Akash Sengupta, Wed 2013-10-09, 15:30
Title: Classical Geometry of del Pezzo Surfaces

Abstract: The aim of this talk is to give an introduction to enumerative
aspects of the geometry of del Pezzo surfaces. We will start by recalling
basics of the theory of complex algebraic surfaces, such as intersection
products, blow-ups and minimal models of surfaces. We will describe
embeddings of del Pezzo surfaces into projective spaces and count the
number of lines in the embedded surface.

Thu 2013-10-17 10:30-11:45 Room LH-2
Speaker: Anwesh Ray
Title: Number of Number fields

Abstract: There are finitely many number fields of a fixed degree 'n' and
discriminant bounded by 'X', it's natural to get estimates on this number
in terms of n and X. These numbers have been described by davenport
heilbronn and manjul bhargava for the cases where the degree n is 3 or 4.
The best upper bound involves an especially generic and ingenious
construction due to Ellenberg and Venkatesh (2006) which I will outline in
my talk. Number field counting is a hot topic in number theory and there
are reasons to believe that a much better upper bound will soon be found.


Rohith Varma,  Wed 2013-10-23: POSTPONED

Wed 2013-10-30, 15:30-16:45 Room LH-1.
Speaker: Suratno Basu
Title: Hodge Decomposition of Cohomology Groups

Abstract: Cohomology is a topological invariant of a space. The cohomology
groups of a class of complex manifolds, namely projective manifolds (i.e.,
smooth projective varieties), decompose into a direct sum of smaller
subspaces in certain manner known as Hodge decomposition. This
decomposition is functorial in the sense that if we have a holomorphic map
between two complex manifolds then the induced map on the cohomology groups
preserves the decomposition. This additional structure on cohomology
carries information about the geometry of the manifold. Now if the variety
is not smooth or is an open subvariety of a projective variety then its
cohomology groups may not decompose as a direct sum as above, but we can
give them a filtration such that each successive quotient has a Hodge
decomposition.  This generalization of Hodge decomposition is known as a
mixed Hodge structure. Delinge proved the existence of mixed Hodge
structures for such varieties. In this talk we will briefly explain Hodge
decomposition and analyze how the Mixed Hodge structure arises in some
simple cases.

Thu 2013-11-07 10:30-11:45 Room LH-2
Speaker: Shraddha Srivastava
Title: Polynomial representations of $\mathrm{GL}_n$ and the category of strict
polynomial functors of degree $d$.

Abstract: We will first discuss the polynomial representations of 
$\mathrm{GL}_n$ and define the category of polynomial functors of degree d :
$\mathrm{Rep}\Gamma^d_K$, where K is a field. We will see this category 
equivalent to the category of polynomial representations of $\mathrm{GL}_n$
of degree $d$, when $n \geq d$. We define an internal tensor product in
$\mathrm{Rep}\Gamma^d_K$. By the above equivalence of categories, this defines a
tensor product on polynomial representations of 
$\mathrm{GL}_n$ of the same degree. There is also an internal Hom, with
the usual Hom-tensor adjunction. Using these two bifunctors with the
exterior power as an argument, we get a version of Koszul duality.


Thu 2013-11-15 10:30-11:45 Room LH-2
Speaker: Ronno Das
Title: Automorphisms of Oriented Surfaces

Abstract: We will discuss the Nielsen-Thurston classification of
automorphisms of surfaces. Every homeomorphism of the (2-)torus is, up to
isotopy, either periodic, a power of a Dehn twist, or among what are called
the Anosov homeomorphisms, stretching and shrinking by the same factor
along two transverse directions. We will state the analogous classification
of homeomorphisms of (orientable) genus g surfaces for g>1, and define the
important construction of laminations via hyperbolic structures. We will
see how the pseudo-Anosov homeomorphisms can be characterized via their
action on laminations.