Title / Titlu: Syzygies, invariants and classification problems in algebraic geometry and topology / Syzygy-uri, invarianti si probleme de clasificare in geometria algebrica si topologie

Contract no. PCE 5/02.02.2021

Starting date / Data de inceput: January 4, 2021 / 4 ianuarie 2021

Ending date / Data finalizarii: December 31, 2023 / 31 decembrie 2023

Budget / Buget: 1 198 032 RON

Experienced team members / Cercetatori cu experienta:

• Marian Aprodu (project leader / director de proiect)

• Vasile Brînzănescu

• Alexandru Dimca

• Laurențiu Maxim

• Marius Vlădoiu

• Dragoș Ștefan (September 1st, 2021 - March 16, 2023)

Young team members / Tineri cercetatori:

• Filip Chindea, postdoc

• Laura Filimon, PhD student / doctorand

• Roxana Cojman, PhD student / doctorand (as of September 1st, 2021)

Short description / Scurta descriere: 

The present project addresses some classification problems in algebraic geometry and topology. Invariants, either of numerical, geometric or algebraic nature, play a key role. At the core of proposal lies the concept of syzygy, a mathematical term invented by J. J. Sylvester to underline the existence of quadratic forms that are linearly related. The results obtained in the frame of the project underline the role of homological methods in the computation of those invariants.

Acest proiect se ocupa de unele probleme de clasificare din geometria algebrica si topologie. Invariantii, fie ei de natura numerica, geometrica sau algebrica joaca un rol esential. In centrul proiectului se afla conceptul de syzygy, un termen matematic inventat de J. J. Sylvester pentru a pune in evidenta existenta formelor patratice ce sunt legate prin relatii liniare. Rezultatele obtinute in cadrul proiectului subliniaza rolul metodelor omologice in calculul acestor invarianti.

Research objectives / Obiectivele proiectului:

(1) Syzygies of projective varieties and Koszul modules, algebra meets geometry and topology. / Syzygy-urile varietatilor proiective si module Koszul.

(2) Projective hypersurfaces, Jacobian syzygies and associated vector bundles. / Hipersuprafete proiective, syzygy-uri Jacobiene si fibrate vectoriale asociate.

(3) Jump loci and (abelian) duality spaces: properties, realization, applications. / Locuri de salt si spatii de dualitate: proprietati, realizare, applicatii.

Activities in 2021 / Activitati in 2021:

Publications / Publicatii:

• Marian Aprodu, Indranil Biswas, Sorin Dumitrescu, Sebastian Heller. On the monodromy map for the logarithmic differential systems, Bulletin de la Société Mathématique de France 150 (3), (2022) 543-568 DOI:10.24033/bsmf.2854 link

• Alexandru Dimca, Brian Harbourne, Gabriel Sticlaru, On the Bounded Negativity Conjecture and singular plane curves, Moscow Mathematical Journal 22 (2022), 427–450. DOI:10.17323/1609-4514-2022-22-3-427-450  link

• Alexandru Dimca, Gabriel Sticlaru, Waring ranks of sextic binary forms via geometric invariant theory, Transformation Groups, DOI 10.1007/s00031-022-09774-0. link

• Alexandru Dimca, Gabriel Sticlaru, Ramblings on the freeness of affine hypersurfaces. Bulletin Mathématique de la Société de Sciences Mathématiques de Roumanie Tome 65 (113) (2022), 233–251. link

• Alexandru  Dimca, Marek Janasz, Piotr Pokora, On plane conic arrangements with nodes and tacnodes. Innovations in Incidence Geometry 19 (2022), 47–58. DOI: 10.2140/iig.2022.19.47  link

• Alexandru  Dimca, Gabriel Sticlaru, On the birationality of the Hessian maps of quartic curves and cubic surfaces. Atti della Accademia Nazionale dei Lincei. Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 4, pp. 885–891. DOI: 10.4171/RLM/991 link

• Yongqiang Liu, Laurenţiu Maxim, Botong Wang. Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár, Journal für die reine und angewandte Mathematik, vol. 2021, no. 781, 2021, pp. 1-18.  DOI: 10.1515/crelle-2021-0055 link

