CNCS-UEFISCDI project PN-III-P4-ID-PCE-2016-0030

Title: Moduli Spaces and Homological Methods in Geometry and Topology

Contract no. 5/19.07.2017

Starting date: July 12, 2017

Ending date: December 31, 2019

Budget: 850.000 RON

Experienced team members:

1. Marian Aprodu

2. Vasile Brînzănescu

3. Alexandru Dimca (hired in June 2018)

4. Laurențiu Maxim

5. Ștefan Papadima*

6. Dragoș Ștefan (hired in December 2017)

7. Marius Vlădoiu (hired in June 2018)

Young team members:

1. Filip Chindea, PhD student

2. Ștefan Deaconu, PhD student (hired in December 2017)

3. Laura Filimon, PhD student (hired in December 2017)

* We sadly announce the passing away of our colleague, Ștefan Papadima, on January 10, 2018.

Short description:

The study of moduli spaces is a ubiquitous trend in modern mathematics, and a difficult subject. Their local structure, alias deformation theory, turns out to be more tractable: deformation functors are usually well–defined, even if the moduli space is not. The same phenomenon appears in deformation theory with (co)homological constraints. This subtler problem appears frequently, when the points P of the moduli space M come with a cohomology theory H. The need to understand the variation in M of H(P) leads to the local analysis of the embedded homology jump loci of M with respect to H. This setup encompasses a rich variety of important situations. The proposed research is an exploration of several problems related to homology jumps loci and geometry of moduli spaces. The research goals are linked to the senior team members’ scientific interests, and rely on a close collaboration between them. The proposed research objectives will enhance our knowledge on syzygy theory and its connections with vector bundles and geometry of projective varieties, will substantially improve the understanding of the fruitful geometry–topology interactions, and will provide a fertile ground for potential applications in field theory and string theory.

Research objectives:

1. Syzygies and the geometry of complex projective varieties

2. Geometry and topology of moduli spaces of objects associated to a variety

3. Jump loci in geometry and topology

Activities in 2017:

Publications:

1. Marian Aprodu, Andrea Bruno, Edoardo Sernesi. A characterization of bielliptic curves via syzygy schemes. Journal of Pure and Applied Algebra 223 (2019) 4690-4700, DOI: 10.1016/j.jpaa.2019.02.01.

2. Vasile Brînzănescu, Andreea C. Nicoară. Relating Catlin and D'Angelo q-types. preprint arxiv:1707.08294v2, submitted manuscript.

3. Filip Chindea. ACM line bundles on elliptic ruled surfaces. Manuscripta Mathematics 161, (2020) 213–222, DOI: 10.1007/s00229-018-1082-4 .

4. Yongqiang Liu, Laurențiu Maxim, Botong Wang. Topology of subvarieties of complex semi-abelian varieties. International Mathematics Research Notices (published online); DOI: 10.1093/imrn/rnaa242.

5. Ștefan Papadima, Alexander I. Suciu. Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds. to appear in Journal of the Institute of Mathematics of Jussieu, DOI:10.1017/S1474748018000063.

6. Ștefan Papadima, Alexander I. Suciu. Naturality properties and comparison results for topological and infinitesimal embedded jump loci. Advances in Mathematics 350 (2019), 256-303, DOI: 10.1016/j.aim.2019.04.058.

Dissemination:

1. Marian Aprodu. Mini-course on Syzygies, three lectures at 25th National School on Algebra, September 3-9, 2017, FMI Bucuresti 2017

2. Marian Aprodu. Green’s conjecture and vanishing of Koszul modules, NoGAGS 2017 Conference, Humboldt Univ. Berlin, November 16-17, 2017

3. Marian Aprodu. Anularea modulelor Koszul si aplicatii, IMAR joint Algebraic Geometry - Topology seminar, December 7, 2017

4. Filip Chindea. ACM line bundles on elliptic ruled surfaces. 25th National School on Algebra, September 3-9, 2017, FMI Bucuresti 2017

5. Laurențiu Maxim. Stratified Morse theory: an overview. two lectures in the Topology and Singularities Seminar, University of Wisconsin-Madison, November 15, November 29, 2017.

