Optimális transzport

Az MTA–RI Lendület Optimális Transzport és Kvantum Információ-geometria Kutatócsoport klasszikus optimális transzporttal, kvantum-információelmélettel, valamint az előbbiek szinergikus kapcsolatából született rendkívül friss tudományterülettel, a kvantum optimális transzport elméletével foglalkozik. A klasszikus Wasserstein-terek szorosan kapcsolódnak az optimális transzport elméletéhez, és az utóbbi években rendkívül fontosnak bizonyultak mind a matematikában (valószínűségelmélet, metrikus geometria, pénzügyi matematika), mind az alkalmazott tudományokban (képfeldolgozás, gépi tanulás). A kvantum Wasserstein-terek és a kvantum optimális transzport elmélete éppen kialakulóban van, rengeteg különböző és érdekes megfogalmazást vizsgálnak, rengeteg a nyitott probléma ezen a területen. A csoport célja klasszikus Wasserstein-terek és entrópikus távolságokkal ellátott kvantum állapotterek alapvető geometriai jellemzőinek (geodetikusok, metrikus tenzormező, baricentrumok) meghatározása, illetve részvétel a nemkvadratikus kvantum optimális transzport elméletének felépítésében.

Publications 

[1] Z. M. Balogh, G. Kiss, T. Titkos, D. Virosztek. Isometric rigidity of the Wasserstein space over the plane with the maximum metric. Preprint, arXiv:2411.07051 (2024). Link(s): arXiv

[2] Z. M. Balogh, E. Ströher, T. Titkos, D. Virosztek, Rigid and flexible Wasserstein spaces. Preprint, arXiv:2502.09364 (2025). Links: arXiv

[3] Z. M. Balogh, T. Titkos, D. Virosztek. Isometric rigidity of the Wasserstein space W_1(G) over Carnot groups. Preprint, arXiv:2305.05492 (2023). Link(s): arXiv

[4] Z. M. Balogh, T. Titkos, D. Virosztek. Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group. Preprint, arXiv:2303.15095 (2023). Link(s): arXiv

[5] G. Bunth. On Quantum Rényi Divergences. PhD thesis, BME, 2024. Link(s): online version

[6] G. Bunth, J. Pitrik, T. Titkos, D. Virosztek. On the metric property of quantum Wasserstein divergences. Physical Review A 110 (2024), 022211. Link(s): arXiv,  published version

[7] G. Bunth, J. Pitrik, T. Titkos, D. Virosztek. Wasserstein distances and divergences of order p by quantum channels. Preprint, arXiv:2501.08066 (2025). Links: arXiv

[8] M. Cúth, M. l. Doucha, T. Titkos. Isometries of Lipschitz-free Banach spaces. J. Lond. Math. Soc. 110 (2024), e70000. Link(s): arXiv, published version 

[9] G.P. Gehér, A. Hrušková, T. Titkos, D. Virosztek. Isometric rigidity of Wasserstein spaces over Euclidean spheres. J. Math. Anal. Appl. 542 (2025), 128810. Link(s): arXiv, published version 

[10] G.P. Gehér, J. Pitrik, T. Titkos, D. Virosztek. Quantum Wasserstein isometries on the qubit state space. J. Math. Anal. Appl. 522 (2023), 126955. Link(s): arXiv, published version

[11] G.P. Gehér, T. Titkos, D. Virosztek. Isometric rigidity of Wasserstein tori and spheres. Mathematika 69 (2023), 20-32. Link(s): arXiv, published version

[12] G.P. Gehér, T. Titkos, D. Virosztek. On isometries of Wasserstein spaces. RIMS Kôkyûroku Bessatsu B93 (2023), 239-250. Link(s): published version 

[13] G.P. Gehér, T. Titkos, D. Virosztek. On the exotic isometry flow of the quadratic Wasserstein space over the real line. Linear Algebra Appl. 693 (2024), 39-48. Link(s): published version 

[14] G.P. Gehér, T. Titkos, D. Virosztek, The isometry group of Wasserstein spaces: the Hilbertian case. J. London Math. Soc. 106 (2022), 3865-3894. Link(s): arXiv, published version  

