Kolloquium Wilhelm Killing: Prof. Dr. Piotr Achinger (IMPAN): Homotopy types in algebraic geometry
Donnerstag, 27.06.2019 16:30 im Raum M5
Algebraic topology studies algebraic invariants of topological spaces,
typically identifying spaces which are "homotopy equivalent." Homotopy
theory proved essential in the study of topological spaces; when
applied to algebraic varieties, its ideas turn out to be even more
powerful, as the resulting invariants have far more structure, often
coming from either complex analysis (Hodge theory) or arithmetic (an
action of a Galois group). In particular, as predicted e.g. by the
Hodge conjecture, the Tate conjecture, or Grothendieck?s section
conjecture, they should be able to detect algebraic cycles and
Over the field of complex numbers, an algebraic variety gives rise to
a nice topological space to which one can apply methods of algebraic
topology directly. From the point of view of arithmetic and algebraic
geometry over more general fields, it is of key importance to
construct algebraically defined topological invariants. Probably the
most successful development to this date achieving this goal is the
introduction of the etale topology by Artin and Grothendieck, and the
etale homotopy type of Artin and Mazur.
I will give a gentle introduction to this circle of ideas, explain
several ways in which one can associate a homotopy type to an
algebraic variety, and discuss how the resulting objects witness
phenomena quite foreign to classical algebraic topology of nice
Kolloquium Wilhelm Killing: Dr. André Hendriques (University of Oxford): Three approaches to conformal field theory
Donnerstag, 04.07.2019 16:30 im Raum M5
There exist three mathematical formalizations of the concept of chiral conformal field theory: Vertex operator algebras, Conformal nets, and Segal CFTs.
These three notions are expected/conjectured to be equivalent (provided appropriate qualifiers are added). In this overview talk, I will explain, based on examples, what are Vertex operator algebras, Conformal nets, and Segal CFTs. I will describe how one is supposed to go from one formalism to the other, and where the main difficulties lie.
Kolloquium Wilhelm Killing: Prof. Dr. Katrin Wendland (Universität Freiburg): Moonshine Phenomena
Donnerstag, 11.07.2019 16:30 im Raum M5
Moonshine', in Mathematics, refers to surprising and deep connections
between finite group theory and the theory of so-called modular forms.
The first known instance of Moonshine is Monstrous Moonshine, where the
coefficients of the modular j-function are identified as dimensions of representations
of the Monster group. This was first observed by John McKay in 1987; Richard
Borcherds received a Fields Medal for his proof of the resulting Moonshine
Conjectures in 1998. More than a decade later, Tohru Eguchi, Hiroshi Ooguri
and Yuji Tachikawa proposed 'Mathieu Moonshine', which links the largest
Mathieu group to topological invariants of K3 surfaces, yielding the Fourier
coefficients of a certain elliptic modular form. Conformal field theory turns
out to be key to every known instance of moonshine.
The talk will give an introduction to solved and unsolved mysteries of these
two types of Moonshine.