Scientific report:

• romanian

• english

Activities in 2018:

Foreign visitors:

• Preda Mihăilescu, Goettingen U., Germany (March 2018)

• Cai Tianxin, Zhejiang U., China (March 2018)

• Cristian Vay, U. Clermont-Ferrand, France (March 2018)

• Gavril Farkas, HU Berlin, Germany (May 2018)

Publications:

1. Marian Aprodu, Gavril Farkas, Ștefan Papadima, Claudiu Raicu, Jerzy Weyman. Topological invariants of groups and Koszul modules. preprint arXiv:1806.01702.

2. Marian Aprodu, Gavril Farkas, Ștefan Papadima, Claudiu Raicu, Jerzy Weyman. Koszul modules and Green's conjecture. Inventiones Mathematicae, Volume 218, Issue 3 (2019) 657-720, DOI:10.1007/s00222-019-00894-1

3. Alexandru Dimca, Gabriel Sticlaru. Plane curves with three syzygies, plus-one generated curves, and nearly cuspidal curves. preprint arXiv:1810.11766, to appear in Geometriae Dedicata.

4. Yongqiang Liu, Laurențiu Maxim, Botong Wang. Mellin transformation, propagation, and abelian duality spaces. Advances in Mathematics 335 (2018), 231 - 260, DOI:10.1016/j.aim.2018.07.012.

5. Yongqiang Liu, Laurențiu Maxim, Botong Wang. Generic vanishing for semi-abelian varieties and integral Alexander modules. Mathematische Zeitschrift 293 (2019) 629–645, DOI: 10.1007/s00209-018-2194-y.

6. Laurențiu Maxim, Morihiko Saito, Joerg Schuermann. Spectral Hirzebruch-Milnor classes of singular hypersurfaces. to appear in Mathematische Annalen (published online), DOI:10.1007/s00208-018-1750-4.

7. Laurențiu Maxim, Shoji Yokura. Homological congruence formulae for characteristic classes of singular varieties. European Journal of Mathematics, Volume 5, Issue 4 (2019) 1267–1292, DOI:10.1007/s40879-018-0305-2.

Dissemination:

1. Marian Aprodu. Green’s conjecture and vanishing of Koszul modules. KU Leuven, Number Theory and Algebraic Geometry Seminar, April 25, 2018.

2. Marian Aprodu, Green’s conjecture and vanishing of Koszul modules. Moduli Spaces in Algebraic Geometry and Applications, Campinas, Bresil, July 26 – 31, 2018.

3. Marian Aprodu. Modules de Koszul et syzygies de courbes rationnelles cuspidales. Inst. Math. Jussieu, Paris, November 15, 2018.

4. Marian Aprodu, Koszul modules and syzygies of rational cuspidal canonical curves. Algebraic Geometry Conference, Haeundae, South Korea, November 31 – December 1, 2018.

5. Marian Aprodu. Koszul modules and syzygies of rational cuspidal canonical curves. KIAS-KAIST Workshop in Mathematics, Seoul, South Korea, December 10, 2018.

6. Vasile Brînzănescu. Co-Higgs bundles over ruled surfaces. Topology and Geometry: a Conference in Memory of Ștefan Papadima, Bucharest May 28 - 31, 2018.

7. Filip Chindea. The geometry of derived categories of K3s. Workshop for Young Researchers in Mathematics – Bucharest, Romania, May 17 - 18, 2018.

8. Alexandru Dimca. Free curves, nearly free curves and plus-one generated curves. Algebraic Geometry Seminar, IMAR, November 1, 2018.

9. Laurențiu Maxim. Fundamental groups of quasiprojective manifolds. Georgia Southern University Colloquium – Statesboro, Georgia, USA, March 9, 2018.

10. Laurențiu Maxim. Mellin transformation, propagation and abelian duality. “A day of singularities” – Northeastern University, Boston, New England, USA, April 20, 2018.

11. Laurențiu Maxim. Topology of very affine manifolds. Special Session on “Singularities of Spaces and Maps” at the AMS Spring Eastern Sectional Meeting – Boston, New England, USA, April 21 - 22, 2018.

12. Laurențiu Maxim. Topology of very affine manifolds. Workshop for Young Researchers in Mathematics – Bucharest, Romania, May 17 - 18, 2018.

13. Laurențiu Maxim. Mellin transformation, propagation and abelian duality. Topology and Geometry: a Conference in Memory of Ștefan Papadima, Bucharest May 28 - 31, 2018.

14. Laurențiu Maxim. Perverse Sheaves on Semi-abelian Varieties: Structure and Applications. MPS Conference on Singularities: Geometric, Topological, and Analytic Aspects – Simons Institute, New York City, New York, USA, August 13 - 17, 2018.