[15] G. Kiss, T. Titkos. Isometric rigidity of Wasserstein spaces: the graph metric case. Proc. Amer. Math. Soc. 150 (2022), 4083-4097. Link(s): arXiv, published version

[16] J. Maas, S. Rademacher, T. Titkos, D. Virosztek (Eds.). Optimal transport on quantum structures. Bolyai Society Mathematical Studies, Volume 29 (2024). Link(s): published version

[17] M. Mosonyi, G. Bunth, and P. Vrana. Geometric relative entropies and barycentric Rényi divergences. Linear Algebra Appl. 699 (2024), 159–276. Link(s): arXiv, published version

[18] L. Portinale, S. Rademacher, D. Virosztek. Limit theorems for empirical measures of interacting quantum systems in Wasserstein space. Preprint arXiv:2312.00541 (2023). Link(s): arXiv

[19] A. Ramanantoanina, T. Titkos. Lattice properties of strength functions. Acta Sci. Math. (Szeged) 90 (2024), 679-689. Link(s): published version

[20] R. Simon, D. Virosztek. Preservers of the p-power and the Wasserstein means on 2×2 matrices.  Electron. J. Linear Algebra 39 (2023), 395-408. Link(s): arXiv, published version

[21] R. Simon, D. Virosztek, Isometries of the qubit state space with respect to quantum Wasserstein distances. Linear Algebra Appl. 714 (2025), 1-14. Link(s): arXiv, published version 

[22] G. Toth, J. Pitrik. Quantum Wasserstein distance based on an optimization over separable states. Quantum 7 (2023), 1143. Link(s): arXiv, published version

[23] D. Virosztek, Operator means, barycenters, and fixed point equations. Acta Sci. Math. (Szeged) 90 (2024), 391-408. Link(s): arXiv, published version 

Conference and seminar talks 

2025

[1] 26th Conference of the International Linear Algebra Society (ILAS), Kaohsiung, Taiwan, invited talks, 23-27/06/2025 (T. Titkos and D. Virosztek — upcoming)

[2] 17th Workshop on Numerical Ranges and Numerical Radii, Taichung, Taiwan, invited talks, 19-21/06/2025 (T. Titkos and D. Virosztek — upcoming)

[3] Dynamics of Density Operators Workshop, IPAM, UCLA, Los Angeles, USA, invited talk, 28/04–02/05/2025 (D. Virosztek — upcoming)

[4] Analysis Seminar at Politecnico di Milano, Dipartimento di Matematica, Milano, Italy, invited talk 27/03/2025 (T. Titkos  — upcoming)

[5] Analysis and Applied Mathematics Seminar, Bocconi University, Milano, Italy, invited talk, 04/03/2025 (T. Titkos)

2024

[6] Quantum Optimal Transport: Theory and Applications (QOTTA) Workshop, Palazzone di Cortona, Italy, invited and contributed talks, 1-6/09/2024 (J. Pitrik, T. Titkos, D. Virosztek)

[7] 35th International Workshop on Operator Theory and its Applications, University of Kent, Canterbury, UK, talks, 15/08/2024 (T. Titkos and D. Virosztek)

[8] 2nd Analysis Mathematica Conference, Renyi Institute, Budapest, Hungary, invited talk, 31/07/2024 (D. Virosztek)

[9] 9th European Congress of Mathematics, Sevilla, Spain, talk, 16/07/2024 (T. Titkos)

[10]  Summer School in Mathematics, Eötvös Loránd University, invited lecture series, 17-21/06/2024 (T. Titkos and D. Virosztek) 

[11] Optimal transport minicourse at the Analysis Seminar of the Rényi Institute, three lectures, April-May of 2024 (T. Titkos and D. Virosztek)

[12] International Workshop on Matrix Analysis and Quantum Information Theory (MAA&QIT2024), Hue University, Hue City, Vietnam, invited (online) talk, 05/06/2024 (D. Virosztek)

[13] Submodularity seminar (org.: László Lovász et al.), Renyi Institute, Budapest, Hungary, survey talk, 30/04/2023 (T. Titkos)

[14] Discrete and Continuous Structures in Mathematics Workshop, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, invited talk, 26/04/2024 (D. Virosztek)

