Title of Seminar: Algebraic Geometry Preprint Seminar 2022
Title of Talk: Newton-Okounkov bodies and Picard numbers on surfaces
Speaker: Aditya Subramaniam, TIFR, Mumbai
Date: January 11, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this talk, we will discuss a preprint of Fernández-Nickel-Roé 'Newton-Okounkov bodies and Picard numbers on surfaces' https://arxiv.org/pdf/2101.05338v1.pdf Newton-Okounkov bodies were introduced by A. Okounkov as a tool in representation theory; later Kaveh-Khovanskii and Lazarsfeld-Mustata developed a general theory with applications to both convex and algebraic geometry. In this preprint, the authors study the shapes of all Newton-Okounkov bodies of a given big divisor on a surface S with respect to all rank 2 valuations of K(S). They obtain upper bounds for, and in many cases determine exactly, the possible numbers of vertices of these bodies. The upper bounds are expressed in terms of Picard numbers. They also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor determines the Picard number of S, and proves that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.
Title of Seminar: Number Theory Seminar
Title of Talk: On the Wiles-Lenstra-Diamond numerical criterion for freeness of modules over complete intersections
Speaker: Chandrashekhar Khare, University of California, Los Angeles
Date: January 11, 2023
Time: 14:30:00 Hours
Venue: AG-77
Abstract: I will talk about recent work with Srikanth Iyengar and Jeff Manning on a higher codimension version of the Wiles--Lenstra--Diamond numerical criterion (the original version is in codimension $0$). The original version played a key role in Wiles?s work on the modularity of semistable elliptic curves over the rationals. I will sketch some (conditional) applications of the commutative algebra we develop to proving integral $R=T$ theorems in positive defect (as arise when considering $2$-dimensional Galois representations over imaginary quadratic fields, a defect one situation), and other questions/perspectives the work leads to. There is an unconditional application to proving an analog of the Jacquet--Langlands correspondence for Hecke algebras acting on the cohomology of Shimura curves with coefficients in weight one sheaves. As these Hecke algebras typically have a lot of torsion, such results cannot be deduced from the classical Jacquet--Langlands correspondence for classical weight one forms.
Title of Talk: Newton-Okounkov bodies and Picard numbers on surfaces
Speaker: Aditya Subramaniam, TIFR, Mumbai
Date: January 11, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this talk, we will discuss a preprint of Fernández-Nickel-Roé 'Newton-Okounkov bodies and Picard numbers on surfaces' https://arxiv.org/pdf/2101.05338v1.pdf Newton-Okounkov bodies were introduced by A. Okounkov as a tool in representation theory; later Kaveh-Khovanskii and Lazarsfeld-Mustata developed a general theory with applications to both convex and algebraic geometry. In this preprint, the authors study the shapes of all Newton-Okounkov bodies of a given big divisor on a surface S with respect to all rank 2 valuations of K(S). They obtain upper bounds for, and in many cases determine exactly, the possible numbers of vertices of these bodies. The upper bounds are expressed in terms of Picard numbers. They also conjecture that the set of all Newton-Okounkov bodies of a single ample divisor determines the Picard number of S, and proves that this is the case for Picard number 1, by an explicit characterization of surfaces of Picard number 1 in terms of Newton-Okounkov bodies.
Title of Seminar: Number Theory Seminar
Title of Talk: On the Wiles-Lenstra-Diamond numerical criterion for freeness of modules over complete intersections
Speaker: Chandrashekhar Khare, University of California, Los Angeles
Date: January 11, 2023
Time: 14:30:00 Hours
Venue: AG-77
Abstract: I will talk about recent work with Srikanth Iyengar and Jeff Manning on a higher codimension version of the Wiles--Lenstra--Diamond numerical criterion (the original version is in codimension $0$). The original version played a key role in Wiles?s work on the modularity of semistable elliptic curves over the rationals. I will sketch some (conditional) applications of the commutative algebra we develop to proving integral $R=T$ theorems in positive defect (as arise when considering $2$-dimensional Galois representations over imaginary quadratic fields, a defect one situation), and other questions/perspectives the work leads to. There is an unconditional application to proving an analog of the Jacquet--Langlands correspondence for Hecke algebras acting on the cohomology of Shimura curves with coefficients in weight one sheaves. As these Hecke algebras typically have a lot of torsion, such results cannot be deduced from the classical Jacquet--Langlands correspondence for classical weight one forms.