15. Dragoș Ștefan. On the Poincaré- Birkhoff-Witt Theorem. Seminari di Algebra e Geometria Algebrica, U. Torino, Italy, May 31, 2018.

16. Marius Vlădoiu. Hypergraph encodings of arbitrary toric ideals. Commutative Algebra Seminar, Purdue University, USA, September 19, 2018.

17. Marius Vlădoiu. Hypergraph encodings of arbitrary toric ideals. Algebraic Geometry and Commutative Algebra Seminar, Notre Dame University, USA, October 31, 2018.

Other achievements:

The paper Koszul modules and Green's conjecture, by Marian Aprodu, Gavril Farkas, Ștefan Papadima, Claudiu Raicu, and Jerzy Weyman will be the subject of a talk given by Rob Lazarsfeld (Stony Brook U., SUA) at the AMS Current Events Bulletin. According to the official site:

"The Current Events Bulletin Session at the Joint Mathematics Meetings, begun in 2003, is an event where the speakers do not report on their own work, but survey some of the most interesting current developments in mathematics, pure and applied. The wonderful tradition of the Bourbaki Seminar is an inspiration, but we aim for more accessible treatments and a wider range of subjects."

http://www.ams.org/meetings/lectures/current-events-bulletin

Scientific report:

• romanian

• english

Activities in 2019:

Foreign visitors:

• Andrea Bruno, Univ. Roma Tre, Italy (October 2019)

• Edoardo Sernesi, Univ. Roma Tre, Italy (October 2019)

Publications:

1. Ștefan Papadima, Alexander I. Suciu. Infinitesimal finiteness obstructions. Journal of the London Mathematical Society (2) 99 (2019) 173-193, DOI: 10.1112/jlms.12169.

2. Alexandru Dimca, Gabriel Sticlaru. Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves. Journal of Pure and Applied Algebra 223 (2019), no. 12, 5055--5066, DOI: 10.1016/j.jpaa.2019.03.009.

3. Sonja Petrović, ApostolosThoma, Marius Vladoiu. Hypergraph encodings of arbitrary toric ideals. Journal of Combinatorial Theory, Series A 166 (2019) 11–41, DOI: 10.1016/j.jcta.2019.02.017.

4. Marian Aprodu, Gianfranco Casnati, Laura Costa, Rosa Maria Miro-Roig, Monserrat Teixidor i Bigas. Theta divisors and Ulrich bundles on geometrically ruled surfaces. Annali di Matematica Pura ed Applicata, Volume 199 (2020) 199–216, DOI: 10.1007/s10231-019-00873-6.

5. Marian Aprodu, Sukmoon Huh, Francesco Malaspina, Joan Pons-Llopis. Ulrich bundles on smooth projective varieties of minimal degree. Proceedings of the American Mathematical Society, Volume 147, Number 12, (2019) 5117–5129, DOI:10.1090/proc/14640.

6. Alexandru Dimca. On the minimal value of global Tjurina numbers for line arrangements. European Journal of Mathematics 6 (2020) 817-828, DOI: 10.1007/s40879-019-00373-0.

7. Marian Aprodu, Yeongrak Kim. On the Borisov-Nuer conjecture and the image of the Enriques-to-K3 map. Mathematische Nachrichten, Volume 293, Issue 6 (2020) 1044-1052, DOI:10.1002/mana.201900226.

8. Yongqiang Liu, Laurentiu Maxim, Botong Wang. Perverse sheaves on semi-abelian varieties - a survey of properties and applications. European Journal of Mathematics 6 (2020) 977-997, DOI: 10.1007/s40879-019-00340-9.

9. Alexandru Dimca, Rodrigo Gondim, Giovanna Ilardi. Higher order Jacobians, Hessians and Milnor algebras. Collectanea Mathematica 71, (2020) 407-425, DOI: 10.1007/s13348-019-00266-1.

10. Laurentiu Maxim, Jose Rodriguez, Botong Wang. Euclidean distance degree of the multiview variety. SIAM Journal on Applied Algebra and Geometry, 4(1) (2020) 28-48, DOI: 10.1137/18M1233406.

11. Laurentiu Maxim, Jose Rodriguez, Botong Wang. Euclidean distance degree of projective varieties. International Mathematics Research Notices rnz266 (2019), DOI: 10.1093/imrn/rnz266.

