Topology and geometry of manifolds (TiGR)
Master seminar 2021/2022
Organisers: Jarosław Buczyński and Krzysztof Ziemiański
Time of meetings
Place of meetings
MIMUW, room 3240
|7.10.2021|| J.B. i K.Z.
|Organisation, presentation of the subject and planning of schedule,|
then presentations on Master theses in preparation:
Twierdzenie Grauerta-Fischera w dodatniej charakterystyce (Grauert-Fischer Theorem in positive characteristic)
Oswojona grupa podstawowa dla analitycznych przestrzeni adycznych (Tame fundamental group for adic analytic spaces)
|Presentations on Master and BSc theses or other projects|
Linie na rozmaitościach rzutowych (lines on projective varieties)
Rozkład Heegarda, homologie Heegarda-Floera, trysekcje czterowymiarowych rozmaitości (Heegard decomposition, Heegard-Floer homology, and trisections of four dimensional manifolds)
Geometryczna teoria grup i Z-struktury (Geometric grup theory and Z-structures)
Kongruencje dla szeregów Laurent (Congruences for Laurent series)
Struktura algebroidu Atiyah (Sturcture of Atiyah's algebroid)
|21.10.2021||Kuba Krawczyk||Resolution of singularities: Newton's method|
We are studying:
- Resolutions of singularities in algebraic geometry.
- Lectures on Resolution of Singularities (János Kollár),
hard copy available at IMPAN library, signature 74.026
- The Hironaka Theorem on resolution of singularities (Herwig Hauser), Bulletin of the AMS, 2003
- 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
- Resolution of Singularities - Seattle Lecture (János Kollár)
- 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:
- resolving of plane curve singularities: Wall, Singular points of plane curves
- Examples of surface singularities, see some of the chapters of Reid Chapters on algebraic surfaces, or Matsuki Introduction to the Mori program
- topology of the blow up of a submanifold in a manifold (in the book of Griffiths-Harris)
- problems in positive characteristics
Further topic suggestions
Please contact the Principal Investigator of the respective grant if you are interested.
- 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).
- 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
- 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
- Spherical varieties. Brion has good and brief introductions to this topic:
(about actions of algebraic groups on manifolds/varieties)
(about spherical varieties)
I would have to look up some further readings and the base book.
- 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.
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: