Description

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.

Go to day
  • Monday, 16 May 2022
    • 14:30 - 16:00
      Time zone: You can choose your time zone (in place of Europe/Rome) on the top right of the page and the programme time will be modified accordingly.
      • 14:30 Orientation and Introduction 10'
        Speaker: Lothar GOETTSCHE (ICTP, Italy), Jugal Kishore VERMA (Indian Institute of Technology Bombay, India)
        Material: Video
      • 14:40 Instructions 15'
        Tutors: Kriti GOEL (University of Utah, USA), Kyle MADDOX (University of Kansas, USA)
        Material: Video
      • 14:55 Introduction of Lecturer 5'
        Speaker: Marilina ROSSI (University of Genoa, Italy)
      • 15:00 Where does Tight Closure Come From? 45'
        Speaker: Craig HUNEKE (University of Virginia, USA)
        Material: Video
      • 15:45 Q and A session 10'
      • 15:55 Break 5'
    • 16:00 - 16:50
      • 16:00 An Overview of Tight Closure of Ideals - Part 1 50'
        Speaker: Neil EPSTEIN (George Mason University, USA)
        Material: Video
  • Wednesday, 18 May 2022
    • 15:00 - 16:50
      • 15:00 An Overview of Tight Closure of Ideals - Part 2 50'
        Speaker: Neil EPSTEIN (George Mason University, USA)
      • 15:50 Break 10'
      • 16:00 An Overview of Tight Closure of Ideals - Part 3 50'
        Speaker: Neil EPSTEIN (George Mason University, USA)
  • Monday, 23 May 2022
    • 15:00 - 16:50
      • 15:00 An Overview of Tight Closure of Ideals - Tutorial 50'
        Tutor: Kriti GOEL, University of Utah, USA
        Speaker: Neil EPSTEIN (George Mason University, USA)
      • 15:50 Break 10'
      • 16:00 Test Ideals - Part 1 50'
        Speaker: Kevin TUCKER (University of Illinois, USA)
  • Wednesday, 25 May 2022
    • 15:00 - 16:50
      • 15:00 Test Ideals - Part 2 50'
        Speaker: Kevin TUCKER (University of Illinois, USA)
      • 15:50 Break 10'
      • 16:00 Test Ideals - Part 3 50'
        Speaker: Kevin TUCKER (University of Illinois, USA)
  • Monday, 30 May 2022
    • 15:00 - 16:50
      • 15:00 Test Ideals - Tutorial 50'
        Tutor: Kenta SATO, Kyushu University, Japan
        Speaker: Neil EPSTEIN (George Mason University, USA)
      • 15:50 Break 10'
      • 16:00 Direct Summands of Regular Rings - Part 1 50'
        Speaker: Linquan MA (Purdue University, USA)
  • Wednesday, 1 June 2022
    • 15:00 - 16:50
      • 15:00 Direct Summands of Regular Rings - Part 2 50'
        Speaker: Linquan MA (Purdue University, USA)
      • 15:50 Break 10'
      • 16:00 Direct Summands of Regular Rings - Part 3 50'
        Speaker: Linquan MA (Purdue University, USA)
  • Monday, 6 June 2022
    • 15:00 - 16:50
      • 15:00 Direct Summands of Regular Rings - Tutorial 50'
        Tutor: Cheng MENG, Purdue University, USA
        Speaker: Linquan MA (Purdue University, USA)
      • 15:50 Break 10'
      • 16:00 F-rational Rings and Rational Singularities - Part 1 50'
        Speaker: Florian ENESCU (Georgia State University, USA)
  • Wednesday, 8 June 2022
    • 15:00 - 16:50
      • 15:00 F-rational Rings and Rational Singularities - Part 2 50'
        Speaker: Florian ENESCU (Georgia State University, USA)
      • 15:50 Break 10'
      • 16:00 F-rational Rings and Rational Singularities - Part 3 50'
        Speaker: Florian ENESCU (Georgia State University, USA)
  • Monday, 13 June 2022
    • 15:00 - 16:50
      • 15:00 F-rational Rings and Rational Singularities - Tutorial 50'
        Tutor: Kyle MADDOX, University of Kansas, USA
        Speaker: Florian ENESCU (Georgia State University, USA)
      • 15:50 Break 10'
      • 16:00 Introduction to Hilbert-Kunz Multiplicity - Part 1 50'
        Speaker: Vijaylaxmi TRIVEDI (Tata Institute of Fundamental Research, India)
  • Wednesday, 15 June 2022
    • 15:00 - 16:50
      • 15:00 Introduction to Hilbert-Kunz Multiplicity - Part 2 50'
        Speaker: Vijaylaxmi TRIVEDI (Tata Institute of Fundamental Research, India)
      • 15:50 Break 10'
      • 16:00 Introduction to Hilbert-Kunz Multiplicity - Part 3 50'
        Speaker: Vijaylaxmi TRIVEDI (Tata Institute of Fundamental Research, India)
  • Monday, 20 June 2022
    • 15:00 - 16:50
