Nijenhuis deformation of L$_{\infty}$-structures
Classification of topological 4-manifolds with infinite fundamental group, 2
This talk will be the second part of a survey of ideas and results related to the classification of topological 4-manifolds up to homeomorphism or s-cobordism. In this talk I will assume that the fundamental group is infinite, and discuss (i) geometrically 2-dimsnsional groups, (ii) minimal 4-manifolds with a given fundamental group, and (iii) open problems.
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
Classification of topological 4-manifolds with finite fundamental group
This talk will be the first part of a survey of ideas and results related to the classification of topological 4-manifolds up to homeomorphism or s-cobordism. In this talk I will assume finite fundamental group, and discuss (i) the algebraic topology of 4-manifolds, (ii) stable versus unstable classification, and (iii) applications to algebraic surfaces.
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
An approach to the topological classification of 4-manifolds II
The starting point is Freedman's disc theorem for 4-manifolds with "good" fundamental group. This has two immediate consequences: The topological s-cobordism theorem and topological surgery in dimension 4. The first reduces the topological classification to the classification up to s-cobordism, the second can be used in constructing topological 4-manifolds. I will introduce into my modified surgery theory which allows a simplified approach to both problems. In particular the topological 4-dimensional Poincare conjecture will follow as well as the characterization of manifolds homeomorphic to R4.
An approach to the topological classification of 4-manifolds I
The starting point is Freedman's disc theorem for 4-manifolds with "good" fundamental group. This has two immediate consequences: The topological s-cobordism theorem and topological surgery in dimension 4. The first reduces the topological classification to the classification up to s-cobordism, the second can be used in constructing topological 4-manifolds. I will introduce into my modified surgery theory which allows a simplified approach to both problems. In particular the topological 4-dimensional Poincare conjecture will follow as well as the characterization of manifolds homeomorphic to $R^4$.
http://people.mpim-bonn.mpg.de/teichner/Math/4-Manifolds.html
q-series and quantum modular forms
While the theory of mock modular forms has seen great advances in the last decade, questions remain. We revisit Ramanujan's last letter to Hardy, and prove one of his remaining conjectures as a special case of a more general result. Quantum modular forms, defined by Zagier, as well as Dyson's combinatorial rank function, the Andrews-Garvan crank function, and mock theta functions, all play key roles. Along these lines, we also show that the Rogers-Fine false theta functions, functions that have not been well understood within the theory of modular forms, specialize to quantum modular forms. This is joint work with K. Ono (Emory U.) and R.C. Rhoades (Stanford U.).
Betti numbers of large hyperkähler manifolds
Braneworlds and elliptic functions
Mathieu moonshine in string theory
Feynman graphs, periods, and modular forms
Spectral Presheaves as Generalised Gelfand Spectra for Nonabelian Unital C*-algebras
To each unital C*-algebra, we associate a generalised Gelfand
spectrum in the form of the spectral presheaf. It is shown that this
assignment is contravariantly functorial, generalising the Gelfand
spectrum functor to nonabelian unital C*-algebras. We show that the
spectral presheaf is 'locally dual' to the Bohrification of a C*-algebra
in the sense of Heunen, Landsman and Spitters. Moreover, it is
demonstrated that the spectral presheaf determines the C*-algebra up to
Jordan isomorphisms in many cases, and that time evolution of a quantum
system can be formulated in terms of flows on the spectral presheaf.
Hopf monads
Hopf monads generalize Hopf algebras to a non-braided setting, that is,
to arbitrary monoidal categories. The initial motivation to introduce
the notion of Hopf monad was to understand the Drinfeld-Joyal-Street
categorical center in Hopf algebraic terms. Such a description is useful
in quantum topology for comparing the Turaev-Viro and Reshetikhin-Turaev
invariants of 3-manifolds.
Poincarè-Bendixson theory for parabolic holomorphic foliations by curves
The classical Poincar\'e - Bendixson theory describes the way a trajectory of
a vector field on the real plane behaves when both accumulating, or not to the
singular locus of a vector field in question. We describe, in the first approximation,
the way a leaf with contractible holonomy of a parabolic holomorphic foliation by
curves on the complex manifold approaches the singular locus of the foliation.
Rational points on singular curves, with applications to the Brauer-Manin obstruction on surfaces
Singular curves over number fields have properties very different from those of smooth curves:
such a curve can have the trivial Brauer group, contain infinitely many adelic points, but only
finitely many rational points or none at all. Nevertheless, finite descent explains all counterexamples
to the Hasse principle on singular curves provided all the geometric irreducible components are rational.
Singular curves can be used to construct surfaces which are counterexamples to the Hasse
principle not explained by the Brauer-Manin obstruction, even when applied to etale covers.
This is a joiont work with Yonatan Harpaz.
A theorem of Soule and end of proof
A differential graded approach to derived differential geometry, Part 5
tba
Calculation of Riemann's zeta function via interpolating determinants
Using intensive computer calculations, the author empirically discovered
unusual methods for calculating high-precision approximations to the non-trivial zeroes
of Riemann's zeta function, its values and values of its derivative on the whole
complex plane. So far no theoretical explanation to these phenomena is known.
dg-categories lecture 4 (after Toen)
dg-categories lecture 3 continued (after Toen)
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