, J. Math. Anal. Appl. 516 (2022) 21 pp. (pdf)
11. The truncated moment problem on the union of parallel lines, Linear Algebra Appl. 649 (2022) 186-239. (pdf)
10. The truncated Hamburger moment problems with gaps in the index set, Integ. Equ. Oper. Theory 93 (2021) 36 pp. (pdf)
9. with A. Bhardwaj: The tracial moment problem on quadratic varieties, J. Math. Anal. Appl. 498 (2021) 39 pp. (pdf)
8. with J. Cimprič: A local-global principle for linear dependence in enveloping algebras of Lie algebras, Linear Algebra Appl. 596 (2020) 230-255. (pdf)
7. with I. Klep, S. McCullough, K. Šivic: There are many more positive maps than completely positive maps, Int. Math. Res. Not. 11 (2019) 3313-3375. (pdf)
6. with A. Bhardwaj: The singular bivariate quartic tracial moment problem, Complex Anal. Oper. Theory 12 (2018) 1057-1142. (pdf)
5. Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets, J. Math. Anal. Appl. 445 (2017) 32-80. (pdf)
4. Matrix Fejér-Riesz theorem with gaps, J. Pure Appl. Algebra 220 (2016) 2533-2548. (pdf,Arxiv-v1)
3. Non-negative spectral measures and representations of C*-algebras, Integ. Equ. Oper. Theory 79 (2014) 219-242. (pdf)
2. with J. Cimprič: Moment problems for operator polynomials, J. Math. Anal. Appl. 401 (2013) 307-316. (pdf)
1. A note on a matrix version of the Farkas lemma, Comm. Algebra 40 (2012) 3420-3429. (pdf)

In preparation

• with S. Yoo: The truncated moment problem on reducible cubic curves II: Hyperbolic type relations.
• with I. Zobovič: Matricial Gaussian quadrature rules: degenerate case.
• with A. Bhardwaj: Flat extensions for cubic curves in Weierstrass form.

Grants

• Slovenian Research and Innovation Agency (ARIS) grant J1-60011. The truncated moment problem via Real algebraic geometry.
  PI January 2025 – December 2027.

Upcoming Conferences

Conferences and Talks

• IWOTA 2025, 14-18 July, 2025, Twente, Netherlands (coorganizer of the session Moment problems and applications).
• A gap between positive polynomials and sums of squares in various settings, SIAM Conference on Applied Algebraic Geometry (AG25), 7-11 July, 2025, Madison, Wisconsin, U.S.
• 9th Linear Algebra Workshop, 2-6 June, 2025, Portoroz, Slovenia (member of the organizing committee).
• Matrix Polynomials and Matrix Positiv/Nichtnegativstellensätze, Winter School, Moments, Non-Negative Polynomials, and Algebraic Statistics, University of Konstanz, Germany, February 2025.
• A gap between positive polynomials and sums of squares in three different settings, Seminar Algebra–Geometrie–Kombinatorik, TU Dresden, Germany, June 2024.
• A gap between positive even quartics and sums of squares ones, MoPAT-24 – Moments and Polynomials: Applications and Theory, University of Konstanz, Germany, March 2024.
• A gap between positive polynomials and sums of squares in various settings, OS Reelle Geometrie und Algebra, University of Konstanz, Germany, February 2024.
• The truncated moment problem on some plane curves, IWOTA 2023, Helsinki, Finland, August 2023.
• The truncated moment problem on some polynomial and rational plane curves, SIAM - AG23, Eindhoven, Netherlands, July 2023.
• A gap between positive maps (resp. copositive matrices) and completely positive ones, Seminar Alpe-Jadran, Zagreb, Croatia, May 2023.
• The truncated moment problem on quadratic, cubic and some higher degree curves, IWOTA 2022, Kraków, Poland, September 2022.
• Solving bivariate truncated moment problems by reduction to the univariate setting, Operator Theory and Beyond 2022, Jagiellonian University, Kraków, Poland, July 2022.
• The bivariate truncated moment problem on quadratic and some higher degree curves, IWOTA 2020, Lancaster, UK, August 2021, online.
• The tracial moment problem on quadratic varieties, ECM 2020, Portorož, Slovenia, June 2021, online.
• The singular bivariate quartic tracial moment problem, IWOTA 2019, Lisbon, Portugal, July 2019.
• There are many more positive maps than completely positive maps, Sums of Squares - Real Algebraic Geometry and its Applications, Innsbruck, Austria, August 2017.
• There are many more positive maps than completely positive maps, LAW 2017, Ljubljana, Slovenia, June 2017.
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets, IWOTA 2016, St Louis, Missouri, USA, July 2016.
• Univariate matrix polynomials positive semidefinite on semialgebraic sets, Summer School on Real Algebraic Geometry and Optimization, Georgia Tech, Atlanta, USA, July 2016.
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets, The University of Auckland, New Zealand, March 2016.
• Matrix Fejér-Riesz theorem with gaps, Algebra and Geometry seminar, The University of Auckland, New Zealand, October 2015.
• Matrix Fejér-Riesz theorem with gaps, IWOTA 2015, Tbilisi, Georgia, July 2015.
• Matrix Fejér-Riesz theorem with gaps, Oberseminar Reelle Geometrie und Algebra, University of Konstanz, Germany, May 2015.

