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SUMMARY:Juergen Herzog (University of Duisberg-Essen\, Germany)
DTSTART:20210729T082500Z
DTEND:20210729T085500Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/1
DESCRIPTION:Title: A short survey of numerical semigroup rings\nby Juergen Her
zog (University of Duisberg-Essen\, Germany) as part of Symposium in honou
r of Dilip Patil\n\n\nAbstract\nIn this lecture\, I will give a short surv
ey on numerical semigroups from a\nviewpoint of commutative algebra. A num
erical semigroup is a subsemigroup\n$S$ of the additive semigroup of non-n
egative integers. One may assume\nthat the greatest common divisor of the
elements of $s$ is one. Then\nthere is an integer $F(S) \\not\\in S$\, suc
h that all integers bigger than\n$F(S)$ belongs to $S$. This number is cal
led the Frobenius number of $S$.\nFor a fixed field\, $K$ one considers th
e $K$-algebra $K[S]$ which is the\nsubalgebra of the polynomial ring $K[t]
$ which is generated over $K$ by\nthe powers $t^s$ with $s\\in S$. This a
lgebra is finitely generated and\nits relation ideal $I(S)$ is a binomial
ideal. In general\, it is hard to\ncompute $I(S)$. I will recall what is
known about this ideal by my own\nwork but also by the work of Bresinsky\,
Delorme\, Gimenez\, Sengupta and\nSrinivasan\, Patil\, and others. The se
migroup ring $K[S]$ is a\nCohen--Macaulay domain\, and by the theorem of K
unz it is Gorenstein if\nand only if the semigroup $S$ is symmetric. Baruc
ci\, Dobbs\, and Fontana\nintroduced pseudo-symmetric numerical semigroups
. This concept was\ngeneralized by Barucci and Fröberg\, who introduced a
lmost symmetric\nnumerical semigroups. The corresponding semigroup ring is
called almost\nGorenstein. Onc can define almost Gorenstein rings not on
ly in dimension\n$1$. A full-fledged theory in this direction has been dev
eloped by Goto\,\nTakahashi and Taniguchi. By considering the trace of the
canonical ideal\nof a numerical semigroup ring one is led to define nearl
y Gorenstein\nnumerical semigroups\, as has been done by Hibi\, Stamate\,
and myself. I\nwill briefly discuss these generalizations of Gorensteiness
and address a\nfew open problems related to this.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jugal K Verma (IIT Bombay\, India)
DTSTART:20210729T090000Z
DTEND:20210729T093000Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/2
DESCRIPTION:Title: On the vanishing of the Chern number of filtrations of ideals\nby Jugal K Verma (IIT Bombay\, India) as part of Symposium in honour o
f Dilip Patil\n\n\nAbstract\nLet $I$ be an $\\mathfrak m$-primary ideal of
a Noetherian local ring $R$.\nLet $\\mathcal F$ be an $I$-good filtration
of ideals. The second Hilbert\ncoefficient $e_1(\\mathcal F)$ of the Hilb
ert polynomial of $\\mathcal F$\nis called its Chern number. We discuss ho
w the vanishing of the Chern\nnumber characterizes Cohen--Macaulay local r
ings\, regular local rings and\n$F$-rational local rings using the $I$-adi
c filtration\, the filtrations\nof the integral closure of powers\, and th
e filtration of the tight\nclosure of powers of a parameter ideal. We prov
ide a partial answer to a\nquestion of Craig Huneke about $F$-rational loc
al rings. \n\nThis is joint work with Saipriya Dubey (IIT Bombay) and Pham
Hung\nQuy (FPT University\, Vietnam).\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Indranath Sengupta (IIT Gandhinagar\, India)
DTSTART:20210729T093500Z
DTEND:20210729T100500Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/3
DESCRIPTION:Title: Some results on numerical semigroup rings\nby Indranath Sen
gupta (IIT Gandhinagar\, India) as part of Symposium in honour of Dilip Pa
til\n\n\nAbstract\nWe will discuss Professor Patil's contribution in the f
ield of numerical\nsemigroups and my association with the subject through
some old and\nrecent results.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajendra V Gurjar (IIT Bombay\, India)
DTSTART:20210729T102000Z
DTEND:20210729T105000Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/4
DESCRIPTION:Title: On $\\mathbb A^1$-fibrations of affine varieties\nby Rajend
ra V Gurjar (IIT Bombay\, India) as part of Symposium in honour of Dilip P
atil\n\n\nAbstract\nWe will begin with the fundamental result of Fujita--M
iyanishi--Sugie that\na smooth affine surface $V$ has log Kodaira dimensio
n $-\\infty$ if and\nonly if $V$ has an $\\mathbb{A}^1$-fibration over a s
mooth curve.\nGeneralizations of this to singular affine surfaces and high
er\ndimensional affine varieties raise non-trivial questions. We will\ndes
cribe some results in these directions. Connection with\nlocally-nilpotent
derivations will be mentioned. Use of topological\narguments for proving
some of these results will be indicated.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Kreuzer (University of Passau\, Germany)
DTSTART:20210729T105500Z
DTEND:20210729T112500Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/5
DESCRIPTION:Title: Differential methods for 0-dimensional schemes\nby Martin K
reuzer (University of Passau\, Germany) as part of Symposium in honour of
Dilip Patil\n\n\nAbstract\nGiven a $0$-dimensional subscheme $X$ in $\\mat
hbb{P}^n$\, the traditional\nway to study the geometry of $X$ is to look a
t algebraic properties of\nits homogeneous coordinate ring $R = K[x_0\,\\l
dots\,x_n]/I_X$ and the\nstructure of the canonical module of $R$.\n\nHere
we introduce and exploit a novel approach: we look at the Kaehler\ndiffer
