Starts 16 May 2022
Ends 25 Jul 2022
Central European Time
ICTP
Strada Costiera, 11 I - 34151 Trieste (Italy)

An ICTP Virtual Meeting

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

Schedule: Mondays and Wednesdays
1.00-3.00 pm GMT (link)
 
Lectures. There will be 3  lectures  and  a  tutorial  on each topic  each week.  The speakers will provide detailed   typed notes of their lectures on the day  they finish their lectures. These notes will contain exercises  which the participants should try to solve  and submit to the tutorial instructor before 12 pm GMT on Fridays.
 
Tutorials. The tutorial on each topic will be held a week after the lectures are delivered. The tutorial instructor will present solutions to problems assigned by the Lecturer that are received from the participants and discuss some  material covered during the lectures. Grades will be assigned to the solutions.

Registration: There is no registration fee.

Organizers

K. MADDOX (University of Kansas, USA), K. GOEL (University of Utah, USA), M.E. ROSSI (University of Genoa, Italy), N. V. TRUNG (Institute of Mathematics, Vietnam), J. K. VERMA (Indian Institute of Technology Bombay, India), Local Organiser: Lothar Goettsche (ICTP)

Co-sponsors