ATMW Singularity Categories in Algebraic Geometry and Commutative Algebra (2013)
Venue: | IIT Madras |
Dates: | 2nd to 12th Jan, 2013 |
Convener(s) | Speakers, Syllabus and Time table | Applicants/Participants |
Name | T.E.Venkata Balaji | Suresh Nayak | Srikanth Iyengar |
Mailing Address | Dept of Maths, IIT-Madras,Chennai 600036, India |
ISI Bangalore, India | Dept of Maths, University of Nebraska-Lincoln, USA |
Schedule:
9:00–9:50 | 10:00–10:50 | 11:30–12:20 | 2:30–4:30 | |
Wed, Jan. 2 | Lecture 1
Manoj |
Lecture 2
Manoj |
Lecture 3
Suresh |
Tutorial
Krishna, Manoj, Srikanth |
Thu, Jan. 3 | Lecture 4
Manoj |
Lecture 5
Suresh |
Lecture 6
Pramath |
Tutorial
Sukhendu, Manoj, Srikanth |
Fri, Jan. 4 | Lecture 7
Srikanth |
Lecture 8
Srikanth |
Lecture 9
Suresh |
Tutorial
TEV, Krishna, Ajay |
Sat, Jan. 5 | Lecture 10
Manoj |
Lecture 11
Srikanth |
Lecture 12
Srikanth |
Tutorial
Krishna, Suresh, Ajay |
Sun, Jan. 6 | Free day | |||
Mon, Jan. 7 | Lecture 13
Suresh |
Lecture 14
Sukhendu |
Lecture 15
Sukhendu |
Tutorial
Krishna, Manoj, Sukhendu |
Tue, Jan. 8 | Lecture 16
Pramath |
Lecture 17
Ajay |
Lecture 18
Sukhendu |
Tutorial
Krishna, Krishna, Ajay |
Wed, Jan. 9 | Lecture 19
Pramath |
Lecture 20
Sukhendu |
Lecture 21
Ajay |
Tutorial
Sukhendu, Manoj, TEV |
Thu, Jan. 10 | Lecture 22
Pramath |
Lecture 23
Ajay |
Lecture 24
Ajay |
Tutorial
TEV, Manoj, Krishna |
Workshop:
Participants will arrive on 1st January, 2013 a Tuesday. The workshop runs from 2nd January through the afternoon of 10th January, the following Thursday. There will be no lectures on 6th January.
A tentative plan for the lectures is as follows:
Wednesday, 2nd January.
1. Dimension and depth for modules over commutative rings; Gorenstein rings.
References: Bruns and Herzog[1], Matsumura [12]
2. Sheaf theory and affine schemes
References: [5]
3. Triangulated categories: definitions and basic properties
References: Neeman [14],
Exercise session: Krull-Remak-Schmidt
Thursday, 3rd January.
4. Coherent sheaves on projective schemes
References: Serre [5, 19]
5. Thick subcategories and quotient categories (abelian and triangulated version); examples (homotopy categories and derived categories)
References: [10, 14]
6. Derived category of a ring and of a scheme
References: [5, 7]
Friday, 4th January.
7. Stable category of MCM modules over Gorenstein rings
References: Buchweitz [2], Herzog [6]
8. MCM modules over hypersurfaces and matrix factorizations
References: Eisenbud [4]
9. Cech cohomology, computations for P^{n} ; Serre duality
References: Serre [19]
Saturday, 5th January.
10. Hypersurfaces of finite representation type
References: Buchweitz, Greuel, and Schreyer [3]
11. Kn ̈rrer periodicity - I
12. Kn ̈rrer periodicity - II
References: Kn ̈rrer [8]
Exercise session: ADE singularities
Monday, 7th January.
13. Derived functors in geometry; Grothendieck duality
References: Huybrechts [7]
14. Singularity categories in geometry; Zariski localization
References: Orlov [15]
15. Geometric version of Kn ̈rrer periodicity
References: Orlov [15]
Tuesday, 8th January.
16. Graded matrix factorizations and graded singularity categories.
References: Orlov [16]
17. Semi-orthogonal decompositions; the derived category of projective space bundles.
References: Huybrechts [7]
18. The derived category of a projective hypersurface and the singularity category of its affine cone - I
Exercise session: the quotient of the category of graded modules by torsion modules, and coherent sheaves on projective space.
Wednesday, 9th January.
19. Duality for singularity categories - I
References: Murfet [13]
20. The derived category of a projective hypersurface and the singularity category of its affine cone - II
References: Orlov [16]
21. Riemann-Roch for singularity categories - I
Thursday, 10th January.
