Research Seminar on Algebraic Topology, Winter Term 2024/25

Julian Holstein, Birgit Richter

In the winter term 2024/25 we will meet roughly every second Monday from 14:15h to 17:45 in 142 (Geomatikum) or online. The precise dates are: 21.10.24, 4.11.24, 18.11.24, 2.12.24, 16.12.24, 6.1.25, 27.1.25

The topic for our working seminar is Deformation Theory. The talks are as follows:

1. Introduction to the deformations of algebras (Yannick Hoyer)
2. More deformations of algebras and A_\infty algebras (Sascha Jevnewitsch): Whitherspoon 5.3 (reinterpret formal deformations in terms of the Maurer-Cartan equation, define deformation quantization of Poissen algebras. Then introduce A_\infty algebras following Keller and describe how they can arise from deformations (Keller 3.2, Witherspoon 7).
3. Some deformations of categories (Kristoffer Rasmussen): Introduce curved A_\infty categories (aka A_{[0, \infty[}-categories and their Hochschild complex (Lowen 2.1-2.3 & 2.7) and describe how elements in Hochschild cohomology give deformations as curved A_\infty algebras (Lowen 4.11). Mention as motivation the main topic of the article: When deforming an algebra A we can ask if we can deform the objects of the derived category to objects in the derived category of the deformed algebra. This is exactly obstructed by curvature.
4. Dg Lie algebras and L_\infty algebras: Give a brief introduction to dg Lie algebras, mention L_\infty algebras, consider the shifted Hochschild cochains as an example (Witherspoon 7.5, 7.6).
5. Deformation theory after Lurie I: Define formal moduli problems (focusing on the deformation context of commutative algebras) and their tangent complexes, define deformation theories and state their relation (Lurie 1.3.12).
6. Deformation theory after Lurie II: Explain Luries Theorem 2.0.2 classifying deformations over commutative algebras in terms of dg Lie algebras. Explain the Chevalley-Eilenberg complex and Koszul duality.

References:

• Sarah Witherspoon, Hochschild Cohomology for Algebras, Graduate Studies in Mathematics 204, American Mathematical Society, 2019, (available on the author's webpage)
• Bernhard Keller, Introduction to A-infinity algebra and modules, Homology, Homotopy and Applications, vol.3, No.1, 2001, pp.1--35
• Wendy Lowen: Hochschild cohomology, the characteristic morphism and derived deformations, Compositio Mathematica, Volume 144, Issue 6, November 2008, pp. 1557--1580
• Jacob Lurie, Derived Algebraic Geometry X: Formal Moduli Problems

If you want to give a talk in the working seminar, send an email to us. If you are interested in participating, then please register for the mailing list of the seminar by sending "subscribe" to topologieseminar.math-request@lists.uni-hamburg.de