## Seminar Darstellungstheorie

Das Seminar Darstellungstheorie findet normalerweise dienstags, 14.15 - 15.15 Uhr in G.13.18 statt.

Beteiligte Dozenten: Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth.

07.05.2024 | Niels Feld | Perverse homotopy heart of stable motivic homotopy and Milnor-Witt cycle modules |

14.05.2024 | Julian Quast | On local Galois deformation rings |

28.05.2024 | Shuji Saito | Pro-cdh descent for algebraic K-theory of derived schemes |

04.06.2024 | Dzoara Nunez Ramos | Derived functors as triangulated functors (research seminar) |

18.06.2024 | Mattia Tiso | Classical cotangent complex (research seminar) |

25.06.2024 | Lucas Ruhstorfer | The Alperin-McKay conjecture and blocks of maximal defect |

02.07.2024 | Julian Reichardt | Noncommutative Perspective on Quotient Varieties |

09.07.2024 | J. Miquel Martínez | On the reduction theorems for counting conjectures |

16.07.2024 | Heng Xie | Hermitian K-theory of Grassmannians |

### Niels Feld: Perverse homotopy heart of stable motivic homotopy and Milnor-Witt cycle modules

In the nineties, Voevodsky proposed a radical unification of algebraic and topological methods. The amalgam of algebraic geometry and homotopy theory that he and Fabien Morel developed is known as motivic homotopy theory. Roughly speaking, motivic homotopy theory imports methods from simplicial homotopy theory and stable homotopy theory into algebraic geometry and uses the affine line to parameterize homotopies. Voevodsky developed this theory with a specific objective in mind: prove the Milnor conjecture. He succeeded in this goal and won the Fields Medal for his efforts in 2002.

In this talk, I will present an ongoing project in collaboration with Frédéric Déglise and Fangzhou Jin where we realize Ayoub's conjectural program showing that the heart of the motivic stable homotopy category over appropriate base schemes can be related to a suitable version of relative Milnor-Witt modules.

### Julian Quast: On local Galois deformation rings

In joint work with Vytautas Paškūnas, we show that the universal framed deformation ring of an arbitrary mod p representation of the absolute Galois group of a p-adic local field valued in a possibly disconnected reductive group G is flat, local complete intersection and of the expected dimension. In particular, any such mod p representation has a lift to characteristic 0. The work extends results of Böckle, Iyengar and Paškūnas in the case G=GL_n. We give an overview of the proof of this main result.

### Shuji Saito: Pro-cdh descent for algebraic K-theory of derived schemes

Thanks to a celebrated theorem of Kerz-Strunk-Tamme, algebraic K-theory satisfies the pro-excision for abstract blowup squares of noetherian schemes. The latter property fails for non-noetherian schemes in general. A main result in this talk affirms that algebraic K-theory satisfies the property for (not necessarily noetherian ) qcqs derived schemes. As an application, we deduce Weibel’s vanishing of negative K-theory for qcqs derived schemes. In the noetherian case, this is due also to Kerz-Strunk-Tamme. This is a joint work with Shane Kelly and Georg Tamme.

### Lucas Ruhstorfer: The Alperin-McKay conjecture and blocks of maximal defect

For blocks of maximal defect the Alperin-McKay conjecture can be seen as a blockwise refinement of the McKay conjecture. I will explain what additional block-theoretic problems arise in this situation and report on recent progress on this question. This is joint work with Britta Späth.

### Julian Reichardt: Noncommutative Perspective on Quotient Varieties

Let X be a smooth affine variety with ring of regular functions A and G a finite group acting on X. Understanding the relation between the geometry of X and the group action of G on X naturally leads to considering the quotient variety X/G. However, this quotient might be singular, complicating the computation of its cohomology. Instead of working with the subring of G-invariants of A, we consider the noncommutative crossed product algebra of A and G. This allows to make full use of the assumptions on X.

Assuming we work over the complex numbers, the Hochschild--Kostant--Rosenberg theorem generalises to crossed product algebras arising as above. More precisely, their Hochschild, cyclic, and periodic homology are given in terms of algebraic cohomologies of the associated quotient varieties. We show that this extends to arbitrary base fields of characteristic 0. For simplicity we focus on the case of Hochschild homology. We then sketch how to apply this result to affine varieties in positive characteristic. Here it was recently shown by R. Meyer et al. that the construction of Monsky--Washnitzer cohomology may be rephrased in the framework of bornological modules and their associated bornological Hochschild homology. We briefly explain how our generalisation of the Hochschild--Kostant--Rosenberg theorem may then be used to compute the bornological Hochschild homology of the bornological completions of crossed product algebras.

### J. Miquel Martinez: On the reduction theorems for counting conjectures

Some of the main conjectures in the last 50 years in the representation theory of finite groups predict that we can count global invariants locally (with respect to the prime *p*). The paradigmatic example is the McKay conjecture (recently solved by M. Cabanes and B. Späth), which was followed by fundamental conjectures of Alperin and Dade. In 2007, the McKay conjecture was reduced to a problem for simple groups, and this was followed by reduction theorems for the other conjectures. Shortly after, it has been observed that the conditions required in these reduction theorems in fact imply stronger versions of the conjectures. In this talk, we discuss the reduction theorems, the stronger versions that they imply and the surprising consequences obtained from them.

### Heng Xie: Hermitian K-theory of Grassmannians

Hermitian K-theory, a cohomology theory, has found interesting applications in recent works. Through a series of works by Asok, Fasel, and Srinivas, the obstruction class for splitting algebraic vector bundles is shown to reside in Hermitian K-theory under certain conditions. Moreover, Hermitian K-theory aids in understanding an unsolved problem on the composition of quadratic forms posed by Hurwitz in 1898. However, not many computations have been done in Hermitian K-theory. Pushforward is a powerful computational tool in cohomology theory. Recently, we developed pushforward in Hermitian K-theory via Grothendieck's residue complex. Additionally, we have proven the base change, projection, and excess intersection formulas. These tools allow us to compute the Hermitian K-theory of Grassmannians, leading to a special class of Young diagrams that we call buffalo-check Young diagrams. This is joint work with Tao Huang.