• Yongqiang Liu, Laurenţiu Maxim, Botong Wang. Perverse sheaves on semi-abelian varieties. Selecta Mathematica 27 (2021), article 30; DOI: 10.1007/s00029-021-00635-4 link

• Laurentiu Maxim, Laurentiu Paunescu, Mihai Tibar. Vanishing cohomology and Betti bounds for complex projective hypersurfaces. Annales de l'Institut Fourier 72 (2022) no. 4, pp. 1705-1731; DOI: 10.5802/aif.3486. link

• Eva Elduque, Christian Geske, Moises Herradon-Cueto, Laurenţiu Maxim, Botong Wang. Mixed Hodge structures on Alexander modules. Memoirs of the American Mathematical Society (accepted) link

• Laura Filimon. On translated rank-2 Brill-Noether loci on regular surfaces. Archiv der Mathematik 118, 271–281 (2022). DOI:10.1007/s00013-021-01700-2   link

• Shmuel Onn, Apostolos Thoma, Marius Vlădoiu. Asymptotic behavior of Markov complexity of matrices.  arXiv:2210.01686, Journal of Pure and Applied Algebra (accepted) link

Dissemination / Diseminare:

• Marian Aprodu. Koszul modules, resonance varieties and applications. Algebraic Geometry Seminar, University of Wisconsin-Madison (online), February 12, 2021.

• Marian Aprodu. Invarianți topologici și varietăți de rezonanță. Conferinta Cercetarii Stiintifice din Romania, Academia Română (CCSAR-2021), November 22-23, 2021. 

• Vasile Brinzanescu. Gauduchon metrics and stability, (plenary lecture), International Virtual Conference, Mathematics & IT: Research and Education, MITRE 2021, Chisinau, Republic of Moldova, July 01-03, 2021.

• Alexandru Dimca. On the Tjurina number of projective plane curves, TIBAR60 conference, University of Wisconsin-Madison (online), September 10-11, 2021. 

• Laurentiu Maxim. Hodge theory on Alexander modules,  University of Western Ontario Geometry and Combinatorics Seminar (online), March 8, 2021. 

• Laurentiu Maxim. Topology of complex projective hypersurfaces, University of Lille Colloquium, Lille, France,  June 18, 2021. 

• Laurentiu Maxim. Topology of complex projective hypersurfaces, at the Institute of Mathematics of the Romanian Academy, Topology Seminar (online), June 18, 2021. 

• Laurentiu Maxim. Topology of complex projective hypersurfaces, University of Wisconsin-Madison Geometry & Topology Seminar, Madison, WI, USA, October 15, 2021. 

• Laurentiu Maxim. On the topology of aspherical complex projective manifolds, Conference on Compactifications, Configurations, and Cohomology, Northeastern University, Boston, USA, October 22-24, 2021. 

• Laura Filimon. Locuri Brill-Noether pe suprafețe regulate. Comunicări științifice pentru studenți, Faculty of Mathematics and Informatics, University of Bucharest, May 8, 2021.

Scientific report / Raport stiintific:

• Romanian

• English

Activities in 2022 / Activitati in 2022:

Publications / Publicatii:

• Marian Aprodu, Gavril Farkas, Claudiu Raicu, Jerzy Weyman. Koszul modules with vanishing resonance in algebraic geometry. arXiv:2201.00839, Selecta Mathematica (accepted) link

• Vasile Brinzanescu, Andreea Nicoara. Relating Catlin and D’Angelo q-types. Journal of Mathematical Analysis and Applications 527, Issue 1, Part 1, 127349 (2023). DOI: 10.1016/j.jmaa.2023.127349 link

• Vasile Brinzanescu, Cristina Maria Sandu. Algebraic methods for Bloch-Iserles Hamiltonian systems,  Bulletin Mathématique de la Société de Sciences Mathématiques de Roumanie, Tome 65(113), No. 2, 2022, 181-189. link

• Alexandru Dimca, Piotr Pokora. On conic-line arrangements with nodes, tacnodes, and ordinary triple points. Journal of Algebraic Combinatorics  56, 403–424 (2022), DOI 10.1007/s10801-022-01116-3. link