ential algebra $\\Omega_{R/K}$ which is the exterior algebra over\nthe Kae
hler differential module $\\Omega^1_{R/K}$ of $X$. Based on a\ncareful exa
mination of the embedding of R into its normal closure and the\ncorrespond
ing embedding of $\\Omega^1_{R/K}$\, we provide new bounds for\nthe regula
rity index of the Kaehler differential module and connect it\nto the geome
try of $X$ in low embedding dimensions.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leslie Roberts (Queen's University\, Canada)
DTSTART:20210729T113000Z
DTEND:20210729T120000Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/6
DESCRIPTION:Title: Ideal generators of projective monomial curves in $\\mathbb{P}^
3$\nby Leslie Roberts (Queen's University\, Canada) as part of Symposi
um in honour of Dilip Patil\n\n\nAbstract\nI discuss ideal generators of p
rojective monomial curves of degree $d$ in\n$\\mathbb{P}^3$\, based on the
paper P. Li\, D.P. Patil and L. Roberts\,\nBases and ideal generators for
projective monomial curves\,\nCommunications in Algebra\, 40(1)\, pages 1
73--191\, 2012\, which was my last\npaper with Dilip. I also discuss more
recent observations by Ping Li and\nmyself.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreedevi Masuti (IIT Dharwad\, India)
DTSTART:20210730T043000Z
DTEND:20210730T050000Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/7
DESCRIPTION:Title: The Waring rank of binary binomial forms\nby Shreedevi Masu
ti (IIT Dharwad\, India) as part of Symposium in honour of Dilip Patil\n\n
\nAbstract\nIt is well known that every form $F$ of degree $d$ over a fiel
d can be\nexpressed as a linear combination of $d$th powers of linear form
s. The\nleast number of summands required for such an expression of $F$ is
known\nas the Waring rank of $F$. Computing the Waring rank of a form is
a\nclassical problem in mathematics. In this talk\, we will discuss the Wa
ring\nrank of binary binomial forms. This is my joint work with L. Brusten
ga.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parnashree Ghosh (ISI\, Kolkata\, India)
DTSTART:20210730T050500Z
DTEND:20210730T053500Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/8
DESCRIPTION:Title: Homogeneous locally nilpotent derivations of rank 2 and 3 on $k
[X\, Y\, Z]$\nby Parnashree Ghosh (ISI\, Kolkata\, India) as part of S
ymposium in honour of Dilip Patil\n\n\nAbstract\nIn this talk we will disc
uss homogeneous locally nilpotent derivations\n(LND) on $k[X\, Y\, Z]$ whe
re $k$ is a field of characteristic $0$. For a\nhomogeneous locally nilpot
ent derivation $D$ on the polynomial ring in\nthree variables we will see
how the $\\deg_D$ values of the linear terms\nare related and see a conseq
uence on the rank $3$ homogeneous derivations\nof degree $\\leq 3$.\n\nFur
ther we will discuss homogeneous locally nilpotent derivations of rank\n$2
$ and give a characterization of the triangularizable derivations among\nt
hose. We will also see the freeness property of a homogeneous\ntriangulari
zable LND on $k[X\, Y\, Z]$.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kriti Goel (IIT Gandhinagar\, India)
DTSTART:20210730T054500Z
DTEND:20210730T061500Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/9
DESCRIPTION:Title: On Row-Factorization matrices and generic ideals\nby Kriti
Goel (IIT Gandhinagar\, India) as part of Symposium in honour of Dilip Pat
il\n\n\nAbstract\nThe concept of Row-factorization (RF) matrices was intro
duced by A.\nMoscariello to explore the properties of numerical semigroups
. For\nnumerical semigroups $H$ minimally generated by an almost arithmeti
c\nsequence\, we give a complete description of the RF-matrices associated
\nwith their pseudo-Frobenius elements. We use the information from\nRF-ma
trices to give a characterization of the generic nature of the\ndefining i
deal of the semigroup. Further\, when $H$ is symmetric and has\nembedding
dimension 4 or 5\, we prove that the defining ideal is minimally\ngenerate
d by RF-relations.\n\nThis is joint work with Om Prakash Bhardwaj and Indr
anath Sengupta.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neena Gupta (ISI\, Kolkata\, India)
DTSTART:20210730T062000Z
DTEND:20210730T065000Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/10
DESCRIPTION:Title: On $2$-stably isomorphic four dimensional affine domains\n
by Neena Gupta (ISI\, Kolkata\, India) as part of Symposium in honour of D
ilip Patil\n\n\nAbstract\nA famous theorem of Abhyankar--Eakin--Heinzer pr
oves that if $A$ is a one\ndimensional ring containing $\\mathbb{Q}$ and $
n \\ge 1$ be such that the\npolynomial ring in $n$-variables over $A$ is i
somorphic to the polynomial\nring in $n$ variables over $B$ for some ring
$B$\, then $A \\cong B$. This\ndoes not hold in higher dimensional rings i
n general. In this connection\nthe following question arises:\n\nIf $A[X\,
Y] \\cong B[X\,Y]$\, does this imply $A \\cong B$?\n\nIn this talk we shal
l present four-dimensional seminormal\naffine domains over ${\\mathbb C}$
for which the above question does not\nhold. \n\nThis is joint work with P
rofessor T. Asanuma.\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dilip Patil (IISc\, Bangalore)
DTSTART:20210729T121000Z
DTEND:20210729T122500Z
DTSTAMP:20260422T212557Z
UID:CommutativeAlgebra/11
DESCRIPTION:Title: Reminiscences\nby Dilip Patil (IISc\, Bangalore) as part o
f Symposium in honour of Dilip Patil\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CommutativeAlgebra/11/
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