Seminar Algebra und Topologie
Das Seminar findet normalerweise am Mittwoch in F.13.11, 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Tobias Schmidt, Britta Späth, Kay Rülling, Matthias Wendt.
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.
Vivien Picard: The construction problem for logarithmic Hodge numbers modulo an integer (Masterkolloquium)
The only linear relations, that the Hodge numbers of smooth projective varieties over an algebraically closed base field of positive characteristic satisfy, are the ones induced by connectedness and by Serre duality. We want to investigate if these are also the only linear relations for the logarithmic Hodge numbers. For this, we will work with ordinary varieties. We will focus on the inner Hodge numbers.
Yuqing Shi: Higher enveloping algebras
The universal enveloping algebra functor assigns to a Lie algebra over a field k a unital associative algebra over k, characterised by the property that it sends free Lie algebras to free associative algebras. Using the theory of operads, one can generalise this classical construction to a functor assigning to a free Lie algebra a free Eₙ-algebra, for every natural number n. This construction is known as “higher enveloping algebras”, originally due to Knudsen. In this talk I will introduce the higher enveloping algebra functors using the self Koszul duality of the Eₙ-operad and discuss a relationship between Lie algebras and Eₙ-algebras in some stable infinity categories. As a consequence we obtain the universal property of the Bousfield—Kuhn functor.
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.
Finn Wiersig: Towards a p-adic analytic Riemann-Hilbert Correspondence
The Riemann-Hilbert correspondence establishes a profound connection between certain differential equations on a given complex analytic manifold and its topology. This result had numerous interesting applications; for example, it was applied by Beilinson-Bernstein and Brylinski-Kashiwara in the 1980s to study the representation theory of semisimple algebraic groups. This talk explores ongoing research leveraging these concepts to investigate representation theory appearing in the p-adic local Langlands program, focusing on progress towards a p-adic analytic Riemann-Hilbert correspondence.
Morten Lüders: p-adic Milnor K-theory
We explain a joint work with Matthew Morrow on p-adic Milnor K-theory. Our main theorem is a comparison of mod p^r Milnor K-groups of p-henselian local rings with the Milnor range of a newly defined syntomic cohomology theory by Bhatt, Morrow and Scholze. We begin by putting our result into context. Then we sketch the proof which builds on an analysis of a filtration on Milnor K-groups and a new technique called the left Kan extension from smooth algebras. Finally, we mention applications to the Gersten conjecture.
Tariq Syed: Motivic cohomology of cyclic coverings
The generalized Serre question asks whether algebraic vector bundles over topologically contractible smooth affine complex varieties are always trivial or not. Many examples of topologically contractible smooth affine complex varieties are given by so-called cyclic coverings. In this talk, we present new results on the motivic cohomology of such cyclic coverings which might indicate a positive answer to the generalized Serre question in low dimensions.
Thomas Geisser: Brauer and Neron-Severi groups of surfaces over finite fields
For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behavior of the zeta-function at 1 to other invariants of the surface. We give a refinement which equates invariants only depending on the Brauer group to invariants only depending on the Neron-Severi group. We also give estimates of the terms appearing in the formula. This implies, for example, the largest Brauer group of an abelian surface over the field of order q=p2r has order 16q, and the largest Brauer group of a supersingular abelian surface over a prime field is 36.
Alexander Ziegler: Computing mod p Chow rings of classifying spaces given by p-groups
An affine group scheme of finite type over the complex numbers gives rise to a motivic classifying space that can be approximated using schemes.
By taking the Chow ring in each step of the approximation, we get an algebraic analogue of group cohomology called the Chow ring of the classifying space. It turns out that these are often determined by the representation theory of the underlying group, at least in low degrees. After modding out a prime number p, we also gain a lot of control over higher degrees. Thus mod p Chow rings of classifying spaces grant (comparatively) easy access to examples of mod p Chow rings of schemes.
In this talk, I will report on a novel computational tool based on GAP and Sage that can be used to compute previously unknown examples of mod p Chow rings given by p-groups.
Quentin Posva: Singularities of 1-foliations
1-foliations are sheaves of vector fields on varieties in positive characteristic which can be used to understand purely inseparable morphisms. Their local theory can be quite pathological, reflecting the singularities that appears as normalized cyclic covers of degree divisible by the characteristic. In this talk I will present a resolution theorem for 1-foliations on surfaces. If time permits I will also describe how to implement the singularities of the MMP for 1-foliations and explain which local behavior this allows.
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.