• Alexandru  Dimca, Giovanna Ilardi, Gabriel Sticlaru. Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves, Journal of Algebra, Volume 615, (2023)  pages 77-102, DOI 10.1016/j.jalgebra.2022.10.019. link

• Alexandru  Dimca, Giovanna  Ilardi. Lefschetz properties of Jacobian algebras and singularities of hyperplane sections. arXiv:2202.02233, Annali della Scuola Normale Superiore di Pisa (accepted) link 

• Alexandru  Dimca, Piotr Pokora. Maximizing curves viewed as free curves. International Mathematics Research Notices (2023), rnad042, DOI: 10.1093/imrn/rnad042  link

• Alexandru  Dimca, Piotr Pokora. Hunting for Miyaoka-Kobayashi curves. arXiv:2209.14847, submitted. link

• Yongqiang Liu, Laurentiu Maxim, Botong Wang. Non-abelian Mellin transformations and applications. Forum of Mathematics, Sigma, Vol. 10, 2022, e100, 1-17; DOI: 10.1017/fms.2022.91. link

• Eva Elduque, Moises Herradon-Cueto, Laurentiu Maxim, Botong Wang. Alexander modules, Mellin transformations and variations of mixed Hodge structures. Advances in Mathematics Volume 411, Part A 2022, 108798, DOI:10.1016/j.aim.2022.108798. link

• Dragos Stefan. The spectral sequence associated to a Hopf action and applications, work in progress.

• Dimitra Kosta, Apostolos Thoma, Marius Vladoiu. On the strongly robust property of toric ideals, Journal of Algebra, vol. 616, 2023, 1-25. DOI:10.1016/j.jalgebra.2022.11.002  link 

Dissemination / Diseminare:

• Marian Aprodu. Koszul modules and applications, Bandoleros 2022 Sixth Algebraic Geometry Meeting, Ankara, May 11 – 13, 2022.

• Marian Aprodu. Resonance and vector bundles, Resonance, topological invariants of groups, moduli, Humboldt-Universitaet zu Berlin, november 15 – 18, 2022

• Alexandru Dimca. Curbe maximale si curbe libere, Seminar at the Institute of Mathematics of the Romanian Academy, November10, 2022.

• Alexandru Dimca.  Proprietati Lefschetz pentru algebre Milnor, Seminar at Universitatea Ovidius Constanta, November21, 2022.

• Alexandru Dimca.  De la topologie la algebra comutativa, Seminar at the Institute of Mathematics of the Romanian Academy, November 25, 2022.

• Laurentiu Maxim. On a question of Bobadilla and Kollár. Topology Seminar at the Institute of Mathematics of the Romanian Academy, Bucharest, Romania, December 16, 2022.

• Laurentiu Maxim. Geometric and topological aspects of optimization.  Applied CATS Seminar at the KTH Royal Institute of Technology, Stockholm, Sweden, December 12, 2022.

• Laurentiu Maxim. On variants of the Singer-Hopf conjecture in complex geometry. Algebraic Geometry Seminar at Humboldt University, Berlin, Germany, November 30, 2022.

• Laurentiu Maxim. On the topology of aspherical projective manifolds and related questions. Arithmetic Geometry Seminar at Humboldt University, Berlin, Germany, November 29, 2022.

• Laurentiu Maxim. On the topology of aspherical projective manifolds and related questions. Joint Seminar on Complex Algebraic Geometry and Complex Analysis (Bochum - Essen - Köln - Münster - Wuppertal), Wuppertal, Germany, November 25, 2022.

• Laurentiu Maxim. On the topology of aspherical projective manifolds and related questions. Algebra and Geometry Seminar at the University of Bologna, Bologna, Italy, November 22, 2022.

• Laurentiu Maxim.  On the topology of aspherical projective manifolds and related questions. University of Regensburg SFB Lecture, Regensburg, Germany, 8 November 2022.

• Laurentiu Maxim. Singularities and Optimization. KU Leuven Number Theory and Algebraic Geometry Seminar, Leuven, Belgium, 19 October 2022.

