Semi-algebraic differential forms
Freitag, 25.10.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Modifying a construction by Hardt, Lambrechts, Turchin and Volić, we will present (\(\bmathbb{Q}\)-)semi-algebraic differential forms and explain connections to period numbers.
The pro-etale homotopy type
Freitag, 10.1.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
After reviewing the classical construction of the etale homotopy type by Artin-Mazur we define a pro-etale analogue for the pro-etale site of a scheme. An important difference between the etale and pro-etale site of a scheme is that the latter has enough weakly contractible objects. Using this fact we prove that the pro-etale homotopy type of a qcqs scheme is determined by a single split affine weakly contractible hypercovering. Lastly and if time permits, we discuss the pro-etale homotopy type of a field.
Random Generation: from Groups to Algebras
Freitag, 17.1.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, with emphasis on finite simple groups. In this talk, based on joint work with Aner Shalev, we study similar notions for finite and profinite associative algebras.\n\nLet \(k\) be a finite field. Let \(A\) be a finite associative algebra over \(k\), and let \(P(A)\) be the probability that two random elements of \(A\) will generate it. It is known that, if \(A\) is simple, then \(P(A) \bto 1\) as \(|A| \bto \binfty\). We extend this result for larger classes of finite associative algebras. For \(A\) simple, we estimate the growth rate of \(P(A)\) and find the best possible lower bound for it. We also study the random generation of \(A\) by two special elements.\n\nNext, let \(A\) be a profinite algebra over \(k\). We show that \(A\) is positively finitely generated if and only if \(A\) has polynomial maximal subalgebra growth. Related quantitative results are also\nobtained.
CoHAs of Weighted Projective Lines
Freitag, 7.2.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The Cohomological Hall Algebra (CoHA), introduced by Kontsevich--Soibelman, is a cohomological analogue of the Ringel--Hall algebra. While for a symmetric quiver this algebra is free (super) symmetric, there were essentially no other examples that were known explicitly in terms of generators and relations, until Franzen--Reineke computed the CoHA of (regular representations) of the 2-Kronecker quiver. The 2-Kronecker quiver is derived equivalent to the projective line and regular representations correspond to torsion sheaves; thus Franzen--Reineke's algebra can also be seen as the CoHA of torsion sheaves on \(\bmathbb{P}^1\). We extend this computation to CoHAs of torsion sheaves on the so-called weighted projective lines. As a special case, we also obtain the CoHAs of regular representations of all other extended Dynkin quivers (satisfying a certain natural condition on the Euler form).