We cordially invite you to the (online) symposium commemorating the superannuation of Professor Dilip P. Patil (IISc Bangalore).

**Program Committee:** Apoorva Khare (IISc Bangalore), Ravi A. Rao (NMIMS, Mumbai), Jugal K. Verma (IIT Bombay)

**Technical Committee:** Kriti Goel (IIT Gandhinagar), Shreedevi Masuti (IIT Dharwad), R. Venkatesh (IISc Bangalore), Jugal K. Verma (IIT Bombay)

The programme schedule for the symposium is as follows:

**Date:** 29th July, 2021 (Thursday)

**Venue:** Zoom (online) + YouTube (live-streaming)

Time | Speaker & Title |
---|---|

1.55 pm - 2.25 pm | Jürgen Herzog A short survey on numerical semigroups [slides] [video] |

2.30 pm - 3.00 pm | Jugal K. Verma The Chern number of an $I$-good filtration of ideals [slides] [video] |

3.05 pm - 3.35 pm | Indranath Sengupta Some results on numerical semigroup rings [slides] [video] |

3.35 pm - 3.50 pm | Break |

3.50 pm - 4.20 pm | Rajendra V. Gurjar $\mathbb{A}^1$-fibrations on affine varieties [slides] [video] |

4.25 pm - 4.55 pm | Martin Kreuzer Differential methods for $0$-dimensional schemes [slides] [video] |

5.00 pm - 5.30 pm | Leslie Roberts Ideal generators of projective monomial curves in $\mathbb{P}^3$ [slides] [video] |

**Date:** 30th July, 2021 (Friday)

**Venue:** Zoom (online) + YouTube (live-streaming)

Time | Speaker & Title |
---|---|

10.00 am - 10.30 am | Shreedevi Masuti The Waring rank of binary binomial forms [slides] [video] |

10.35 am - 11.05 am | Parnashree Ghosh Homogeneous locally nilpotent derivations of rank $2$ and $3$ on $k[X,Y,Z]$ [slides] [video] |

11.05 am - 11.15 am | Break |

11.15 am - 11.45 am | Kriti Goel On Row-Factorization matrices and generic ideals [slides] [video] |

11.50 pm - 12.20 pm | Neena Gupta On $2$-stably isomorphic four dimensional affine domains [slides] [video] |

Each lecture will be of 30 minutes with 5 minutes break for Q&A and change of speaker.

**Speaker:** Jürgen Herzog (Universität Duisburg–Essen, Germany)

**Title:** A short survey on numerical semigroups

**Abstract:**
In this lecture I will give a short survey on numerical semigroups from a
viewpoint of commutative algebra. A numerical semigroup is a subsemigroup
$S$ of the additive semigroup of non-negative integers. One may assume
that the greatest common divisor of the elements of $s$ is one. Then
there is an integer $F(S) \not\in S$, such that all integers bigger than
$F(S)$ belong to $S$. This number is called the Frobenius number of $S$.
For a fixed field $K$ one considers the $K$-algebra $K[S]$ which is the
subalgebra of the polynomial ring $K[t]$ which is generated over $K$ by
the powers $t^s$ with $s\in S$. This algebra is finitely generated and
its relation ideal $I(S)$ is a binomial ideal. In general it is hard to
compute $I(S)$. I will recall what is known about this ideal by my own
work but also by the work of Bresinsky, Delorme, Gimenez, Sengupta and
Srinivasan, Patil and others. The semigroup ring $K[S]$ is a
Cohen–Macaulay domain, and by the theorem of Kunz it is Gorenstein if
and only if the semigroup $S$ is symmetric. Barucci, Dobbs and Fontana
introduced pseudo-symmetric numerical semigroups. This concept was
generalized by Barucci and Fröberg, who introduced almost symmetric
numerical semigroups. The corresponding semigroup ring is called almost
Gorenstein. One can define almost Gorenstein rings not only in dimension
$1$. A full-fledged theory in this direction has been developed by Goto,
Takahashi and Taniguchi. By considering the trace of the canonical ideal
of a numerical semigroup ring one is led to define nearly Gorenstein
numerical semigroups, as has been done by Hibi, Stamate and myself. I
will briefly discuss these generalizations of Gorensteiness and address a
few open problems related to this.

**Speaker:** Jugal K. Verma (IIT Bombay, India)

**Title:** The Chern number of an $I$-good filtration of ideals

**Abstract:**
Let $I$ be an $\mathfrak m$-primary ideal of a Noetherian local ring $R$.
Let $\mathcal F$ be an $I$-good filtration of ideals. The second Hilbert
coefficient $e_1(\mathcal F)$ of the Hilbert polynomial of $\mathcal F$
is called its Chern number. We discuss how the vanishing of the Chern
number characterizes Cohen–Macaulay local rings, regular local rings and
$F$-rational local rings using the $I$-adic filtration, the filtrations
of the integral closure of powers, and the filtration of the tight
closure of powers of a parameter ideal. We provide a partial answer to a
question of C. Huneke about $F$-rational local rings.

(This is joint work with Saipriya Dubey (IIT Bombay) and Pham Hung Quy (FPT University, Vietnam).)