• Laurentiu Maxim. Singularities and Optimization. MPI-Oberseminar at the Max Planck Institute, Bonn, Germany, 13 October 2022.

• Laurentiu Maxim. Singularities and Optimization. Conference "Singularity theory and its applications'', RIMS, Kyoto University, Kyoto, Japan (hybrid), 3-5 October 2022.

• Laurentiu Maxim.  On the Singer-Hopf conjecture and its variants. University of Heidelberg Topology Seminar, Heidelberg, Germany, 20 September 2022.

• Laurentiu Maxim.  On the Singer-Hopf conjecture and its variants. Institute of Mathematics of the Romanian Academy, Topology Seminar, Bucharest, Romania, 1 July 2022.

• Laurentiu Maxim.  Geometry and Topology of Optimization. Institute of Mathematics of the Romanian Academy, Algebraic Geometry Seminar, Bucharest, Romania, 30 June 2022.

• Laurentiu Maxim.  Geometry and Topology of Optimization. Centro Internazionale per la Ricerca Matematica (CIRM), Trento, Italy, 6 June 2022.

• Laurentiu Maxim.  Geometry and Topology of Optimization. University of Lille Algebraic Geometry Seminar, Lille, France, 24 May 2022.

Visiting scientists / Invitati:

• Alexandru Suciu (Northeastern University, USA), June 12-26, 2022

Scientific report / Raport stiintific:

• Romanian

Activities in 2023 / Activitati in 2023:

Publications / Publicatii:

• Marian Aprodu, Gavril Farkas, Claudiu Raicu, Alexander I. Suciu. Reduced resonance schemes and Chen ranks. preprint arXiv:2303.07855 link

• A. Dimca, G. Ilardi. On the duals of smooth projective complex hypersurfaces, preprint arXiv:2301.06952. Publicacions Matematiques (accepted) link

• A. Dimca, G. Ilardi, P. Pokora, G. Sticlaru. Construction of free curves by adding lines to a given curve. Results in Mathematics 79, Article number: 11 (2024) DOI: 10.1007/s00025-023-02036-9 link

• A. Dimca, P. Pokora, G. Sticlaru. On the Alexander polynomials of conic-line arrangements. preprint arXiv:2305.01450 link

• A. Dimca. On free and plus-one generated curves arising from free curves by addition-deletion of a line. preprint arXiv: 2310.08972. link

• Alexandru Dimca. Unexpected curves in P2, line arrangements, and minimal degree of Jacobian relations. Journal of Commutative Algebra 15 (2023),15–30. DOI: 10.1216/jca.2023.15.15 link

• Sylvain E. Cappell, Laurenţiu Maxim, Jörg Schürmann, Julius L. Shaneson. Equivariant toric geometry and Euler-Maclaurin formulae. preprint arXiv:2303.16785 link

• Laurentiu Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu. Linear optimization on varieties and Chern-Mather classes. preprint arXiv:2208.09073 link

• Laurentiu Maxim, Morihiko Saito, Ruijie Yang. Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities. preprint arXiv:2302.00970 link

• Laurentiu Maxim, Ruijie Yang. Higher Du Bois and higher rational singularities of hypersurfaces. preprint arXiv:2301.09084 link

• Laurentiu Maxim. On singular variants of the Singer-Hopf Conjecture. Mathematische Nachrichten 296 (11) (2023) pp.5232-5241  DOI: 10.1002/mana.202200322 link

• Laurentiu Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu. Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels Involution conjecture. Communications on Pure and Applied Mathematics (published online) DOI:10.1002/cpa.22156 link

• Laurentiu Maxim, Mihai Tibar. Euclidean distance degree and limit points in a Morsification.  Advances in Applied Mathematics 152 (2024) article 102597. DOI: 10.1016/j.aam.2023.102597 link

• Christos Tatakis, Apostolos Thoma, Marius Vladoiu. The Markov complexity of book graphs. Linear Algebra and its Applications 661 (2023) 51–78. DOI: 10.1016/j.laa.2022.12.012 link

• Filip Chindea. ACM bundles on abelian surfaces of Picard rank 4.  Bulletin Mathématique de la Société de Sciences Mathématiques de Roumanie, Tome 66 (114), No. 4, 2023, p. 447-450.