12. Eva Elduque, Laurentiu Maxim. Higher order degrees of affine plane curve complements. Indiana University Mathematics Journal 70 (2021), no. 1, 179-211; DOI: 10.1512/iumj.2021.70.8297.

13. Alexandru Dimca, Gabriel Sticlaru. On the jumping lines of bundles of logarithmic vector fields along plane curves. Publicacions Mathematiques 64(2): 513-542 (2020). DOI: 10.5565/PUBLMAT6422006.

14. Alexandru Dimca, Gabriel Sticlaru. Jacobian syzygies and plane curves with maximal global Tjurina numbers. preprint arXiv: 1901.05915.

15. Alexandru Dimca, Gabriel Sticlaru. Waring rank of symmetric tensors, and singularities of some projective hypersurfaces. preprint arXiv: 1902.01351.

16. Alexandru Dimca. Versality, bounds of global Tjurina numbers and logarithmic vector fields along hypersurfaces with isolated singularities. preprint arXiv:1904.00686.

17. Alexandru Dimca, Gabriel Sticlaru. The 0-th Fitting ideal of the Jacobian module of a plane curve. preprint arXiv: 1904.07823.

18. Armando Cerminara, Alexandru Dimca, Giovanna Ilardi. On the Hilbert vector of the Jacobian module of a plane curve. preprint arXiv:1905.04055.

19. Alexandru Dimca. On complex supersolvable line arrangements. preprint arXiv: 1907.12497.

20. Takuro Abe, Alexandru Dimca, Gabriel Sticlaru. Addition-deletion results for the minimal degree of logarithmic derivations of arrangements. preprint arXiv:1908.06885.

21. Laurentiu Maxim, Jose Rodriguez, Botong Wang. Defect of Euclidean distance degree. preprint arXiv:1905.06758.

22. Alessandro Ardizzoni, Paolo Saracco, Dragoș Ștefan. PBW-deformations of graded rings. submitted manuscript.

23. Filip Chindea. ACM bundles on abelian surfaces of Picard rank 4. submitted manuscript.

24. Vasile Brînzănescu, C.-M. Sandu. Algebraic Methods for Solutions of Bloch-Iserles Hamiltonian Systems. in preparation.

25. Apostolos Thoma, Marius Vlădoiu. The bouquet toric algebra and the Markov complexity of Z^{mxn} matrices. in preparation.

26. Apostolos Thoma, Marius Vlădoiu. The defining matrices of self-dual projective toric varieties. in preparation.

27. Abdenacer Makhlouf, Dragoș Ștefan. On deformations of algebraic structures. to appear in Israel Journal of Mathematics.

Books:

1. Laurentiu Maxim. Intersection Homology & Perverse Sheaves, with Applications to Singularities. Graduate Texts in Mathematics, Vol. 281, Springer, 2019; ISBN: 978-3030276430. https://www.springer.com/us/book/9783030276430

Dissemination:

1. Marian Aprodu. Koszul modules and Green's conjecture. Algebraic Geometry Seminar, TU Muenchen, Germany, May 16, 2019.

2. Marian Aprodu. Koszul Modules and Applications. IMAR70 Conference, Bucharest, October 4-5, 2019.

3. Marian Aprodu. Koszul Modules and Applications. International Conference on APPLIED AND PURE MATHEMATICS, Iaşi, Romania, October 31 - November 3, 2019.

4. Marian Aprodu. Green’s conjecture and vanishing of Koszul modules. Workshop on Algebraic Geometry, KIAS Seoul, South Korea, 19-21 November 2019.

5. Marian Aprodu. Green’s conjecture and vanishing of Koszul modules. Hyperplane Arrangements and Singularities (Hyper-JARCS), University of Tokyo, Japan, December 2-6, 2019.

6. Alexandru Dimca. On the global Tjurina number of complex projective plane curves. Algebraic Geometry Seminar, University of Genoa, Italy, February 28, 2019.

7. Alexandru Dimca. On the Jacobian syzygies. "Hyperplane Arrangements: Recent Advances and Open Problems" Conference, Institute of Mathematics of Hanoi, Vietnam, March 11-23, 2019.

8. Alexandru Dimca. On the Waring decomposition of tensors. "Hyperplane Arrangements: Recent Advances and Open Problems" Conference, Institute of Mathematics of Hanoi, Vietnam, March 11-23, 2019.

9. Alexandru Dimca. On the generators of the module of Jacobian syzygies. "Real and complex singularities" Conference, Cargèse, France, April 29-May 3, 2019.

