Time: Monday 21st September 4 to 5pm (joining time 3.50pm)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Algebraic Cycles and Modular Forms
Abstract: There are many instances when special sub-varieties of Shimura
varieties give rise to modular forms. One such is the theorem of Gross and
Zagier linking Heegner divisors with coefficients of modular forms. We
discuss a generalisation of this theorem to higher codimensional cycles
which implies the existence of certain motivic cycles in the universal
families over these Shimura varieties. In special cases we construct some
examples which have applications to another conjecture of Gross and Zagier
on algebraicity of values of Greens functions.
Time:
5:30pm
Description:
Date and Time: 22 September 2020, 5:30 pm IST (joining time : 5:15 pm IST
- 5:30 pm IST)
Google meet link: meet.google.com/sdz-bspz-uhu
Speaker: Shunsuke Takagi, University of Tokyo
Title: F-singularities and singularities in birational geometry - Part 1
Abstract: F-singularities are singularities in positive characteristic
defined using the Frobenius map and there are four basic classes of
F-singularities: F-regular, F-pure, F-rational and F-injective
singularities. They conjecturally correspond via reduction modulo $p$ to
singularities appearing in complex birational geometry. In the first talk,
I will survey basic properties of F-singularities. In the second talk, I
will explain what is known and what is not known about the correspondence
of F-singularities and singularities in birational geometry. If the time
permits, I will also discuss its geometric applications.
Time:
10:30am
Description:
Speaker: Dr. Brett Parker, Monash University.
Time: 10:30 AM, IST, 24 September 2020 (gate open: 10:20 AM).
Title: Tropical counts of Gromov-Witten invariants in dimension 3.
Abstract: Tropical curves appear when we study holomorphic curves under
certain degenerations, or relative to normal-crossing divisors. In many
cases, there is a correspondence between counting tropical curves and
Gromové‚‘itten invariants. In complex dimension 3, this correspondence has
the wonderful feature that each tropical curve corresponds to
Gromové‚‘itten invariants counting curves in all genus. I will illustrate
some examples of this correspondence, including some interesting examples
counting Gromov-Witten invariants in log Calabi-Yau manifolds, where our
tropical curves live in a 3 dimensional integral affine space with
singularities along a 1-dimensional locus.
Time:
4:00pm-5:00pm
Description:
Date and Time: Friday 25 September, 04.00pm - 05.00pm
Google Meet link: https://meet.google.com/mvd-txng-kgf
Speaker: Sarjick Bakshi, CMI
Title: GIT quotients of Grassmannian and smooth quotients of Schubert
varieties
Abstract: The Geometric invariant theory (GIT) quotients of the
Grassmannian variety and its subvarieties lead to many interesting
geometric problems. Gelfand and Macpherson showed that the GIT quotient of
n-points in {\mathbb P}^{r-1} by the diagonal action of PGL(r,\mathbb{C})
is isomorphic to the GIT quotient of Gr_{r,n} with respect to the
T-linearized line bundle {\cal L}(n \omega_r). Howard, Milson, Snowden and
Vakil gave an explicit description of the generators of the ring of
invariants for n even and r=2 using graph theoretic methods. We give an
alternative approach where we study the generators using Standard monomial
theory and we will establish the projective normality of the quotient
variety for odd n and r=2.
Let r < n be positive integers and further suppose r and n are coprime. We
study the GIT quotient of Schubert varieties X(w) in the Gr_{r,n}
admitting semistable points for the action of T with respect to the
T-linearized line bundle {\cal L}(n \omega_r). We give necessary and
sufficient combinatorial conditions for w for which the GIT quotient of
the Schubert variety is smooth.
Time:
5:30pm
Description:
Date and Time: 25 September 2020, 5:30 pm IST (joining time : 5:15 pm IST
- 5:30 pm IST)
Google meet link: meet.google.com/sdz-bspz-uhu
Speaker: Shunsuke Takagi, University of Tokyo
Title: F-singularities and singularities in birational geometry - Part 2
Abstract: F-singularities are singularities in positive characteristic
defined using the Frobenius map and there are four basic classes of
F-singularities: F-regular, F-pure, F-rational and F-injective
singularities. They conjecturally correspond via reduction modulo $p$ to
singularities appearing in complex birational geometry. In the first talk,
I will survey basic properties of F-singularities. In the second talk, I
will explain what is known and what is not known about the correspondence
of F-singularities and singularities in birational geometry. If the time
permits, I will also discuss its geometric applications.