Seminars Sorted by Series

This is the introductory talk for a semester-long learning seminar on the notion of solid modules and the resulting 6-functor formalism for coherent cohomology.

Steven will cover the material from the first two lectures by Peter Scholze (available electronically at http://people.mpim-bonn.mpg.de/scholze/Condensed.pdf). In particular, he will introduce the notion of condensed set and condensed abelian group...

Quasi-coherent sheaves cannot support a six-functor formalism in the same way that e.g. \ell-adic sheaves do. Clausen and Scholze have shown that this can be fixed by extending the category of quasi-coherent sheaves to `solid sheaves’. In this talk...

This talk is an introduction to analytic rings. Some examples and non-examples will be covered. Some nice properties of the module categories will be proved.

We will discuss the notion of solid abelian groups.

We will globalize the notion of solid modules to an arbitrary scheme/discrete adic space.

We will discuss the 6-functor formalism for solid quasi-coherent sheaves.

The topic of the learning seminar this semester is "Analytic K-theory''. More precisely, the goal is to use condensed mathematics to obtain a notion of quasi-coherent sheaves/complexes in rigid analytic geometry, develop Efimov's approach to K...

Define the Grothendieck group of a commutative ring. Then define K1 and K2 and 
state Matsumoto’s theorem. Explain the definitions of higher K-theory in terms of the 
+-construction and in terms of the∞-group completion of groupoid of finite...

Introduce the notion of stable compactly generated $\infty$-category and discuss its properties and some examples. In particular, prove that the $\infty$-category of small idempotent complete stable $\infty$-categories is equivalent to the $\infty$...

Introduce the notion of Verdier sequences of presentable stable $\infty$-categories and then discuss the behavior of dualizable $\infty$-categories in Verdier sequence; in particular, discuss Thomason's trick. Introduce the notion of localizing...

Recall the notion of condensed analytic ring, then introduce its animated variant and discuss its properties. Explain the construction of induced analytic structures on animated condensed rings. Introduce the notion of steady maps of analytic rings...

Introduce the notion of nuclear object in a symmetric monoidal stable ($\infty$-)category and discuss its properties. In particular, prove that if the monoidal unit is compact, then an object is dualizable if and only if it is compact and nuclear...

For a general complete Huber pair $(A,A^+)$, define the analytic ring $(A,A^+)_\blacksquare$. Prove that the associations $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}((A,A^+)_\blacksquare)$, $\mathrm{Spa}\, (A,A^+)\mapsto \mathcal{D}^\omega((A,A^+)_...

Define the category of solid quasi-coherent sheaves $\mathcal{D}_\blacksquare(X)$ on a rigid-analytic varity $X$. Show that this assignment comes with a natural $6$-functor formalism. Finally, prove the Grothendieck--Serre duality in the rigid...

Prove that for a qcqs analytic adic space $X$, the category of nuclear sheaves on $X$ is dualizable in two steps. First, establish the result locally over affinoid Tate adic spaces using the notion of weak proregularity and then prove a general...

Review the notion of cd-structure for a Grothendieck topology and explain how it helps check the sheaf conditions. Introduce and discuss the Nisnevich topology for schemes and analytic adic spaces; in particular, discuss their equivalent...

Introduce the notion of hypersheaf and discuss its basic properties and equivalent characterizations. In particular, discuss the notions of homotopy and cohomological dimensions and their relation to hyperdescent. Sketch the proof of the fact for...

Explain the formulation of the Grothendieck–Riemann–Roch theorem for analytic adic spaces: go through [And23, pp. 32-38] and define all relevant objects and maps. Before explaining the construction of the Chern class map, define the sheaf KU∧p on...

Prove the Grothendieck–Riemann–Roch theorem: first prove [And23, Satz 6.12] and then explain the sketch of the proof on [And23, p. 38] by proving the relevant statements from the second half of [CS22, Lecture 15].

Abstract: The talk is based on joint work with Michael Groechenig and Johan de Jong.
We single out integrality properties of the topological fundamental group of smooth quasi-projective complex varieties which rely on the Langlands program (both...

Abstract: Report on a paper written during the pandemic joint with Minseon Shin and Max Lieblich. Some key words: Brauer groups, nontorsion elements of $\mathrm{H}^2(X, \mathbf{G}_m)$, resolution property, Jouanolou devices, and derived categories...

Abstract:

(joint work with Martin Kalck) Periods are complex numbers obtained by integrating algebraic differential forms over $\mathbf{Q}$ over a domain of algebraic nature. This includes numbers like $\pi$, $\log(2)$ or the values of the Riemann...

Abstract: Vanishing theorems in complex algebraic geometry, such as the Kodaira vanishing theorem, play an important role in understanding the structure of complex algebraic varieties. Ultimately, they rely on critical input from Hodge theory. After...

Abstract: The aim of this talk is to present some results on isometries of even, unimodular lattices and automorphisms of K3 surfaces.

Abstract: It is well-known that for any finite group $G$, there exists a closed $3$-manifold $M$ with $G$ as a quotient of the fundamental group of $M$. However, we can ask more detailed questions about the possible finite quotients of $3$-manifold...

Abstract: The Deligne--Simpson problem asks for a criterion of the existence of connections on an algebraic curve with prescribed singularities at punctures. We give a solution to a generalization of this problem to $G$-connections on $\mathbb{P}^1$...

Abstract: Emmy Noether was a central figure in the development of abstract algebra in the early 20th century. Her ideas were profoundly influential, touching nearly every corner of mathematics. In this talk, I'll discuss how those ideas have taken...

Abstract: For studies on singularities over the field of characteristic zero, we can use many convenient tools: resolutions of singularities, Bertini’s theorem (generic smoothness), cohomology vanishing of Kodaira type, and so on. However, in...

Abstract: To understand the birational geometry of a projective variety $X$, one seeks to describe all rational contractions from $X$. From an algebraic perspective, information about all these contractions are encoded in the ring formed by all...