Birational geometry is a major branch of algebraic geometry, focused on the study of function fields of algebraic varieties.
Among its main classical questions, going back to Italian algebraic geometry, is to determine whether or not an algebraic variety is rational. Many results we obtained by Iskobskikh Manin, Griffiths - Clemens, Artin - Martin.
Fresh ideas continue to enrich this branch of mathematics and to inspire further study of rationality properties. The Merkurjev-Suslin theorem, connecting the Brauer group with K-theory, and via the Bloch–Ogus sequence to Chow groups, proved particularly fruitful in the work of Colliot-Thélène–Sansuc, Colliot-Thélène–Raskind, Parimala–Suresh, Kahn, Pirutka, Saito, and Voisin. At least for the last item, there is a totally unexpected connection: Mirror Symmetry. It originated in physics as a duality between N = 2 superconformal quantum field theories. In 1990, Maxim Kontsevich gave an interpretation of
this duality in a consistent, powerful mathematical framework, called Homological Mirror Symmetry (HMS). The ideas put forth by Kontsevich in his Fields medal and ICM addresses have led to dramatic developments: they created a frenzy of activity in the mathematical community which has led to a remarkable synergy of diverse mathematical disciplines, notably symplectic geometry, algebraic geometry, and category theory. There are currently three directions motivated by HMS that
should lead to progress on rationality problems:
1) The idea of homological projective duality developed by Kuznetsov. Thomas and Addington connected this duality with the work of Hassett on rationality of cubic fourfolds and showed that in this instance, categorial and Hodge-theoretic descriptions are in fact equivalent.
2) The idea of categorical birational invariants, such as gaps of Orlov spectra and phantoms, introduced by Orlov, Katzarkov, and their collaborators.
3) The idea that stable rationality and related questions about algebraic cycles can be studied via the monodromy of Landau-Ginzburg models arising in HMS, proposed by Katzarkov and his group.
A semester on Advances in Birational Geometry – Vienna, April – May, 2017 – will serve as a perfect opportunity to disseminate new result and present new techniques.
The programme contains three workshops. And there will be weekly talks and series of lectures as well.
Talk by Victor Przyjalkowski (Laboratory of Mirror Symmetry HSE & Steklov Institute, Moscow):
Log Calabi-Yau compactifications of Landau-Ginzburg models
Thursday, April 6, 2017, 11:00 a.m.
Talk by David Favero (U of Alberta, Edmonton): Categorical Crepant Resolutions via LG models Tuesday, April 18, 2017, 11:00 a.m. Flyer (pdf)
Talk by Johanna Knapp (TU Vienna): (Non-)birational Calabi-Yaus from physics Thursday, April 20, 2017: 11:00 a.m. Flyer (pdf)
In the course of this programme this time the Vienna - Budapest Seminar takes place at the ESI on Friday, April 21, 2017: 2:00 p.m. Yuri Tschinkel (Courant Institute, New York): Rational points and rational varieties 3:30 p.m. Anton Mellit (IST Austria): Cohomology of character varieties For abstracts (only), see http://www.renyi.hu/~agdt/wienbp.html
Workshop 1 "Recent developments in rationality questions", April 24 – 28, 2017: schedule (pdf)
Workshop 2 "Categorical approach to rationality", May 2 – 5, 2017: schedule (pdf)
Talk by Carlos Simpson (U of Nice):
Moduli spaces of stable bundles for intermediate values of c_2
Tuesday, May 9, 2017: 2:00 p.m.
Talk by Alexander Efimov (RAS, Moscow):
t-structures on triangulated categories with geometric exceptional collections
Tuesday, May 9, 2017: 3:30 p.m.