Categorical Quantum Mechanics: An Introduction
Freitag, 23.4.21, 10:30-11:30, virtueller Raum 404
Building a categorical semantics for quantum protocols has been an ongoing endeavour since the seminal paper of Abramsky and Coecke in 2004. I will give an overview of this novel approach to studying physical models within the wider framework of Process Theories. We will see how the mathematics of dagger compact categories comes together\nwith the internal algebraic structures of special commutative dagger Frobenius algebras to give a rigorous graphical calculus for qubits in the form of the ZX-calculus. Then we will discuss decoherence (the\nquantum-classical transition) from a categorical viewpoint where I will present some recent work on a generalisation of this transition to categories generated by the actions of Galois groups.
Which properties of the canonical class depend only on its first Chern class?
Freitag, 30.4.21, 10:30-11:30, virtueller Raum 404
Given a projective variety X with mild singularities and a line bundle L on it, it is a natural question to determine which properties of L are encoded by its first Chern class. I will argue that most of the interesting properties of the canonical bundle of X, such as its effectivity or semiampleness, are indeed almost always encoded by its first Chern class. The results are a consequence of the Minimal Model Program and Hodge theory, and are new even on surfaces. This is joint work with Thomas Peternell.
Higher dimensional slope inequalities
Freitag, 7.5.21, 10:30-11:30, virtueller Raum 404
onsider a family of varieties f: X-> T, where T is a curve. We prove several inequalities about the slope of f, which are generalisations of Xiao and Cornalba-Harris inequalities in the case where X is a surface. We then apply our results to the KSB moduli space of stable varieties to study the ample cone of such spaces.\nThe talk is based on a joint work with Giulio Codogni and Filippo Viviani.
Highly connected 7-manifolds and non-negative sectional curvature
Freitag, 14.5.21, 10:30-11:30, virtueller Raum Lasker
A six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.
Valuation rings in the context of Algebraic Geometry
Donnerstag, 20.5.21, 11:15-12:15, online: kasparov
Cox rings of algebraic stacks
Freitag, 21.5.21, 10:30-11:30, virtueller Raum Lasker
In this talk, I will discuss the construction of Cox rings on algebraic\nstacks. Recall that the Cox ring consists of all global sections of\ndivisors on a given space. Here the definition of the multiplicative\nstructure is a bit subtle. But it turns out that such a structure\nalways exists, and moreover, its (non-)uniqueness can be measured by an\nExt-group. This talk is based on a joint work with Elena Martinengo and\nFabio Tonini.
A surface and a threefold with equivalent singularity categories
Freitag, 11.6.21, 10:30-11:30, virtueller Raum Lasker
Liquid Tensor Experiment -- a Progress Report
Donnerstag, 17.6.21, 14:15-15:15, online: vGK1821
In December 2020, Peter Scholze posed a challenge to formally verify\nthe main theorem on liquid R-vector spaces,\nwhich is part of his joint work with Dustin Clausen on condensed\nmathematics.\nI took up this challenge with a team of mathematicians\nto verify the theorem in the Lean proof assistant.\nHalf a year later, we have finished the main technical ingredient of\nthis challenge. In this talk I will report on the progress we've made\nand what remains to be done and discuss our experience formalizing\ncutting edge research. No prior knowledge of Lean or liquid mathematics\nis assumed.
On the Boucksom-Zariski decomposition for irreducible symplectic varieties and bounded negativity
Freitag, 25.6.21, 11:00-12:00, virtueller Raum Lasker
Birational geometry of foliations
Freitag, 2.7.21, 10:30-11:30, virtueller Raum Lasker
I will try to explain, by means of some examples and recent results, how the classical framework of the Minimal Model Program has been extended to the case of foliation, in particular in low dimension, as well as, how it has been used to initiate a systematic study and classification of foliations from a birational view point. \nThe talk will feature joint work with C. Spicer.
Frobenius kernels for automorphism group schemes
Freitag, 9.7.21, 10:30-11:30, virtueller Raum Lasker
We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristics. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with is twisted forms. The result actually holds true for rather general schemes, under the assumption that the Frobenius kernel has large isotropy group at the generic point. This is joint work with Nikolaos Tziolas.\n
Cohen-Lenstra-Martinet heuristics on class groups of number fields
Freitag, 16.7.21, 10:30-11:30, virtueller Raum Lasker
In the 1980s Cohen and Lenstra proposed a probabilistic model\nfor the behaviour of class groups of quadratic number fields. A few\nyears later, it was generalised by Cohen and Martinet to class groups\nof more general families of number fields. Recently, in joint work with\nLenstra we disproved these conjectures -- in two completely different\nways, and in joint work with Lenstra and Johnston we have offered a\ncorrected version. In my talk I will give an overview of this work.
The effective model structure and infinity-groupoid objects
Freitag, 23.7.21, 10:30-11:30, virtueller Raum Lasker
I will discuss a construction of a new model structure on\nsimplicial objects in a countably lextensive category (i.e., a category\nwith well behaved finite limits and countable coproducts). This builds\non previous work on a constructive model structure on simplicial sets,\noriginally motivated by modelling Homotopy Type Theory, but now\napplicable in a much wider context. This is joint work with Nicola\nGambino, Simon Henry and Christian Sattler.\n