8 research outputs found
Tropical Geometry: new directions
The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that
have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative
geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress
in more traditional areas of tropical research, such as tropical
moduli spaces, tropical homology and tropical correspondence theorems
Homotopy Theory
Algebraic topology in general and homotopy theory in particular is in an exciting period of growth and transformation, driven in part by strong interactions with algebraic geometry, mathematical physics, and representation theory, but also driven by new approaches to our classical problems. This workshop was a forum to present and discuss the latest result and ideas in homotopy theory and the connections to other branches of mathematics. Central themes of the workshop were derived algebraic geometry, homotopical invariants for ring spectra such as topological Hochschild homology, interactions with modular representation theory, group actions on spaces and the closely-related study of the classifying spaces of groups
Singularities (hybrid meeting)
Singularity theory concerns local and global structure of singularities of (algebraic) varieties and maps. As such, it combines tools from algebraic geometry, complex analysis, topology, algebra and combinatorics
Gromov-Witten invariants in complex-oriented generalised cohomology theories
Given a closed symplectic manifold , we construct Gromov-Witten-type
invariants valued both in (complex) -theory and in any complex-oriented
cohomology theory which is -local for some Morava
-theory . We show that these invariants satisfy a version of the
Kontsevich-Manin axioms, extending Givental and Lee's work for the quantum
-theory of complex projective algebraic varieties. In particular, we prove a
Gromov-Witten type splitting axiom, and hence define quantum -theory and
quantum -theory as commutative deformations of the corresponding
(generalised) cohomology rings of ; the definition of the quantum product
involves the formal group of the underlying cohomology theory. The key
geometric input to these results is a construction of global Kuranishi charts
for moduli spaces of stable maps of arbitrary genus to . On the algebraic
side, in order to establish a common framework covering both ordinary
-theory and -local theories, we introduce a formalism of `counting
theories' for enumerative invariants on a category of global Kuranishi charts.Comment: 63 pages, 2 figure
Annual register. 1919-20
Imprint varies in volumes preceding 1893/94