9:30am» Neeraj Kumar- Time:
- 9:30am - 10:25am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Neeraj Kumar
Title: Linear resolutions of monomial ideals - II
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
- Description:
- Speaker: Kriti Goel
Title: Huneke-Itoh Intersection Theorem and its Consequences - III
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:
- 11:00am - 12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: A Sum Product theorem over finite fields
Abstract: Let A be a finite subset of a field F. Define A+A and AA to
be the set of pairwise sums and products of elements of A,
respectively. We will see a theorem of Bourgain, Katz and Tao that
shows that if neither A+A nor AA is much bigger than A, then A must be
(in some well-defined sense) close to a subfield of F.
- Time:
- 4:00pm - 5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Dr. Rashmi Tiwari, Department of Mathematics IIT Bombay
Title: SOME ASPECTS OF MOMENTS OF ORDERED STATISTICS
Abstract: Recurrence relations of moments which are useful to reduce
the amount of direct computations quite considerably and usefully
express the higher order moments of order statistics in terms of the
lower order moments and hence make the evaluation of higher order
moments easy. We have derived recurrence relations for single, double
(product) and higher moments of various ordered random variables, like
ordinary order statistics, progressively censored order statistics,
generalized order statistics and dual generalized order statistics
from some specific continuous distributions. It also deals with
L-moments and TL-moments which are analogous of the ordinary moments.
We have derived L-moments and TL-moments for some continuous
distributions. These results have been applied to find the L-moment
estimators and TL-moment estimators of the unknown parameters for some
specific continuous distributions.
- Time:
- 3:00pm - 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Dr. Satya P Singh,
Department of Statistics University of Haifa Israel
Title: Some issues in the design of experiments with ordered
experimental treatments
Abstract: There are many situations where one expects an ordering
among K>2 experimental groups or treatments. Although there is a
large body of literature dealing with the analysis under order
restrictions, surprisingly very little work has been done in the
context of the design of experiments. Here, we provide some key
observations and fundamental ideas which can be used as a guide for
designing experiments when an ordering among the groups is known in
advance. Designs maximizing power as well as designs based on single
and multiple contrasts are discussed. The theoretical findings are
supplemented by numerical illustrations.
- Time:
- 3:30pm
- Location:
- Room 215, Department of Mathematics
- Description:
- Title: Kodaira's theorem: criterion for embedding a compact Kahler
manifold in projective space (Lecture 2)
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.
- 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.
- Time:
- 11:30am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Free Resolutions of Monomial Ideals III.
Abstract: We continue the study of resolutions of monomial ideals.
We start with a short proof of the exactness of the Koszul complex. We
then generalize this to free resolutions of any monomial ideal. We'll
conclude with the proof of the Hilbert syzygy theorem and some more
examples of monomial ideals.
- Time:
- 2:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Tropical geometry of curves
Abstract: Perhaps surprisingly, the study of degenerate curves plays a
crucial role in our understanding of a general smooth curve. One of the
first successes of this idea was the theory of limit linear series
developed by Griffiths and Harris which they used to prove the
Brill-Noether theorem. The analogous theory for degenerate curves of
non-compact type falls in the realm of tropical geometry where it takes the
shape of metric graphs (or tropical curves) and divisors on them. This
leads to a rich interplay between graph theory and algebraic geometry of
curves. After explaining the central ideas we will discuss some
applications to Brill-Noether theory and curves of large theta
characteristic.
- Time:
- 4:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Developments in Fractional Dynamical Systems
Abstract: Fractional calculus (FC) is witnessing rapid development in
recent past. Due to its interdisciplinary nature, and applicability it has
become an active area of research in Science and Engineering. Present talk
deals with our work on fractional order dynamical systems (FODS), in
particular on local stable manifold theorem for FODS. Further we talk on
bifurcation analysis and chaos in the context of FODS. Finally we
conjecture a generalization of Poincare-Bendixon for fractional systems.
- Time:
- 9:30am - 10:25am
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Title: Linear resolutions of monomial ideals - IV
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:
- Title : Asymptotic prime divisors.
Abstract : Consider a Noetherian ring R and an ideal I of R. Ratliff asked
a question that what happens to Ass(R/I^n) as n gets large ? He was able
to answer that question for the integral closure of I. Meanwhile Brodmann
answered the original question, and proved that the set Ass(R/I^n)
stabilizes for large n.
We will discuss the proof of stability of Ass(R/I^n). We will also
give an example to show that the sequence is not monotone. The aim of
this
series of talk to present the first chapter of S. McAdam, Asymptotic
prime divisors, Lecture Notes in Mathematics 1023, Springer-Verlag,
Berlin, 1983.
