Topology and geometry of manifolds (TiGR)

Master seminar 2021/2022

Organisers: Jarosław Buczyński and Krzysztof Ziemiański

Time of meetings

Thursdays, 8:30-10:00

Place of meetings

MIMUW, room 3240

Schedule:

We are studying:

• Resolutions of singularities in algebraic geometry. Literature:
  1. Lectures on Resolution of Singularities (János Kollár), hard copy available at IMPAN library, signature 74.026
  2. The Hironaka Theorem on resolution of singularities (Herwig Hauser), Bulletin of the AMS, 2003
  3. Introduction to resolution of singularities (Mircea Mustaţă), this is a chapter (pp405-449) in the book Analytic and Algebraic Geometry: Common Problems, Different Methods, hard copy available at IMPAN library, signature 79.398
  4. Resolution of Singularities - Seattle Lecture (János Kollár)
  5. Resolution of Singularities (Steven Dale Cutkosky), draft available on author's webpage hard copy available at IMPAN library, signature 71.955
• Suggested order of work: everyone should read the begining of Hauser's article, introduction and Chapter 0. We commence presentations with the resolutions of curves following chapter 1 of Kollár's book. Then we will probably deal with surfaces.
• Related material:
  1. resolving of plane curve singularities: Wall, Singular points of plane curves
  2. Examples of surface singularities, see some of the chapters of Reid Chapters on algebraic surfaces, or Matsuki Introduction to the Mori program
  3. topology of the blow up of a submanifold in a manifold (in the book of Griffiths-Harris)
  4. problems in positive characteristics

Further topic suggestions

1. Coxeter groups, following the books: Hiller, Howard - Geometry of Coxeter groups, Huphreys, Coxeter groups. This is related to homogeneous spaces and group actions on manifolds. I find this topic very useful, although a little boring and tedious (very algebraic).
2. Cremona Groups – another classic topic, we can study the plane case, or algebraic properties of the group or its geometric properties or its dynamical properties. - Cantat – The Cremona group https://claymath.org/sites/default/files/canat.pdf - Cantat Déserti Xie Three chapters on Cremona groups https://arxiv.org/abs/2007.13841 In addition: - Cantat Lamy Normal subgroups in the Cremona group https://arxiv.org/abs/1007.0895 - Dolgatchev Finite subgroups of the plane Cremona group https://arxiv.org/abs/math/0610595 - Blanc Relations in the Cremona group over perfect fields https://arxiv.org/abs/1011.4432
3. Spherical varieties. Brion has good and brief introductions to this topic: https://www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf (about actions of algebraic groups on manifolds/varieties) https://link.springer.com/chapter/10.1007/978-1-4612-3702-0_3 (about spherical varieties) I would have to look up some further readings and the base book.
4. another proposal would be to try real algebraic and semi-algebraic geometry. I think there is no course in this topic at our University, but it is fairly large and interesting topic I am not familiar with. I would have to discuss with some people from UJ (Kraków) what are the possible books and references.

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Announcements:

Some NCN grants have a possibility of awarding a scholarships for writing MSc thesis. Grants that we know about and might have this possibility (and whose topics is related to the interests of the seminar) are:

• Piotr Achinger's grant, "Deformacje i degeneracje rozmaitości algebraicznych".
• Agnieszka Bodzenta's grant, "Kategorie dokładne, abelowe i triangulowalne w algebrze i geometrii"
• Jarosław Buczyński's grant, "Complex contact manifolds and geometry of secants (CoCoMaGeS)".
• Kunal Dutta's grant, "Probabilistic tools for high-dimensional geometric inference, topological data analysis and large-scale networks".
• Joachim Jelisiejew's grant, "Generalizations and applications of Bialynicki-Birula decomposition".
• Karol Palka's grant, "Modern methods in affine algebraic geometry".
Please contact the Principal Investigator of the respective grant if you are interested.

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