10. Alexandru Dimca. Free plane curves. "Graded modules over polynomial rings with applications to free divisors" 27th National School on Algebra, Bucharest, May 20-24, 2019.

11. Alexandru Dimca. Nearly free curves versus rational cuspidal curves. "Graded modules over polynomial rings with applications to free divisors" 27th National School on Algebra, Bucharest, May 20-24, 2019.

12. Alexandru Dimca. On supersolvable line arrangements. "Hyperplane Arrangements and Reflection Groups" Conference, Hannover, Germany, September 23-27, 2019.

13. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. The 14-th Kagoshima Algebra-Analysis-Geometry Seminar, Kagoshima, Japan, February 14, 2019.

14. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. Geometry Seminar, Vanderbilt University, Nashville, USA, February 22, 2019.

15. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. NYU - Shanghai Colloquium, Shanghai, China, March 11, 2019.

16. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. Center for Applied Mathematics, Tianjin University, Tianjin, China, April 25, 2019.

17. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. Courant Institute Colloquium, New York University, New York, USA, April 29, 2019.

18. Laurențiu Maxim. Measuring the complexity of hypersurface singularities in algebraic geometry and topology. Basque Center for Applied Mathematics, Bilbao, Spain, May 21, 2019.

19. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. Soochow University, Suzhou, China, June 4, 2019.

20. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. Algebraic Geometry Seminar, East China Normal University, Shanghai, China, June 6, 2019.

21. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. Algebraic Geometry Seminar, Institute of Mathematics of the Romanian Academy, Bucharest, Romania, June 13, 2019.

22. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. “Singularity Workshop”, University of Illinois at Chicago, Chicago, Illinois, July 28-29, 2019.

23. Laurențiu Maxim. Euclidean distance degree and the multiview conjecture. "Non-Isolated Singularities and derived Geometry: A celebration of the 60th birthday of David Massey”, Conference, Cuernavaca, Mexico, July 29-August 3, 2019.

24. Laurențiu Maxim. Measuring the complexity of hypersurface singularities in algebraic geometry and topology. "Conference on Hodge Theory in Honor of Donu Arapura’s 60th Birthday”, Madison, USA, September 13, 2019.

25. Vasile Brînzănescu. Algebraic methods for solutions of Bloch-Iserles Hamiltonian Systems. Conference "Transient Transcendence in Transylvania", Brașov, May 13-17, 2019.

26. Vasile Brînzănescu. Moduli spaces of vector bundles. The Ninth Congress of Romanian Mathematicians, Galați "Dunărea de Jos" University, June 28 - July 3, 2019.

27. Vasile Brînzănescu. Moduli of vector bundles as integrable Hamiltonian Systems. Bucharest Conference on Geometry and Physics, IMAR Bucharest, September 2-6, 2019.

28. Vasile Brînzănescu. Algebraic methods for solutions of Bloch-Iserles Hamiltonian Systems. The Third Romanian-Turkish Mathematics Colloquium, Ovidius University of Constanța, September 18-22, 2019.

29. Filip Chindea. Stability conditions on K3 surfaces. "Graded modules over polynomial rings with applications to free divisors" 27th National School on Algebra, Bucharest, May 20-24, 2019.

30. Marius Vlădoiu. Strongly robust toric ideals. Algebra Seminar, Georgia State University, Atlanta, USA, February 22, 2019.

31. Marius Vlădoiu. Toric ideals through applications. Colloquium Seminar, Georgia State University, Atlanta, USA, February 22, 2019.

32. Marius Vlădoiu. Strongly robust toric ideals. Algebra and Combinatorics Seminar, Tulane University, New Orleans, USA, March 14, 2019.

33. Marius Vlădoiu. Hypergraph encodings of arbitrary toric ideals. Discrete Mathematics Seminar, University of Delaware, Newark, USA, April 16, 2019.

34. Marius Vlădoiu. Minimal graded free resolutions (I). "Graded modules over polynomial rings with applications to free divisors" 27th National School on Algebra, Bucharest, May 20-24, 2019.

35. Marius Vlădoiu. Minimal graded free resolutions (II). "Graded modules over polynomial rings with applications to free divisors" 27th National School on Algebra, Bucharest, May 20-24, 2019.

Other achievements:

Laurențiu Maxim was awarded the "Van Vleck Prize" by the University of Wisconsin, for excellence in research and promise of significant future achievement.

Scientific report:

• romanian

• english