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

16. with S. Yoo: The truncated moment problem on reducible cubic curves I: Parabolic and Circular type relations
15. with I. Klep, K. Šivic: Cross-positive linear maps, sums of squares and positive polynomials
14. with I. Klep, T. Štrekelj: A random copositive matrix is completely positive with positive probability

Publications

13. The truncated moment problem on curves y=q(x) and yx^ℓ=1, Linear and Multilinear Algebra (2023) 45 pp.
12. The strong truncated Hamburger moment problem with and without gaps, J. Math. Anal. Appl. 516 (2022) 21 pp.
11. The truncated moment problem on the union of parallel lines, Linear Algebra and its Applications 649 (2022) 186--239.
10. The truncated Hamburger moment problems with gaps in the index set, Integ. Equ. Oper. Theory 93 (2021) 36 pp.
9. with A. Bhardwaj: The tracial moment problem on quadratic varieties, J. Math. Anal. Appl. 498 (2021) 39 pp.
8. 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.
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.
6. with A. Bhardwaj: The singular bivariate quartic tracial moment problem, Complex Anal. Oper. Theory 12 (2018) 1057-1142.
5. Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets, J. Math. Anal. Appl. 445 (2017) 32-80.
4. Matrix Fejér-Riesz theorem with gaps, J. Pure Appl. Algebra 220 (2016) 2533-2548.
3. Non-negative spectral measures and representations of C*-algebras, Integ. Equ. Oper. Theory 79 (2014) 219-242.
2. with J. Cimprič: Moment problems for operator polynomials, J. Math. Anal. Appl. 401 (2013) 307-316.
1. A note on a matrix version of the Farkas lemma, Comm. Algebra 40 (2012) 3420-3429.

In preparation

• with I. Klep, T. Štrekelj: Construction of exceptional copositive matrices
• with A. Bhardwaj: The truncated moment problem on cubic curves in Weierstrass form.
• with S. Yoo: The truncated moment problem on reducible cubic curves II: Hyperbolic type relations.

Talks

• 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

• Linear Algebra: 2023/24
• Mathematics 2 - MCMC methods (1/4): 2022/23-23/24
• Principles of Uncertainty (1/2): 2022/23-23/24
• Numerical Mathematics: 2021/22-23/24, Lecture notes (in Slovene)
• Numerical Methods: 2019/20-23/24, Lecture notes (in Slovene)
• Mathematical Modelling: 2019/20-23/24, Solved exercises from past exams

Previous teaching

• 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

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