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
- with R. Nailwal: The truncated univariate rational moment problem
- with M. Belšak, M. Omladič, M. Vuk: Extreme values of the mass distribution associated with d-quasi-copulas via linear programming
- with I. Klep, K. Šivic: Cross-positive linear maps, sums of squares and positive polynomials
Publications
- 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 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 and its Applications 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)
- with A. Bhardwaj: The tracial moment problem on quadratic varieties, J. Math. Anal. Appl. 498 (2021) 39 pp. (pdf)
- with J. Cimprič: A local-global principle for linear dependence in enveloping algebras of Lie algebras, Linear Algebra and its Applications 596 (2020) 230-255. (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)
- with A. Bhardwaj: The singular bivariate quartic tracial moment problem, Complex Anal. Oper. Theory 12 (2018) 1057-1142. (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)
- Non-negative spectral measures and representations of C*-algebras, Integ. Equ. Oper. Theory 79 (2014) 219-242. (pdf)
- with J. Cimprič: Moment problems for operator polynomials, J. Math. Anal. Appl. 401 (2013) 307-316. (pdf)
- A note on a matrix version of the Farkas lemma, Comm. Algebra 40 (2012) 3420-3429. (pdf)
In preparation
- with S. Yoo: Bivariate Truncated Moment Sequences with the Column Relation XY=X^m + q(X) and q of degree m-1.
- with M. Kummer: Positive polynomials and the truncated moment problem on plane cubics.
- with A. Bhardwaj: Flat extensions for cubic curves in Weierstrass form.
- with S. Yoo: The truncated moment problem on reducible cubic curves II: Hyperbolic type relations.
- with I. Klep, T. Štrekelj: Construction of exceptional copositive matrices
Talks
- 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
- Numerical Methods: 2019/20-24/25, Lecture notes (in Slovene)
- Principles of Uncertainty (1/2): 2022/23-24/25
- Mathematics 2 - MCMC methods (1/4): 2022/23-24/25
Previous teaching
- 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, Nov 2024-
Master Thesis
- 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
- 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
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2013 - 2017: PhD in Mathematics, University of Ljubljana. Thesis: Contributions to a noncommutative real algebraic geometry. Supervisors: Prof. Jaka Cimprič and Prof. Igor Klep.
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2011 - 2013: MSc in Mathematics, University of Ljubljana. Thesis: Moment problems for matrix and operator polynomials (in Slovene). Supervisor: Prof. Jaka Cimprič
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2008 - 2011: BSc in Mathematics, University of Ljubljana. Thesis: Matrix generalizations of Farkas lemma (in Slovene). Supervisor: Prof. Jaka Cimprič