[15] Trojan Math Seminar, Troy University, Troy, USA, invited (online) talk, 07/03/2024 (D. Virosztek)

[16] Quantum Glue Meeting, UPV/EHU Seminar, Bilbao, On quantum Wasserstein Distance, 14/02/2024 (J. Pitrik)

[17] Bilbao Analsyis and PDE Seminar (BCAM), On quantum Wasserstein distance, 25/01/2024 (J. Pitrik)

[18] 51th Winter School in Abstract Analysis, Vlachovice, Czech Republic, talk, 15/01/2024 (D. Virosztek)

2023

[19] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 30/11/2023 (D. Virosztek)

[20] Optimal transport: classical and quantum. Two invited lecture series at the Niigata University, Niigata, Japan, November of 2023 (D. Virosztek)

[21] Recent Progress in Operator Theory and its Applications Workshop, RIMS Kyoto, Japan, invited talk, 07/11/2023 (D. Virosztek)

[22] Submodularity seminar (org.: László Lovász et al.), Renyi Institute, Budapest, Hungary, survey talk, 10/10/2023 (D. Virosztek) 

[23] Æsy to define, Hard to analyse: First conference on Lipschitz-free spaces, Centre Diocésain Besançon, France, invited talk, 20/09/2023 (T. Titkos)

[24] Séminaire de Mathématiques et Colloquium, Institut de Recherche en Informatique, Mathématiques, Automatique et Signal, Mulhouse, France, invited talk, 18/09/2023 (T. Titkos) 

[25] International Workshop on Matrix Analysis and its Applications, Quy Nhon, Vietnam, talk, 08/07/2023 (J. Pitrik).

[26] Focused Workshop on Moffat's Magnetic Relaxation Problem (org.: László Székelyhidi Jr.), Erdos Center, Budapest, Hungary, invited talk, 05/07/2023 (D. Virosztek)

[27] 50th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 11/01/2023 (T. Titkos and D. Virosztek)

2022

[28] Real and Abstract Analysis Seminar, Charles University, Prague, Czech Republic, invited talk, 23/11/2022 (T. Titkos)

[29] "Kutszem" Research Seminar, Renyi Institute, Budapest, Hungary, invited seminar talk, 21/11/2022 (D. Virosztek) 

[30] Analysis Seminar, Eötvös Loránd University, Budapest, Hungary, invited seminar talk, 21/11/2022 (D. Virosztek) 

[31] AMS-SMF-EMS joint international meeting, Special Session on Quantitative Geometry of Transportation Metrics, Grenoble, France, invited talk, 18/07/2022 (D. Virosztek)

[32] 24th Conference of the International Linear Algebra Society (ILAS), Galway, Ireland, invited talks, 20-24/06/2022 (T. Titkos and D. Virosztek)

[33] Budapest Graduate Seminar, Renyi Institute, invited lectures, 20/04/2022 (T. Titkos and D. Virosztek)

[34] Mathematical Colloquium, University of Bern, Switzerland, invited seminar talk, 04/04/2022 (D. Virosztek)

[35] Institute Seminar, Institute for Solid State Physics and Optics, Wigner Research Center for Physics, 22/02/2022 (J. Pitrik)

[36] 49th Winter School in Abstract Analysis, Svratka, Czech Republic, talks, 09/01/2022 (T. Titkos and D. Virosztek)

2021

[37] Analysis Seminar, Renyi Institute, Budapest, Hungary, talk, 09/12/2021 (D. Virosztek)

[38] "Research on preserver problems on Banach algebras and related topics" Workshop at RIMS Kyoto, Japan, invited talk (online), 27/10/2021 (D. Virosztek)

[39] "Kutszem" Research Seminar, Renyi Institute, Budapest, Hungary, talk, 18/10/2021 (D. Virosztek)

Események

Optimal Transport on Quantum Structures semester - 2022. szeptember-december

Sajtómegjelenések

Magyar Katolikus Rádió Kerengő című műsorában elhangzott interjú - 2021. október 19.

MTA Lendületesek  - interjú Virosztek Dániellel - 2023. január 11.