**Speaker:** Indranath Sengupta (IIT Gandhinagar, India)

**Title:** Some results on numerical semigroup rings

**Abstract:**
We will discuss Professor Patil’s contribution in the field of numerical
semigroups and my association with the subject through some old and
recent results.

**Speaker:** Rajendra V. Gurjar (IIT Bombay, India)

**Title:** $\mathbb{A}^1$-fibrations on affine varieties

**Abstract:**
We will begin with the fundamental result of Fujita–Miyanishi–Sugie that
a smooth affine surface $V$ has log Kodaira dimension $-\infty$ if and
only if $V$ has an $\mathbb{A}^1$-fibration over a smooth curve.
Generalizations of this to singular affine surfaces and higher
dimensional affine varieties raise non-trivial questions. We will
describe some results in these directions. Connection with
locally-nilpotent derivations will be mentioned. Use of topological
arguments for proving some of these results will be indicated.

**Speaker:** Martin Kreuzer (Universität Passau, Germany)

**Title:** Differential methods for $0$-dimensional schemes

**Abstract:**
Given a $0$-dimensional subscheme $X$ in $\mathbb{P}^n$, the traditional
way to study the geometry of $X$ is to look at algebraic properties of
its homogeneous coordinate ring $R = K[x_0, \ldots, x_n]/I_X$ and the
structure of the canonical module of $R$.

Here we introduce and exploit a novel approach: we look at the Kähler differential algebra $\Omega_{R/K}$ which is the exterior algebra over the Kähler differential module $\Omega^1_{R/K}$ of $X$. Based on a careful examination of the embedding of R into its normal closure and the corresponding embedding of $\Omega^1_{R/K}$, we provide new bounds for the regularity index of the Kähler differential module and connect it to the geometry of $X$ in low embedding dimensions.

**Speaker:** Leslie Roberts (Queen’s University, Canada)

**Title:** Ideal generators of projective monomial curves in $\mathbb{P}^3$

**Abstract:**
I discuss ideal generators of projective monomial curves of degree $d$ in
$\mathbb{P}^3$, based on the paper P. Li, D.P. Patil and L. Roberts,
*Bases and ideal generators for projective monomial curves*,
Communications in Algebra, 40(1), pages 173–191, 2012, which was my last
paper with Dilip. I also discuss more recent observations by Ping Li and
myself.

**Speaker:** Shreedevi Masuti (IIT Dharwad, India)

**Title:** The Waring rank of binary binomial forms

**Abstract:**
It is well known that every form $F$ of degree $d$ over a field can be
expressed as a linear combination of $d$th powers of linear forms. The
least number of summands required for such an expression of $F$ is known
as the Waring rank of $F$. Computing the Waring rank of a form is a
classical problem in mathematics. In this talk we will discuss the Waring
rank of binary binomial forms. This is my joint work with L. Brustenga.

**Speaker:** Parnashree Ghosh (ISI Kolkata, India)

**Title:** Homogeneous locally nilpotent derivations of rank $2$ and $3$ on $k[X,Y,Z]$

**Abstract:**
In this talk we will discuss homogeneous locally nilpotent derivations
(LND) on $k[X, Y, Z]$ where $k$ is a field of characteristic $0$. For a
homogeneous locally nilpotent derivation $D$ on the polynomial ring in
three variables we will see how the $\deg_D$ values of the linear terms
are related and see a consequence on the rank $3$ homogeneous derivations
of degree $\leq 3$.

Further we will discuss homogeneous locally nilpotent derivations of rank $2$ and give a characterization of the triangularizable derivations among those. We will also see the freeness property of a homogeneous triangularizable LND on $k[X, Y, Z]$.

**Speaker:** Kriti Goel (IIT Gandhinagar, India)

**Title:** On Row-Factorization matrices and generic ideals

**Abstract:**
The concept of Row-factorization (RF) matrices was introduced by A.
Moscariello to explore the properties of numerical semigroups. For
numerical semigroups $H$ minimally generated by an almost arithmetic
sequence, we give a complete description of the RF-matrices associated
with their pseudo-Frobenius elements. We use the information from
RF-matrices to give a characterization of the generic nature of the
defining ideal of the semigroup. Further, when $H$ is symmetric and has
embedding dimension 4 or 5, we prove that the defining ideal is minimally
generated by RF-relations.

This is joint work with Om Prakash Bhardwaj and Indranath Sengupta.

**Speaker:** Neena Gupta (ISI Kolkata, India)

**Title:** On $2$-stably isomorphic four dimensional affine domains

**Abstract:**
A famous theorem of Abhyankar–Eakin–Heinzer proves that if $A$ is a one
dimensional ring containing $\mathbb{Q}$ and $n \ge 1$ be such that the
polynomial ring in $n$-variables over $A$ is isomorphic to the polynomial
ring in $n$ variables over $B$ for some ring $B$, then $A \cong B$. This
does not hold in higher dimensional rings in general. In this connection
the following question arises:

*If $A[X,Y] \cong B[X,Y]$, does this imply $A \cong B$?*

In this talk we shall present four dimensional seminormal affine domains over ${\mathbb C}$ for which the above question does not hold.

This is a joint work with Professor T. Asanuma.

Last updated: 03 Feb 2023