MATH

Europe/Rome 2022-01-26 16:00:00 2022-01-26 17:00:00 ICTP/IGAP seminar on Algebraic Geometry: The sl(2) action on the Chow ring of an abelian variety The seminar will consist of lecture series by Alina Marian and by Lothar Goettsche (of which the abstracts are below), and talks by Postdocs and Faculty on their research. Toward the cohomology and Chow rings of moduli spaces of sheaves (Alina Marian) A seminar direction will examine the problem of understanding Lie algebra actions on the cohomology and Chow rings of moduli spaces of sheaves interms of the Chern classes of the universal sheaf. One classical action on cohomology is the Lefschetz sl(2) associated with an ample divisor class on a projective variety. For moduli spaces of sheaves over curves and surfaces, Grothendieck's standard conjectures are often known to hold, in particular Lefschetz sl(2) actions should be expressible via algebraic correspondences. Nevertheless there are very few explicit algebraic constructions of the Lefschetz operators. Progress in this direction would have important applications, not least to holomorphic symplectic geometry; the seminar will explain this circle of ideas. --------- Virtual invariants of moduli spaces of sheaves on surfaces and Vafa-Witten invariants (Lothar Goettsche) Another direction of the seminar will be virtual invariants of moduli spaces of sheaves on surfaces and Vafa-Witten invariants. Moduli spaces of sheaves on surfaces with $p_g>0$ tend to be singular, but they carry a so-called perfect obstruction theory, which allows to define virtual versions of the standard topological invariants of smooth varieties, e.g. the virtual Euler number. In 1994 Vafa and Witten gave a formula for "Euler numbers" of moduli spaces of rank 2 sheaves on surfaces. Recently a mathematical definition of these Vafa-Witten invariants was given in arbitrary rank by Tanaka and Thomas in terms of moduli spaces of Higgs pairs. We will review the definition of these invariants and their relation to virtual Euler numbers of moduli spaces of sheaves on surfaces, and show computations of these invariants leading to conjectural generating functions in terms of modular functions. ---------- The second talk on Wednesday 26 January, at 16:00: Alina Marian (Northeastern University) Title: The sl(2) action on the Chow ring of an abelian variety Register in advance for this meeting: https://zoom.us/meeting/register/tJUvd-yurjkrGdHTTDO2mFbC5qPgxAW-B4wj After registering, you will receive a confirmation email containing information about joining the meeting. Online ICTP pio@ictp.it 26 Jan 2022

Europe/Rome 2022-04-11 08:00:00 2022-04-16 22:00:00 AGRA IV (Arithmetic, Analysis and Groups) - Part 2 | (smr 3695) Cabo Frio (Rio de Janeiro) - Brazil ICTP pio@ictp.it 11 Apr 2022 - 16 Apr 2022

Europe/Rome 2022-05-16 16:00:00 2022-07-25 22:00:00 Graduate Course on Tight Closure of Ideals and its Applications | (smr 3707) The aim of the course is to provide training in the foundations of commutative algebra and algebraic geometry in prime characteristic and to present some of the exciting recent developments. The local theory of prime characteristic singularities is a beautiful and historically important subject. Singularities which are defined in terms of the behaviour of the Frobenius endomorphism have been labeled “F-singularities”. The course gives an introduction on the most prominent F-singularity classes that emerged from Hochster–Huneke’s tight closure theory. Since its introduction in the late 1980s, it has had a dramatic effect on the field of commutative algebra. Tight closure gives unified proofs and strong generalisations of many major theorems in commutative algebra, as well stimulated recent proofs of longstanding conjectures. Topics: An overview of tight closure of ideals Test ideals Direct summands F-rational rings and rational singularities Hilbert-Kunz multiplicities Briançon-Skoda Theorems Big Cohen-Macaulay algebras The absolute integral closure Symbolic powers of ideals The localisation problem Uniform Artin-Rees results  Speakers: I. ABERBACH, University of Missouri, USA F. ENESCU, Georgia State University, USA N. EPSTEIN, George Mason University, USA E. GRIFO, University of Nebraska, USA G. LYUBEZNIK, University of Minnesota, USA L. MA, Purdue University, USA T. POLSTRA, University of Virginia, USA I. SWANSON, Purdue University, USA V. TRIVEDI, Tata Institute of Fundamental Research, India K. TUCKER, University of Illinois, USA W. ZHANG, University of Illinois, USA Registration: There is no registration fee.   Online - ICTP pio@ictp.it 16 May 2022 - 25 Jul 2022

Europe/Rome 2022-07-11 08:00:00 2022-07-29 22:00:00 2022 ICTP-EAUMP School: Mathematical Programming and Algorithms | (smr 3725) Nairobi - Kenya ICTP pio@ictp.it 11 Jul 2022 - 29 Jul 2022

Europe/Rome 2022-08-01 08:00:00 2022-08-05 22:00:00 Frontiers in Geometry and Topology Summer School | (smr 3648) An ICTP Hybrid Meeting Organised in partnership with the Clay Mathematics Institute The conference is intended as an appreciation of the contributions of Tomasz Mrowka Geometry and Topology have seen spectacular recent developments, including the solution of classical problems. As a consequence, the established picture of how various parts of these areas fit together is evolving rapidly. On the topological side, among the topics that have progressed are: four-manifolds obtained as knot traces; knot concordance and homology cobordism, Stein fillings. On the Floer homology side, we have powerful new computational methods and new invariants of embedded graphs and webs. On the more analytic side, we have new compactness results and potential new invariants. Topics: Interplay of 3-dimensional and 4-dimensional Topology Floer homology theories and associated invariants Khovanov homology Geometric and analytic aspects of gauge theoretic equations Speakers: J. GREENE, Boston College, USA J. HOM, Georgia Tech, USA F. LIN, Columbia University, USA *J. PARDON, Princeton University, USA T. WALPUSKI, HU Berlin, Germany *To be confirmed Here is the link to the Frontiers in Geometry and Topology Research Conference Registration: There is no registration fee. ICTP ICTP pio@ictp.it 1 Aug 2022 - 5 Aug 2022