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**Organisers: **Jarosław Buczyński and Krzysztof Ziemiański

Thursdays, **8:30-10:00**

**Zoom**

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. |

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 |

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 | ||

20.05.2021 | Jarosław Ławnicki | The Adams operations |

27.05.2021 | Kristian Olechowski | Stiefel-Whitney and Chern classes |

10.06.2021 | Michał Łupiński | Euler and Pontryagin classes |

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

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".