      • 15:00 Introduction to Hilbert-Kunz Multiplicity - Tutorial 50'
        Tutor: Mandira MONDAL, Chennai Mathematical Institute, India
        Speaker: Vijaylaxmi TRIVEDI (Tata Institute of Fundamental Research, India)
      • 15:50 Break 10'
      • 16:00 Briancon-Skoda Theorems - Part 1 50'
        Speaker: Ian ABERBACH (University of Missouri, USA)
  • Wednesday, 22 June 2022
    • 15:00 - 16:50
      • 15:00 Briancon-Skoda Theorems - Part 2 50'
        Speaker: Ian ABERBACH (University of Missouri, USA)
      • 15:50 Break 10'
      • 16:00 Briancon-Skoda Theorems - Part 3 50'
        Speaker: Ian ABERBACH (University of Missouri, USA)
  • Monday, 27 June 2022
    • 15:00 - 16:50
      • 15:00 Briancon-Skoda Theorems - Tutorial 50'
        Tutor: Suprajo DAS, Chennai Mathematical Institute, India
        Speaker: Ian ABERBACH (University of Missouri, USA)
      • 15:50 Break 10'
      • 16:00 Big Cohen-Macaulay Algebras and their Applications - Part 1 50'
        Speaker: Thomas POLSTRA (University of Virginia, USA)
  • Wednesday, 29 June 2022
    • 15:00 - 16:50
      • 15:00 Big Cohen-Macaulay Algebras and their Applications - Part 2 50'
        Speaker: Thomas POLSTRA (University of Virginia, USA)
      • 15:50 Break 10'
      • 16:00 Cohen-Macaulayness of the Absolute Integral Closure in Characteristic p>0 50'
        Speaker: Gennady LYUBEZNIK (University of Minnesota, USA)
  • Monday, 4 July 2022
    • 15:00 - 16:50
      • 15:00 Big Cohen-Macaulay Algebras and their Applications - Tutorial 50'
        Tutor: Mitra KOLEY, ISI, India
        Speaker: Thomas POLSTRA (University of Virginia, USA)
      • 15:50 Break 10'
      • 16:00 Uniform Artin-Rees Results - Part 1 50'
        Speaker: Irena SWANSON (Purdue University, USA)
  • Wednesday, 6 July 2022
    • 15:00 - 16:50
      • 15:00 Uniform Artin-Rees Results - Part 2 50'
        Speaker: Irena SWANSON (Purdue University, USA)
      • 15:50 Break 10'
      • 16:00 Uniform Artin-Rees Results - Part 3 50'
        Speaker: Irena SWANSON (Purdue University, USA)
  • Monday, 11 July 2022
    • 15:00 - 16:50
      • 15:00 Uniform Artin-Rees Results - Tutorial 50'
        Tutor: Kriti GOEL, University of Utah, USA
        Speaker: Irena SWANSON (Purdue University, USA)
      • 15:50 Break 10'
      • 16:00 The Localization Problem - Part 1 50'
        Speaker: Wenliang ZHANG (University of Illinois, USA)
  • Wednesday, 13 July 2022
    • 14:55 - 15:00 Group photo 'virtual'
    • 15:00 - 16:50
      • 15:00 The Localization Problem - Part 2 50'
        Speaker: Wenliang ZHANG (University of Illinois, USA)
      • 15:50 Break 10'
      • 16:00 The Localization Problem - Part 3 50'
        Speaker: Wenliang ZHANG (University of Illinois, USA)
  • Monday, 18 July 2022
    • 15:00 - 16:50
      • 15:00 The Localization Problem - Tutorial 50'
        Tutor: Sudeshna ROY, Tata Institute of Fundamental Research, India
        Speaker: Wenliang ZHANG (University of Illinois, USA)
      • 15:50 Break 10'
      • 16:00 Symbolic Powers of Ideals - Part 1 50'
        Speaker: Eloisa GRIFO (University of Nebraska, USA)
  • Wednesday, 20 July 2022
    • 15:00 - 16:50
      • 15:00 Symbolic Powers of Ideals - Part 2 50'
        Speaker: Eloisa GRIFO (University of Nebraska, USA)
      • 15:50 Break 10'
      • 16:00 Symbolic Powers of Ideals - Part 3 50'
        Speaker: Eloisa GRIFO (University of Nebraska, USA)
  • Monday, 25 July 2022
    • 15:00 - 18:00
      • 15:00 Symbolic Powers of Ideals - Tutorial 50'
        Tutor: Vaibhav PANDEY, University of Utah, USA
        Speaker: Eloisa GRIFO (University of Nebraska, USA)
      • 15:50 Break 10'
      • 16:00 Seminar: Extending Tight Closure to Characteristic 0 and Open Questions in All Characteristics 50'
        Chair: Prof. Craig HUNEKE (University of Virginia, USA)
        
        Abstract and link to register is here: Seminar details
        Speaker: Melvin HOCHSTER (University of Michigan, Ann Arbor, MI, USA)
      • 16:50 Break 10'
      • 17:00 Announcement of Prize Winners 5'
        Speaker: Marilina ROSSI (University of Genoa, Italy)
      • 17:05 Vote of Thanks 5'
        Speakers: Lothar GOETTSCHE (ICTP, Italy), Jugal Kishore VERMA (Indian Institute of Technology, India)