October 9, 2017 - October 13, 2017
Public Access Category: All |
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- Time:
- 11:30am-1:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Madhusudan Manjunath
Title: Free Resolutions of Monomial Ideals II
Abstract: We will study free resolutions of monomial ideals via the
concept of labelled simplicial complexes due to Bayer, Peeva and
Sturmfels. We will derive a criterion for a labelled complex to define a
free resolution. As applications, we will obtain the exactness of the
Koszul complex and the Hilbert syzygy theorem. If time permits, we will
obtain a formula for Betti numbers of a monomial ideal in terms of a
corresponding labelled simplicial complex.
- Time:
- 9:30am-10:25am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Neeraj Kumar
Title: Linear resolutions of monomial ideals - III
Abstract: Consider a graded ideal in the polynomial ring in several
variables. We shall discuss criterion for the graded ideal and its power
to have linear resolution. Then we focus our attention
to study linear resolution of monomial ideals.
Monomial ideals are the bridge between commutative algebra and the
combinatorics. Monomial ideals are also significant because they appear as
initial ideals of arbitrary ideals. Since many properties of an initial
ideal are inherited by its original ideal, one often adopt this strategy
to decipher properties of general ideals. The first talk is meant for
covering the preliminary results on resolution and regularity of monomial
ideal.The aim of this series of talk is to present the result in
arXiv:1709.05055
- Time:
- 10:30am-11:25am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Kriti Goel
Title: Huneke-Itoh Intersection Theorem and its Consequences - IV
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:
- 3:30pm-5:00pm
- Location:
- Room 215, Department of Mathematics
- Description:
- Speaker: Sudarshan Gurjar
Title: Introduction to Higgs bundles
Abstract: A Higgs bundle on a compact Kahler manifold M consists of a
holomorphic vector bundle E together with a holomorphic 1-form with values
in End(E), say \phi, such that \phi^\phi = 0 as a 2-form with values in
End(E). It turns out that there is a one to one correspondence between
irreducible representations of fundamental group of M and stable Higgs
bundles on M with vanishing Chern classes. This can be seen as the
analogue of the Narasimhan-Seshadri theorem connecting irreducible unitary
representations of the fundamental group with stable, flat vector bundles.