Time: Monday 31st August 4 to 5pm (joining time 3.50pm)
Google Meet Link: https://meet.google.com/wnf-ywcy-ozi
Title: Geometric approach for the sheaf of A^1-connected components
Abstract: The $\mathbb{A}^1$-homotopy theory of Morel-Voevodsky attempts
to use
homotopical methods in algebraic geometry by having the affine line play
the role of the unit interval. Analogous to the set of connected
components of a topological space, one associates the sheaf of
$\mathbb{A}^1$-connected components to any variety. However, this sheaf is
extremely difficult to compute since its definition is mired in abstract
machinery. We will discuss how this sheaf may be studied by means using
purely algebro-geometric methods, via the sheaf of ``naively"
$\mathbb{A}^1$-connected components. This approach has been successful in
proving some results about the sheaf of $\mathbb{A}^1$-connected
components by very elementary techniques. We will look at some of these
results and briefly describe the techniques used to prove them. This talk
is based on joint work with Amit Hogadi and Anand Sawant.
Time:
5:30pm-6:30pm
Description:
Date and Time: Tuesday 1st September 2020, 5:30 pm IST - 6:30 pm IST
(joining time : 5:15 pm IST - 5:30 pm IST)
Google Meet link: https://meet.google.com/yqu-mvvy-jrs
Speaker: Matteo Varbaro, University of Genoa
Title: F-splittings of the polynomial ring and compatibly split
homogeneous ideals
Abstract: A polynomial ring R in n variables over a field K of positive
characteristic is F-split. It has many F-splittings. When K is a perfect
field every F-splitting is given by a polynomial g in R with the monomial
u^{p-1} in its support (where u is the product of all the variables)
occurring with coefficient 1, plus a further condition, which is not
needed if g is homogeneous (w.r.t. any positive grading). Fixed an
F-splitting s : R -> R, an ideal I of R such that s(I) is contained in I
is said compatibly split (w.r.t. the F-splittings). In this case R/I is
F-split. Furthermore, by Fedderâ€™s criterion when I is a homogeneous ideal
of R, R/I is F-split if and only if I is compatibly split for some
F-splitting s : R -> R. If, moreover, u^{p-1} is the initial monomial of
the associated polynomial g of s w.r.t. some monomial order, then in(I) is
a square-free monomial idealâ€¦ In this talk I will survey these facts (some
of them classical, some not so classical), and make some examples,
focusing especially on determinantal ideals.
Time:
7:00pm
Description:
The speaker is
Prof. Amritanshu Prasad from IMSc, Chennai. The following are the
details.
Title: Polynomials as Characters of Symmetric Groups.
Time: 7pm, Tuesday, September 1, 2020 (gate opens at 6:45pm).
Google meet link: meet.google.com/prm-feow-zwm.
Phone: (US) +1 740-239-3129 PIN: 706 683 026#
Abstract: Treating the variable $X_i$ as the number of $i$-cycles in a
permutation allows a polynomial in $X_1, X_2,\dotsc$ to be regarded as a
class function of the symmetric group $S_n$ for any positive integer $n$.
We present a simple formula for computing the average and signed average
of such a class function over the symmetric group. We use this formula to
investigate the dimension of $S_n$-invariant and $S_n$-sign-equivariant
vectors in polynomial representations of general linear groups.
This talk is based on joint work with Sridhar P Narayanan, Digjoy Paul,
and Shraddha Srivastava. Some of these results are available in the
preprint available at: http://arxiv.org/abs/2001.04112.
Time:
5:30pm-6:30pm
Description:
Date and Time: Friday 4th September 2020, 5:30 pm IST - 6:30 pm IST
(joining time : 5:15 pm IST - 5:30 pm IST)
Google Meet link: https://meet.google.com/yqu-mvvy-jrs
Speaker: Mandira Mondal, Chennai Mathematical Institute.
Title: Density functions for the coefficients of the Hilbert-Kunz function
of polytopal monoid algebra
Abstract: We shall discuss Hilbert-Kunz density function of a Noetherian
standard graded ring over a perfect field of characteristic $p \geq 0$. We
will also talk about the second coefficient of the Hilbert-Kunz function
and the possibility of existence of a $\beta$-density function for this
coefficient.
Watanabe and Eto have shown that Hilbert-Kunz multiplicity of affine
monoid rings with respect to a monomial ideal of finite colength can be
expressed as relative volume of certain nice set arising from the convex
geometry associated to the ring. In this talk, we shall discuss similar
expression for the density functions of polytopal monoid algebra with
respect to the homogeneous maximal ideal in terms of the associated convex
geometric structure. This is a joint work with Prof. V. Trivedi. We shall
also discuss the existence of $\beta$-density function for monomial prime
ideals of height one of these rings in this context.