Topology and geometry of manifolds (TiGR)

Master seminar 2020/2021

This is out of date description of an old seminar.
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Organisers: Jarosław Buczyński and Krzysztof Ziemiański

Time of meetings

Thursdays, 8:30-10:00

Place of meetings

Zoom

Schedule:

When?	Who?	What?
15.10.2020	J.B. i K.Z. Jarosław Ławnicki, Michał Łupiński, Kristian Olechowski	Organisation, presentation of the subject and planning of schedule, then presentations on Master theses in preparation: Embeddings of curves into surfaces and the complement problem (advisors: Karol Palka, Jarosław Wiśniewski) Singularities of foliations on algebraic surfaces (advisor: Adrian Langer) Derived categories and perverse sheaves (advisor: Agnieszka Bodzenta)
22.10.2020	Mykola Shameiev, Kamil Szubiński, Katarzyna Krawiec, Mateusz Zugaj, Michał Szachniewicz, Piotr Rudnicki	Presentations on Master and BSc theses or other projects: Semiorthogonal decompositions of derived category of coherent sheaves (advisor: Agnieszka Bodzenta) Graph Classes with fmNIP property (advisor: Szymon Toruńczyk) Heegaard's diagrams (advisor: Andrzej Czarnecki) Applications of ultracategories in logic (advisor: Marek Zawadowski) Białynicki-Birula decomposition for $\mathbb{G}_m$ actions on affine schemes (advisor: Joachim Jelisiejew) First glimpse on equivariant cohomology theory
29.10.2020	Oskar Słowik, Mateusz Kobak	Cohomological invariants of space of quatnum states (advisor: Andrzej Weber) Book by Brendon: Section I.1, G-complexes; notes from the talk.
5.11.2020	Paweł Poczobut, Krystian Olechowski	An algebraic variant of Fischer-Grauert's Theorem (advisors: Piotr Achinger, Adrian Langer) Section I.2, Equivariant cohomology theories; notes from the talk.
~~12.11.2020~~	no meeting	On this Thursday the whole department has Wednesday classes
19.11.2020	Piotr Oszer Majid Bigdeli	Berezinian from cohomological point of view (advisor: Andrzej Weber) Equivariant cohomology: definition, coefficient systems
26.11.2020	Kunal Dutta, Mateusz Zugaj	Proposal of MSc project on "Randomization in Computational Geometry and Topology" Examples of equivariant cohomologies
03.12.2020	Mateusz Kobak, Chirantan Mukherjee	Category of finite covering spaces of a connected topologigal space as an example of Galois category (advisor: Andrzej Weber) Products
10.12.2020	Karol Janowicz, Paweł Poczobut	Units in rings of algebraic integers (advisor: Zbigniew Marciniak) Spectral sequence(s)
17.12.2020	Magdalena Karaś, Katarzyna Krawiec	Property A in coarse geometry (advisor: Andrzej Nagórko) Equivariant obstruction theory
07.01.2021	Majid Bigdeli, Jarosław Ławnicki	The Cyclic Homology of the Group Algebras (advisor: Massoud Amini) Primary obstructions
14.01.2021	Makar Veramchuk, Michał Łupiński	Weighted $L_p$ inequalities for maximal function (advisor: Veniamin Krotov) The characteristic class of a map; notes from the talk.
21.01.2021	Chirantan Mukherjee Piotr Rudnicki	Weakly globular double categories and weak units (based on arXiv:2008.11180 by Simona Paoli) Hopf $G$-spaces and Eilenberg-MacLane $G$-complexes
28.01.2021	Jeremi Zglinicki Oskar Słowik	Constant curvature metrics on closed surfaces (advisor: Stefan Jackowski) Function spaces and $G$-spectra
Spring term
04.03.2021	Michał Szachniewicz	we commence the book by Atiyah on K-theory
11.03.2021	Piotr Oszer	Constructions and basic properties of vector bundles
18.03.2021	Kamil Szubiński	G vector bundles
25.03.2021	Magdalena Karaś	Cell structures on Grassmannians
~~1.04.2021~~
8.04.2021	Karol Janowicz	Basic definitions of K-theory
15.04.2021	Mykola Shamaiev	The Fundamental Product Theorem
22.04.2021	Jeremi Zgnilicki	K-theory as a cohomological theory and Bott periodicity
29.04.2021	Dorota Jankowska	(Short lecture) Type of equivariant Khovanov homotopy for some toric knots (Longer lecture) K-theory follow up
6.05.2021		Organisational meeting
13.05.2021	~~Makar Veramchuk~~
20.05.2021	Jarosław Ławnicki	The Adams operations
27.05.2021	Kristian Olechowski	Stiefel-Whitney and Chern classes
~~3.06.2021~~
10.06.2021	Michał Łupiński	Euler and Pontryagin classes

We are studying:

Topological and algebraic K-theory and their equivariant variants. First we read about equivariant cohomologies following three books:
1. Equivariant Cohomology Theories (Glen E. Bredon), available from IMPAN
2. Equivariant Ordinary Homology and Cohomology (Steven R. Costenoble, Stefan Waner), available from IMPAN
3. Introductory Lectures on Equivariant Cohomology (Loring W. Tu)
4. K-theory (M.F.Atiyah), available probably to everyone
5. Vector Bundles and K-Theory (Allan Hatcher), book in preparation, available on the author's webpage
6. Notes on equivariant stable homotopy theory based on lectures of Andrew Blumberg

Further topic suggestions

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".
Karol Palka's grant, "Modern methods in affine algebraic geometry".

Please contact the Principal Investigator of the respective grant if you are interested.