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Titlul proiectului: PROBLEME RECENTE IN TEORIA SUBVARIETATILOR

Tipul proiectului: Proiecte de cercetare pentru tineri cercetatori



Echipa de cercetare a FMI este formata din:

 Director de proiect:
Asist.dr. Adela Mihai

 Assist.prof.dr. Bogdan Suceava ( )
 Prep.drd. Valentin Ghisoiu
 Drd. Simona Decu ( )

Tema face parte din Programul National de Cercetare de Excelenta (CEEX 2005) Modulul II - Proiecte de Dezvoltare a Resurselor Umane pentru Cercetare.

Teoria invariantilor Chen a fost initiata de B.-Y. Chen in 1993. Prima inegalitate a lui Chen furnizeaza o noua conditie necesara ca o varietate riemanniana sa admita o imersie minimala intr-un spatiu euclidian. Ulterior, el a introdus "Δ"-invariantii, cunoscuti azi ca invariantii Chen si a demonstrat inegalitati optime pentru acestia (invarianti intrinseci) in functie de principalul invariant extrinsec (curbura medie).

Mai tarziu s-au obtinut inegalitati de tip Chen si pentru alti invarianti intrinseci, de exemplu: curbura scalara, curbura Ricci, etc.. Aceasta teorie a cunoscut o larga dezvoltare in Belgia, Brazilia, Franta, Japonia, Polonia, Romania, SUA, etc..

Ne propunem sa gasim noi aplicatii ale inegalitatilor lui Chen, in particular obstructii topologice si diferentiale la diverse clase de subvarietati in forme spatiale complexe si respectiv Sasaki.

De asemenea vom construi invarianti specifici pentru anumite clase de spatii Riemann, vom obtine estimari optime ale acestora in functie de curbura medie si vom investiga cazul de egalitate.

In particular, vom continua studiul produselor warped in forme spatiale reale si complexe si avem in vedere demonstrarea de inegalitati optime intre invarianti intrinseci (ca de exemplu functia warping) si invarianti extrinseci.

Pentru ca studiul subvarietatilor puternic minimale in forme spatiale complexe sa fie complet, este important sa furnizam noi exemple de hipersuprafete complexe ce satisfac global conditia de minimalitate puternica. In prezent nu exista nici un articol dedicat acestui concept in generalitatea sa. Constructia unor astfel de hipersuprafete de dimensiune mai mare sau egala cu 3 este un alt obiectiv al acestui proiect.




Algebre Hopf si teme inrudite
Hopf algebras and related topics

  

Grant no. 88/05.10.2011 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI - PN II Idei

The Team:

  • Prof. Dr. Gigel Militaru (the team leader of the project),
  • CS III Sebastian Burciu,
  • CS III Bogdan Ion,
  • AC Ana-Loredana Agore (PhD student - PhD since 1/10/2012),
  • AC Costel Gabriel Bontea (PhD student).

Project Summary:

Hopf algebras were introduced around 1960 arising from algebraic geometry and algebraic topology. The second important development within this field of study started in 1987 with the appearance of the paper Quantum groups by V. Drinfel"d. The project will be focused on both parts (classic and quantum) of the theory of Hopf algebras and has two corresponding general objective each of them with several open problems. Objective I: New structural results for Hopf algebras. There are four open problems related to this objective. Objective II: Categorical methods in quantum groups. Braided, monoidal and fusion categories. There are five open problems related to the second objective.

Annual scientific reports (in romanian): 2011, 2012

Actual annual expenses: 2011, 2012

Published/Accepted papers in ISI journals (the impact factor for 2011 is given):

  1. S. Burciu, S. Natale - Fusion rules of equivariantizations of fusion categories, Journal of Mathematical Physics, 54(2013), 013511; DOI: 10.1063/1.4774293 (IF: 1.291). Link
  2. A. L. Agore, G. Militaru - Classifying complements for Hopf algebras and Lie algebras, Journal of Algebra, 391(2013), 193-208 (IF: 0,613) Link
  3. A. L Agore – Coquasitriangular structures for extensions of Hopf algebras, Glasgow Math. J. 55 (2013), 1-15. (IF: 0.571)
  4. A. L. Agore, C. G. Bontea, and G. Militaru – Classifying coalgebra split extensions of Hopf algebras, J. Algebra Appl. 12 (2013), 1-24. (IF: 0.483)
  5. A. L. Agore, G. Militaru – Unified products and split extensions of Hopf algebras, Contemporary Math. AMS, Vol. 585 (2013), 1-15.
  6. A. L. Agore, S. Caenepeel, G. Militaru – The center of the category of bimodules and descent data for non-commutative rings, J. Algebra Appl. 11 (2012), 1-17. (IF: 0.483)
  7. G. Militaru – Representable functors for corings, Comm. Algebra 40 (2012), no.5, 1766-1796. (IF: 0.347)
  8. A. L. Agore and G. Militaru – Schreier type theorems for bicrossed products, Cent. Eur. J. Math. 10 (2012), no.2, 722-739. (IF: 0.440)
  9. A. L. Agore, G. Militaru – Extending structures for Lie algebras, accepted for publications in Monatshefte für Mathematik (IF: 0,616)
  10. A. L. Agore, G. Militaru – Classifying complements for Hopf algebras and Lie algebras, accepted for publication in Journal of Algebra, DOI: 10.1016/j.jalgebra.2013.06.012 (IF: 0,613)
  11. A.L. Agore, S. Caenepeel, G. Militaru – Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings, in press Applied Cat. Structures – DOI: 10.1007/s10485-012-9294-3. (IF: 0.600)
  12. A.L. Agore, G. Militaru – Extending structures I: the level of groups, in press, Algebr. Represent. Theory, DOI: 10.1007/s10468-013-9420-4. (IF: 0.595)
  13. A. L. Agore, C. G. Bontea, G. Militaru – Classifying bicrossed products of Hopf algebras, in press Algebr. Represent. Theory, DOI:10.1007/s10468-012-9396-5. ( IF: 0.595).
  14. S. Burciu – Kernels of representations of Drinfel’d doubles of finite groups, accepted for publications in Cent. Eur. J. Math. (IF: 0.440)
  15. S. Burciu – Subgroups of odd depth – a necessary condition, accepted for publications in Czechoslovak J. Math. (IF: 0.262)
  16. C. G. Bontea – Classifying bicrossed products of two Sweedler.s Hopf algebras, accepted for publications in Czechoslovak J. Math. (IF: 0.262)

Papers submitted for publication:

  1. S. Burciu – On coideal subalgebras of cocentral Kac algebras and a generalization of Wall’s conjecture, submitted.
  2. S. Burciu – On the irreducible representations of generalized quantum doubles, submitted
  3. A. L. Agore, G. Militaru – Classifying complements for groups. Applications, submitted
  4. A. L. Agore, C. G. Bontea, and G. Militaru – The classification of all crossed products H4 # k[Cn], submitted
  5. A. L. Agore, G. Militaru – The extending structures problem for algebras, submitted.
  6. V. Chari, B. Ion – BGG reciprocity for current algebras, submitted.

Dissemination of the scientific results. Talks given at international conferences:

  1. G. Militaru - Classifying bicrossed products of quantum groups, Algebra Geometry Mathematical Physics, Brno, september 2012.
  2. G. Militaru - Extending structures: the level of groups, Groups and their actions, Bedlewo, July 2012.
  3. A.L. Agore - Bicrossed descent theory of exact factorizations and the number of types of groups of finite order , Groups and their actions, Bedlewo, July 2012
  4. C. Bontea - Classifying crossed product of quantum groups, Algebra Geometry Mathematical Physics(AGMP), Brno, September 2012.
  5. A. L. Agore - Deformations and descent type theory for Hopf algebras, Algebra Geometry Mathematical Physics, Brno, September 2012.
  6. G. Militaru - Classifying bicrossed products. Deformations and descent type theory for quantum groups, Hopf Algebra Workshop, Brussel VUB, 19 March 19, 2012.
  7. A.L. Agore - Classifying exact factorizations of a group through a given subgroup and the number of types of groups of finite order, 2nd Biennial Groups Theory Conference, Dogus University, Istanbul, February 2013.
  8. A.L. Agore - The factorization problem for Lie algebras, International workshop Lie theory and its applications, Varna, Bulgaria, June 2013.
  9. A.L. Agore - Exact factorizations for Hopf algebras. Applications, Joint international meeting of AMS and RMS , Alba-Iulia, Romania, June 2013.
  10. A.L. Agore - Bicrossed products of Hopf algebras, Workshop for young researchers, Constanta, Romania, May 2013.
  11. A.L. Agore - Classifying complements for Hopf algebras, Lie algebras and groups, Workshop "Rings, categories and Hopf algebras", Bucharest, May 2013.
  12. C.G. Bontea - The factorization problem for Hopf algebras, Workshop "Rings, categories and Hopf algebras", Bucharest, May 2013.
  13. G. Militaru - The Extending Structures Problem, 2nd Biennial Groups Theory Conference, Dogus University, Istanbul, February, 2013
  14. G. Militaru - The extending structures for algebras, Workshop "Rings, categories and Hopf algebras", Bucharest, May 2013.
  15. G. Militaru - Extending structures for Lie algebras, International workshop Lie theory and its applications, Varna, Bulgaria, June 2013.
  16. G. Militaru - The extension problem for Hopf algebras, Joint international meeting of AMS and RMS, Alba-Iulia, Romania, June 2013.
  17. S. Burciu - Kernels of representations and coideal subalgebras of Hopf algebras Workshop "Rings, Categories and Hopf Algebras", May 18-19, 2013, Bucharest.

2002 - 2015 -- Facultatea de Matematica si Informatica, Universitatea din Bucuresti
Str. Academiei nr. 14, sector 1, C.P. 010014, Bucuresti, Romania
Tel: (4-021) 314 2863, Fax: (4-021) 315 6990, secretariat fmi.unibuc.ro