• Dimitra Kosta, Apostolos Thoma, Marius Vladoiu. The strongly robust simplicial complex of monomial curves. preprint arXiv:2305.11743 link

Dissemination / Diseminare:

• Marian Aprodu. Resonance and vector bundles. Algebraic Geometry Seminar Humboldt-Universitaet zu Berlin, May 3, 2023

• Marian Aprodu.  Resonance and vector bundles. A Day of Complex Geometry, University of Bucharest, June 19, 2023.

• Marian Aprodu. Applications of Koszul modules and resonance. Geometry Topology and Algebra of Singular Spaces, IMAR Bucharest, July 7-11, 2023

• Marian Aprodu. Koszul Modules, resonance schemes and applications. Algebraic Geometry in l'Aquila, July 18-21, 2023

• Vasile Brinzanescu. Relating Catlin and D’Angelo q-types. The Tenth Congress of Romanian Mathematicians, June 30 – July 5, 2023, Pitesti, Romania

• Alexandru Dimca. Despre dualele hipersuprafetelor proiective complexe netede, Seminar IMAR, Mat 11, 2023.

• Alexandru Dimca. Despre monodromia aranjamentelor de drepte si conice, Seminar IMAR, May 18, 2023. 

• Laurentiu Maxim. A Homological Interpretation of Higher Du Bois and Higher Rational Singularities. Conference on Periods, Shafarevich Maps and Applications, Institute of the Mathematical Sciences of the Americas, University of Miami, Miami, USA, January 30-February 3, 2023.

• Laurentiu Maxim. A Homological Interpretation of Higher Du Bois and Higher Rational Singularities. Conference on Singularities and Algebraic Geometry’, University of Khanh Hoa, Nha Trang, Vietnam, February 6-10, 2023.

• Laurentiu Maxim. On variants of the Singer-Hopf conjecture in complex geometry. Special Session on the Singer-Hopf Conjecture in Geometry and Topology', at the Spring Southeastern AMS Sectional Meeting, Georgia Tech University, Atlanta, USA, March 18-19, 2023.

• Laurentiu Maxim. Lefschetz properties in the constructible context. Workshop on Lefschetz Properties in Algebra, Geometry, Topology and Combinatorics, Fields Institute, Toronto, Canada, May 15-19, 2023

• Laurentiu Maxim. Cohomology of Z-local systems on complex hyperplane arrangement complements. Topology Seminar at the Institute of Mathematics of the Romanian Academy, Bucharest, Romania, June 9, 2023.

• Laurentiu Maxim. Higher du Bois and higher rational singularities of hypersurfaces. at the Special Session on Algebraic, Complex and Differential Geometry and Topology, at the Tenth Congress of Romanian Mathematicians, Pitesti, Romania, June 30-July 5, 2023.

• Laurentiu Maxim. Chern-Mather classes and optimization. Topology Seminar at the Institute of Mathematics of the Romanian Academy, Bucharest, Romania, August 25, 2023.

• Laurentiu Maxim. Chern-Mather classes and optimization. at the Conference on Characteristic Classes and Singular Spaces, Kiel University, Germany, September 18-22, 2023.

• Dragos Stefan. On the Hochschild cohomology ring. Workshop on Hopf Algebras and Tensor Categories, Marburg, May, 21-26 2023 

Visiting scientists / Invitati:

• Alexander Suciu  (Northeastern University, USA), June 25-July 28, 2023 

• Laurențiu Păunescu (University of Sydney, Australia), June 28-July 15, 2023

• Alexander Hof (University of Wisconsin-Madison, USA), June 29-July 12, 2023

• Laura Costa (Universitat de Barcelona, Spain), November 5-8, 2023

Scientific events / Evenimente stiintifice:

• Geometry, Topology and Algebra of Singular Spaces, IMAR Bucharest, July 7-11, 2023

Scientific report / Raport stiintific:

• Romanian