Galois action on knots
My talk is on my attempt in arithmetic topology. I will discuss an
arithmetic structure on knots;
I will show that the absolute Galois group of the rational number
field acts non-trivially on 'the space of knots' in a non-trivial way.
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The title of my talk should be: "Characteristic classes of singular toric varieties
I will discuss the computation of the homology Hirzebruch
characteristic classes of (possibly singular) toric varieties. I will
present two different perspectives for the computation of these
characteristic classes. First, by taking advantage of the torus-orbit
decomposition and the motivic properties of the homology Hirzebruch
classes, we express the latter in terms of the (dual) Todd classes of
closures of orbits. The obtained formula is then applied to weighted
lattice point counting in lattice polytopes. Secondly, in the case of
simplicial toric varieties, we make use of the Lefschetz-Riemann-Roch
theorem in the context of the geometric quotient description of such
varieties. In this setting, we define mock Hirzebruch classes of
simplicial toric varieties and investigate the difference between the
(actual) homology Hirzebruch class and the mock Hirzebruch class. We
show that this difference is localized on the singular locus, and we
obtain a formula for it in which the contribution of each singular cone
is identified explicitly. This is joint work with Joerg Schuermann.
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Green's functions on Riemann surfaces III
Green's functions on Riemann surfaces II
Green's functions on Riemann surfaces I
A theorem of Narasimhan and Seshadri (following Donaldson)
Optimal spherical designs
Fibrations and Adjunctions for quasicategories
Cartesian and co-Cartesian morphisms. Maybe other fibrations in used for adjunctions.
Definition of Adjunction via correspondences (See sect 5.2.1 and 5.2.2 in HTT)
Statement of Prop 5.2.1.3 (HTT) (probably skip the proof)
Prop. 5.2.3.5, sketch of proof.
The Fourier decomposition on the Chow ring of certain hyperkaehler fourfolds
Using a codimension-1 algebraic cycle obtained from the Poincare line bundle, Beauville defined the Fourier transform on the Chow ring of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring CH_*(A) which is compatible with its ring structure. We prove that a similar decomposition exists for certain hyperkaehler fourfolds by using a codimension-2 algebraic cycle representing the Beauville--Bogomolov bilinear form. This is joint work with Mingmin Shen.
Kudla's modularity conjecture for cycles of codimension 2
We establish Kudla's modularity conjecture for the generating function of special cycles of codimension 2. The proof is based on a convergence result for formal Fourier Jacobi expansions. We define such expansions in some simple cases, and outline the approach taken to show that they always converge. Finally, we showcase the connection to previous results by Zhang, which yields the desired proof of Kudla's conjecture.
Topos Theory
Bachelor/Master-Seminar Darstellungstheorie
Yang-Mills stratification versa Harder-Narasimhan stratification
Representation equivalence, characteristic equivalence and commensurability of arithmetic lattices
Gopal Prasad and A. S. Rapinchuk defined a notion of weakly
commensurable lattices in a semisimple group, and gave a
classification of weakly commensurable Zariski dense subgroups. A
motivation was to classify pairs of locally symmetric spaces
isospectral with respect to the Laplacian on functions. For this, in
higher ranks, they assume the validity of Schanuel's conjecture.
In this talk, we observe that if we use the stronger notion of
representation equivalence of lattices, then Schanuel's conjecture can
be avoided. Further, the results are also applicable in a
S-arithmetic setting.
We also introduce a new relation on the class of arithmetic lattices,
stronger than weak commensurability, which we call as characteristic
equivalence, and show that it simplifies some of the arguments used in
Prasad and Rapinchuk (2009) to deduce commensurability type results
from weak commensurability.
Topology of algebraic varieties and perverse sheaves IV
Noncommutative Local Monodromy Theorem
Iterated disc constructions in 4-manifold topology (Semester on 4-manifolds)
English
Deutsch