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.

### Abstracts for July 29

#### Lecture 1 ​

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.

#### Lecture 2​

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).)

#### Lecture 3​

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.

#### Lecture 4​

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.

#### Lecture 5​

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.

#### Lecture 6​

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.

### Abstracts for July 30

#### Lecture 1 ​

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.

#### Lecture 2​

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 $k[X, Y, Z]$.

#### Lecture 3​

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.

#### Lecture 4​

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.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 22 Sep 2021