Seminar on Combinatorial aspects of
commutative algebra and algebraic geometry.
Title: What is a Syzygy?
This talk will be introduction to syzygies: basic theorems, examples and
some early applications. This is the first seminar on this topic.
Time:
4:00pm
Location:
Ramanujan Hall
Description:
Title: Asymptotic estimates on the geometry of Laplace eigenfunctions
Abstract: Given a closed smooth Riemannian manifold M, the Laplace operator
is known to possess a discrete spectrum of eigenvalues going to infinity.
We are interested in the properties of the nodal sets and nodal domains of
corresponding eigenfunctions in the high energy (semiclassical) limit. We
focus on some recent results on the size of nodal domains and tubular
neighbourhoods of nodal sets of
such high energy eigenfunctions (joint work with Bogdan Georgiev).
Time:
9:30am-10:25am
Location:
Ramanujan Hall
Description:
Title: Huneke-Itoh Intersection Theorem and its Consequences - I
Abstract: Huneke and Itoh independently proved a celebrated result on
integral closure of powers of an ideal generated by a regular sequence. As
a consequence of this theorem, one can find the Hilbert-Samuel polynomial
of the integral closure filtration of I if the normal reduction number is
at most 2. We prove Hong and Ulrich's version of the intersection theorem.
Time:
10:30am-11:25am
Location:
Ramanujan Hall
Description:
Title: Tate Resolutions - III
Abstract: Let S be a Noetherian ring, and R = S/I. It is always possible
to construct a differential graded algebra (DG-algebra) resolution of R
over S due to a result of Tate. If R is the residue field of S, then
Gulliksen proved that such a DG-algebra resolution is minimal. We shall
discuss the construction of the Tate resolution in our talk.
Time:
2:30pm-5:30pm
Location:
Venue (tentative): Room 216, Department of Mathematics
Description:
Title: Local Fields
Speaker: Nagarjuna Chary
Venue (tentative): Room 216, Department of Mathematics
Abstract: We will continue with the material in Chapter 1 in Cassels and
Frohlich.
Time:
2:00pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
Title : Hairy balls, fixed points and coffee!!!
Abstract :
Singularities occur naturally everywhere around us, may it be an eye of a
cyclone where there is no wind at all, or the north pole where the
different time zones converge. The purpose of this talk is to study these
patterns mathematically. Hairy ball theorem precisely states that: An even
dimensional sphere does not possess any continuous nowhere vanishing
tangent vector field". The basic notions of tangent vector field,
fundamental groups, some concepts of point set topology will be discussed
(at least intuitively) and then a geometric proof of the theorem will be
studied. It will be followed by a few applications in the end.
Time:
11:00am-12:30pm
Location:
Ramanujan Hall
Description:
Title: Ruzsa's theorem in additive combinatorics
Abstract: We show that in a finite group G of bouded torsion, any set
A \subseteq G such that |A + A| = O(|A|) generates a subgroup H of
size O(|A|). We will introduce some standard techniques in additive
combinatorics to prove this theorem.
Time:
3:30pm
Location:
Room 215, Department of Mathematics
Description:
Title:
Kodaira's theorem: criterion for embedding a compact Kahler manifold in
projective space
Abstract:
Let $M$ be a compact Kahler manifold and $\Omega (M)$ the canonical
$2$-form on $M$. When $M$ is projective $n$-spce $\P^n(\C)$ , $H^2(M,\C)$
is of dimension 1. It follows that for any Kahler metric on the projective
space, the cohomology class $[\Omega (M)$ of the canonical $2$-form is a
multiple of the (unique up to sign) of a generator of $H^2(M,\Z)$. It is
immediate from this that if $M$ is a complex sub-manifold of $\P^n(\C)$ for
some $n$, then for the Kahler metric on $M$ induced from one on $\P^n(\C)$,
it is clear that $[\Omega(M)] \in $\C \cdot H^2(M, Z)$. Kodaira's theorem
is a converse to this fact: If a complex manifold $M$ admits a Kahler
metric such that the class of $\Omega(M)$ is a multiple of an integral
class, then $M$ can be embedded in some projective space. This result was
conjectured by W V D Hodge.