Homepage of the project
"Secant varieties, computational complexity, and toric degenerations"
The project is realized within the
Homing Plus
programme of Foundation for Polish Science,
cofinanced from European Union, Regional Development Fund.
Time span of the project:
May 1st, 2012 -- April 30th, 2014 (24 months)
Main Location:
Institute of Mathematics of Polish Academy of Sciences (IMPAN), Warsaw, Poland.
Participants:
- Jarosław Buczyński, IMPAN (principal investigator)
- Gavin Brown, Loughborough University, UK (international collaborator)
- Kristian Ranestad, University of Oslo, Norway (international collaborator)
- Joachim Jelisiejew, University of Warsaw (MSc student)
- Adam Michalik, University of Warsaw (MSc student)
- Maciej Gałązka, University of Warsaw (MSc student)
- Elisa Postinghel (postdoctoral assistant)
- Ada Boralevi (postdoctoral assistant)
Topics of the project:
We concentrate on the following two subjects:
-
Secant varieties and applications
We study ranks of points in a projective space with respect to a projective variety X and generalisations.
This is a geometric problem originating in the 19th century with many modern applications.
We will approach problems of finding the rank,
finding minimal decompositions and describing defining equations of secant varieties.
-
Calabi-Yau varieties, Fano varieties, and their toric degenerations
Our goal is to understand in details toric degenerations of Calabi-Yau varieties and of Fano varieties.
The idea behind introducing the toric degeneration is to replace the study of fairly complicated geometric object such as Calabi-Yau manifold (or variety, if we allow mild singularities), with the study of its degeneration, which is a union of toric varieties. This degeneration can be described in combinatorial terms, and a lot of geometry of original Calabi-Yau manifold can be read from the degeneration. Similar technique can be useful to construct Fano varieties.
An extract of the research proposal for this grant is available here (.pdf).
It includes the overview of the problems we are going to consider here,
but does not contain details of objectives of the grant.
Student scholarships:
We had a possibility of funding several scholarships
for students in the Master program. A call for the final scholarship had a deadline for applications on July 7th, 2013.
Applications for the first round were accepted before April 14th, 2012.
See here for the old announcement (Spring 2012).
One scholarship has been awarded to Joachim Jelisiejew.
Applications for the second round were accepted before November 11th, 2012.
See here for the old announcement (Fall 2012).
A scholarship has been awarded to Adam Michalik.
Postdoc positions:
Jointly with Institute of Mathematics of the Polish Academy of Sciences
we fund several short-term postdoctoral positions.
The deadline for applications was May 14th, 2012 and the applications were be decided on or before May 24, 2012.
Details and an official announcement are available here (in Polish), or here (in English).
Other projects:
Three other Homing Plus projects are being run in Algebraic Geometry (one project) and Geometric Group Theory (two projects):
Karol Palka is the principal investigator of the project "Embeddings and automorphisms of low-dimensional rational varieties". He has 2 student scholarships available for 2013/2014.
Piotr Nowak is the principal investigator of the project "Rigidity for actions on Banach spaces and applications". He has 2 student scholarships available for 2013/2014.
Piotr Przytycki is the principal investigator of the project "Cubulating groups". He had student scholarships and postdoc positions available.