22. Duality for singularity categories - II
References: Murfet [13]
23. Riemann-Roch for singularity categories - II
24. Riemann-Roch for singularity categories - III
Conference Schedule (11th and 12th January 2013)
9:00–9:50 | 10:00–10:50 | 11:30–12:20 | 2:30–3:20 | 3:30-4:20 | |
Fri, Jan. 11 |
Krause | Singh | Srinivas | Mallick | Dubey |
Sat, Jan. 12 |
Krause | Dell’Aquila | Srinivas | Dubey | - |
Umesh Dubey, TIFR First lecture: Cohen-Macaulay representation type of singularities In this talk we will survey some results describing a coarse classiﬁcation of special classes of singularities. We will also mention Kahn correspondence between CM modules and vector bundles on curves which is used in such classiﬁcations. Second lecture: Localisation theorems in Algebraic Geometry Localisation theorems are important tool in computation of algebraic K-groups proved by Quillen for regular schemes. This was generalised to more general schemes by Thomason and later by Neeman in more categorical setting. Aim of this talk is to describe these results and mention some other localisation theorems proved using theorem of Neeman. If time permits we will mention our |
Henning Krause, University of Bielefeld First lecture: The stable derived category of a noetherian scheme Second lecture: Cohomological length functions |
Vivek Mallick, ISER, Pune. Computing Balmer spectrum: An example In this talk we compute the spectrum of the derived category of G-equivariant coherent sheaves on a projective variety. If time permits we shall also talk about the derived category of coherent sheaves over super-schemes. This is joint work with Dr. Umesh Dubey. |
Anurag Singh, University of Utah The F-pure threshold of a Calabi-Yau hypersurface The F-pure threshold is a numerical invariant of prime characteristic singularities. It constitutes an analogue of a numerical invariant for complex singularities—the log canonical threshold—that measures local integrability. We will discuss the calculation of F-pure thresholds of elliptic curves, and also indicate how this calculation extends to Calabi-Yau hypersurfaces. This is work in progress with Bhargav Bhatt. |
Vasudevan Srinivas, TIFR Introduction to Grothendieck-Witt groups, after Balmer |
References
[1] W. Bruns, J. Herzog, Cohen-Macaulay rings, Revised edition, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1998.
[2] R.-O. Buchweitz, Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Ring, http://hdl.handle.net/1807/16682
[3] R.-O. Buchweitz, G.-M. Greuel, , F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math. 88 (1987), 165–182.
[4] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.
[5] R. Hartshorne, Algebraic Geometry, Graduate Texts Math. 52, Springer-Verlag, New York, 1977.
[6] J. Herzog, Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen–Macaulay–Moduln, Math. Ann. 233 (1978), 21–34.
[7] D. Huybrechts, Fourier-Mukai transforms in Algebraic Geometry, Oxford Mathematical Monographs (2006).
[8] H. Kn ̈rrer, Cohen-Macaulay modules on hypersurface singularities. I. 88 (1987), 153–164.
[9] H. Krause, The stable derived category of a Noetherian scheme, Compositio Math. 141 (2005), 1128–1162.
[10] H. Krause, Derived categories, resolutions, and Brown representability, Interactions between homotopy theory and algebra (Chicago, 2004), Contemp. Math. 436, Amer. Math.
Soc., Providence, RI 2007; 101–139.
http://front.math.ucdavis.edu/0511.5047
[11] G. Leuschke, R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys and Monographs 181, Amer. Math. Soc., Providence, RI 2012.
[12] H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, 1986.
[13] D. Murfet, Residues and duality for singularity categories of isolated Gorenstein singularities, preprint 2010. front.math.ucdavis.edu/0912.1629
[14] A. Neeman, Triangulated categories, Annals of Math. Studies 148, Princeton Univ. Press, Princeton, NJ, 2001.
[15] D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), 227–248.
http://front.math.ucdavis.edu/0302.5304
[16] D. Orlov, Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities; http://front.math.ucdavis.edu/0503.5632
[17] D. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sb. Math. 197 (2006), 1827–1840.
http://front.math.ucdavis.edu/0503630
[18] D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226 (2011), 206–217.
http://front.math.ucdavis.edu/0901.1859
[19] J.-P. Serre, Faisceaux alg ́briques coh ́rents, Ann. of Math. 61 (1955), 197–278.
[20] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.
Sr | Full name | Affiliation | Position |
1 | Mr Narasimha Chary B | CMI, Chennai | PhD student |
2 | Mr Saravanan Raja | Periar University, Salem | PhD student |
3 | Mr Umamaheswaran Arunachalam | Periar University, Salem | PhD student |
4 | Mr Udhayakumar Ramalingam | Periar University, Salem | PhD student |
5 | Mr. Krishanu Dan | IMSc, Chennai | PhD student |
6 | Mr. Anjan Gupta | TIFR, Mumbai | PhD student |
7 | Mr. Shuddhodan Kadattur Vasudevan | TIFR, Mumbai | PhD student |
8 | Mr. Anand Prabhakar Sawant | TIFR, Mumbai | PhD student |
9 | Mr. Gaurab Tripathi | Jadavpur University | PhD student |
10 | Mr. Susobhan Mazumdar | IITM Chennai | PhD student |
11 | Mr. Balakrishnan R | IITM Chennai | PhD student |
12 | Mr. Vivek Sadhu | IIT Bombay | PhD student |
13 | Mr. Seshadri Chintapalli | IMSc Chennai | PhD student |
14 | Mr. Umesh Vanktesh Dubey | TIFR, Mumbai | Post Doc |
15 | Dr. Vivek Mohan Mallick | IISER, Pune | Faculty |
16 | Dr. Yashonidhi Pandey | IISER Mohali | Faculty |
17 | Dr. Jishnu Biswas | ISI Bangalore | Faculty |
18 | Dr. Arvind Nair | TIFR, Mumbai | Faculty |