Teaching

• Mathematics 2 - MCMC methods (1/4): 2022/23-24/25

Previous teaching

• Principles of Uncertainty (1/2): 2022/23-24/25
• Numerical Methods: 2019/20-24/25, Lecture notes (in Slovene)
• Linear Algebra: 2023/24
• Numerical Mathematics: 2021/22-23/24, Lecture notes (in Slovene)
• Mathematical Modelling: 2019/20-23/24, Solved exercises from past exams
• Discrete Structures: 2019/20-21/22, Solved exercises from past exams (in Slovene)
• Calculus: 2019/20-20/21, Solved exercises from past exams (in Slovene), Recordings 2020/21

Student supervision

Postdoctoral fellows

• Rajkamal Nailwal, Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia, November 2024-

PhD Thesis

• Igor Zobovič, Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia, May 2025-

Master Thesis

• Noah Novšak: A software approach to the PPT2 conjecture, MSc, ongoing
• Matic Šutar: The Christoffel-Darboux kernel and its use for data analysis, MSc, ongoing
• Veljko Dudić: Hankel low-rank approximation for data analysis, MSc, ongoing
• Una Sekulović: Algorithms for solving tensor completion problem and its application for reconstruction of images, MSc (2024)
• Mirjam Skobe: Sample rate conversion of a digital signal, MSc (2021)

Bachelor Thesis

• Erazem Stanonik: Tensor algorithms for image classification, BSc (2025)
• Tomaž Jonatan Leonardis: Algorithms for finding Nash equilibrium in bimatrix games, BSc (2024)
• Matej Belšak: Extreme volumes of quasi-copulas, BSc (2024)
• Biljana Vitanova: Principal Component Analysis of Multidimensional data, BSc (2024)
• Ana Poklukar: Matrix Completion Problem Through Optimization on Riemannian Manifolds, BSc (2024)
• Žan Terplan: Use of Lasserre hierarchies in computer graphics, BSc (2024)
• Tadej Jeršin: Distance matrix completions, BSc (2024)
• Anamari Orehar: Statistical analysis of efficiency of solving bimatrix games using LemkeHowson algorithm, BSc (2023)
• Matej Klančar: Algorithms for solving the matrix completion problem, BSc (2023)
• Gregor Kovač: Ray tracing in non-euclidean spaces, BSc (2023)
• Veljko Dudić: Usage of Lasserre hierarchies in game theory, BSc (2022)
• Filip Marušič: Truncated Moment Problems and positive semidefinite matrix completions, BSc (2022)
• Matjaž Ciglič: Inertia of matrix completions, BSc (2022)
• Nace Gorenc: Determining a type of a measure in a nonsingular bivariate quartic tracial moment problem, BSc (2020)

Academic education

• 2013 - 2017: PhD in Mathematics, University of Ljubljana. Thesis: Contributions to a noncommutative real algebraic geometry. Supervisors: Prof. Jaka Cimprič and Prof. Igor Klep.

• 2011 - 2013: MSc in Mathematics, University of Ljubljana. Thesis: Moment problems for matrix and operator polynomials (in Slovene). Supervisor: Prof. Jaka Cimprič

• 2008 - 2011: BSc in Mathematics, University of Ljubljana. Thesis: Matrix generalizations of Farkas lemma (in Slovene). Supervisor: Prof. Jaka Cimprič