- Time:
- 11:00am - 12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Niranjan Balachandran
Title: The Erdos-Heilbronn conjecture
Abstract: The conjecture of Erdos-Heilbronn (1964) states the following:
Suppose G is a a finite group and and we have a G-sequence
(g_1,g_2,...,g_l) of pairwise distinct g_i where l>2|G|^{1/2}, there is a
subsequence (g_{i_1},g_{i_2},..,g_{i_t}) (for some t) such that \prod_{j}
g_{i_j} = 1.
The conjecture is open in its fullness, but has been settled (up to a
constant) in some special cases of groups. We will see the proof of the
E-H conjecture for cyclic groups, by Szemeredi.
- Time:
- 3:00pm - 5:30pm
- Location:
- Room 215, Department of Mathematics
- Description:
Title: Higgs Bundles
Abstract: In this second talk of the series, I will continue the
discussion on Higgs bundles with focus on some of the moduli aspects of
the theory.
- Time:
- 9:30am - 10:25am
- Location:
- Ramanujan Hall
- Description:
- Title: Gotzmann's regularity and persistence theorem
Abstract: Gotzmann's regularity theorem establishes a bound on
Castelnuovo-Mumford regularity using a binomial representation (the
Macaulay representation) of the Hilbert polynomial of a standard graded
algebra. Gotzmann's persistence theorem shows that once the Hilbert
function of a homogeneous ideal achieves minimal growth then it grows
minimally for ever. We start with a proof of Eisenbud-Goto's theorem to
establish regularity in terms of graded Betti numbers. Then we discuss
Gotzmann's theorems in the language of commutative algebra.
- Time:
- 10:30am - 11:25am
- Location:
- Ramanujan Hall
- Description:
- Title : Asymptotic prime divisors - II
Abstract : Consider a Noetherian ring R and an ideal I of R. Ratliff asked
a question that what happens to Ass(R/I^n) as n gets large ? He was able
to answer that question for the integral closure of I. Meanwhile Brodmann
answered the original question, and proved that the set Ass(R/I^n)
stabilizes for large n.
We will discuss the proof of stability of Ass(R/I^n). We will also
give an example to show that the sequence is not monotone. The aim of
this series of talks to present the first chapter of S. McAdam,
Asymptotic prime divisors, Lecture Notes in Mathematics 1023,
Springer-Verlag, Berlin, 1983.
- Time:
- 2:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- The title and abstract of the talk are attached.
- Time:
- 2:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Typical representations for depth-zero representations.
Abstract: We are interested in understanding cuspidal support of smooth
representations of p-adic reductive group via understanding the
restriction of a smooth representation to a maximal compact subgroup. This
is motivated by arithmetic applications via local Langlands
correspondence. We will explain the case of general linear groups and
classical groups.
- Time:
- 3:30pm
- Location:
- Room 215
- Description:
- Title: Higgs Bundles
Abstract: We will define the moduli of Higgs bundles and the Hitchin
fibration, which is a morphism from the moduli of Higgs bundles to an
affine space. Then we will describe the general fibre in terms of the
spectral cover.
- Time:
- 9:30am - 10:25am
- Location:
- Ramanujan Hall
- Description:
- Speaker: Sudeshna Roy
Title: Gotzmann's regularity and persistence theorem - II
Abstract: Gotzmann's regularity theorem establishes a bound on
Castelnuovo-Mumford regularity using a binomial representation (the
Macaulay representation) of the Hilbert polynomial of a standard graded
algebra. Gotzmann's persistence theorem shows that once the Hilbert
function of a homogeneous ideal achieves minimal growth then it grows
minimally for ever. We start with a proof of Eisenbud-Goto's theorem to
establish regularity in terms of graded Betti numbers. Then we discuss
Gotzmann's theorems in the language of commutative algebra.
- Time:
- 10:30am - 11:25am
- Location:
- Ramanujan Hall
- Description:
- Speaker: Provanjan Mallick
Title : Asymptotic prime divisors - III
Abstract : Consider a Noetherian ring R and an ideal I of R. Ratliff asked
a question that what happens to Ass(R/I^n) as n gets large ? He was able
to answer that question for the integral closure of I. Meanwhile Brodmann
answered the original question, and proved that the set Ass(R/I^n)
stabilizes for large n.
We will discuss the proof of stability of Ass(R/I^n). We will also
give an example to show that the sequence is not monotone. The aim of
this series of talks to present the first chapter of S. McAdam,
Asymptotic prime divisors, Lecture Notes in Mathematics 1023,
Springer-Verlag, Berlin, 1983.
- Time:
- 3:30pm
- Location:
- Room 216
- Description:
- Speaker: Rekha Santhanam
We will talk about relative homotopy groups, long exact sequence in
homotopy and cellular approximation theorem.