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, copula theory.

Publications

Commutative Moment Problems

• with S. Yoo: Constructive approach to the truncated moment problem on reducible cubic curves: Hyperbolic type relations
• with M. Kummer: Positive polynomials and the truncated moment problem on plane cubics.
• with L. Fialkow: The pure Y=X^{d} truncated moment problem: part 1.
• with S. Yoo: Bivariate Truncated Moment Sequences with the Column Relation XY=X^m + q(X), with q of degree m-1.
• with R. Nailwal: Moment theory approach to Gaussian quadratures with prescribed nodes
• with R. Nailwal: The truncated univariate rational moment problem, Linear Algebra Appl. 708 (2025) 280-301. (pdf)
• 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)
• The truncated moment problem on curves y=q(x) and yx^ℓ=1, Linear and Multilinear Algebra 72, 12 (2024) 1922-1966. (pdf)
• The strong truncated Hamburger moment problem with and without gaps, J. Math. Anal. Appl. 516 (2022) 21 pp. (pdf)
• The truncated moment problem on the union of parallel lines, Linear Algebra Appl. 649 (2022) 186-239. (pdf)
• The truncated Hamburger moment problems with gaps in the index set, Integ. Equ. Oper. Theory 93 (2021) 36 pp. (pdf)

Commutative Real Algebraic Geometry

• with I. Klep, K. Šivic: Cross-positive linear maps, sums of squares and positive polynomials, Journal of Algebra 688 (2026) 189-243. (pdf)
• with T. Štrekelj: Construction of exceptional copositive matrices, Linear Algebra Appl. 727 (2025) 368-384. (pdf)
• 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)
• 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)

Noncommutative Moment Problems

• with I. Zobovič: Matricial Gaussian quadrature rules: singular case.
• with I. Zobovič: Matricial Gaussian quadrature rules: nonsingular case, Linear Algebra Appl. 731 (2026) 160-185. (pdf)
• with A. Bhardwaj: The tracial moment problem on quadratic varieties, J. Math. Anal. Appl. 498 (2021) 39 pp. (pdf)
• with A. Bhardwaj: The singular bivariate quartic tracial moment problem, Complex Anal. Oper. Theory 12 (2018) 1057-1142. (pdf)
• with J. Cimprič: Moment problems for operator polynomials, J. Math. Anal. Appl. 401 (2013) 307-316. (pdf)

Noncommutative Real Algebraic Geometry

• with S. Sun: Matrix Fejér-Riesz type theorem for a union of an interval and a point, J. Pure Appl. Algebra. 230 (2026) 108173. (pdf)
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets, J. Math. Anal. Appl. 445 (2017) 32-80. (pdf)
• Matrix Fejér-Riesz theorem with gaps, J. Pure Appl. Algebra 220 (2016) 2533-2548. (pdf,Arxiv-v1)
• A note on a matrix version of the Farkas lemma, Comm. Algebra 40 (2012) 3420-3429.

Copula Theory

• with M. Omladič, M. Vuk: Extreme mass distributions for k-increasing quasi-copulas.
• with M. Omladič, M. Vuk: Extreme mass distributions for quasi-copulas, Fuzzy Sets and Systems 527 (2026) 109698. (pdf)
• 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) 109457 (pdf)

Miscellaneous

• with R. Nailwal: Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk
• with J. Cimprič: A local-global principle for linear dependence in enveloping algebras of Lie algebras, Linear Algebra Appl. 596 (2020) 230-255. (pdf)
• Non-negative spectral measures and representations of C*-algebras, Integ. Equ. Oper. Theory 79 (2014) 219-242. (pdf)

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

• IWOTA 2026, 3-7 August, Québec, Canada.
• SIAM Conference on Optimization (OP26), 2-5 June, Edinburgh, United Kingdom.
• GPOTS 2026, 26-30 May, Iowa City, Iowa.

Conferences and Talks

• Spectral Theory Seminar 2026, Matrix Fejér-Riesz theorem with gaps, January 2026, University of Ljubljana, Slovenia.
• Algebra and functional analysis seminar 2025, Positive polynomials and the truncated moment problem on plane cubic curves, October 2025, University of Ljubljana, Slovenia.
• 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.
• Algebra and functional analysis seminar 2024, Positivity and complete positivity meet positive polynomials, April 2024, University of Ljubljana, Slovenia.
• 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): 2022/23-25/26
• Principles of Uncertainty (1/2): 2022/23-25/26

Previous teaching

• 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

• Sara Milovanova: Nuclear norm minimization for data reconstruction, MSc, ongoing
• Ana Poklukar: Existence of copulas with prescribed values using linear programming, MSc, ongoing
• Noah Novšak: A software approach to the PPT2 conjecture, MSc, ongoing
• Veljko Dudić: Hankel low-rank approximation for data analysis, MSc (2025)
• 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