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BEGIN:VEVENT
SUMMARY:Divya Aggarwal (Indraprastha Institute of Information Technology\,
Delhi)
DTSTART:20210212T133000Z
DTEND:20210212T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/1
DESCRIPTION:Title: Enumeration of matrices and splitting subspaces over finit
e fields\nby Divya Aggarwal (Indraprastha Institute of Information Tec
hnology\, Delhi) as part of Online Weekly Research Seminar for Early Caree
r Mathematicians from India\n\nLecture held in Zoom.\n\nAbstract\nWe will
outline some enumeration techniques and discuss their applications in coun
ting various kinds of matrices. We will then introduce the notion of T-spl
itting subspaces and discuss their connections with other areas. Enumerati
on of splitting subspaces for an arbitrary operator T is an open problem.
We will describe some recent progress on it and some future directions for
research.\n\nMeeting ID 926 1140 2828 and the Passcode is the smallest po
sitive integer that can be written as the sum of two cubes in two differen
t ways.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritu Dutta (Dibrugarh University)
DTSTART:20210219T133000Z
DTEND:20210219T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/2
DESCRIPTION:Title: Voting rules: an introduction\nby Ritu Dutta (Dibrugar
h University) as part of Online Weekly Research Seminar for Early Career M
athematicians from India\n\nLecture held in Zoom.\n\nAbstract\nIn this tal
k\, we talk about different voting methods. Their merits and demerits\, ho
w voting rule(s) shaped our democracy\, what are the possibilities to impr
ove our democratic institutions? If time permits\, we will discuss some re
cent voting reforms across the globe.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gauranga K. Baishya (Tezpur University)
DTSTART:20210226T133000Z
DTEND:20210226T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/3
DESCRIPTION:Title: Combinatorial proof of a beautiful Euler-type Identity
\nby Gauranga K. Baishya (Tezpur University) as part of Online Weekly Rese
arch Seminar for Early Career Mathematicians from India\n\nLecture held in
Zoom.\n\nAbstract\nLet a(n) be the number of partitions of n such that th
e set of even parts have exactly one element\, b(n) be the difference betw
een the number of parts in all odd partitions of n and the number of parts
in all distinct partitions of n and c(n) are the number of partitions of
n in which exactly\none part is repeated. One can show that a(n) = b(n) an
d b(n)=c(n) separately. We will prove combinatorially\, the beautiful iden
tity that a(n) = b(n) = c(n). The proof relies on bijections between a set
and a multiset\, where the partitions in the multiset are decorated with
bit strings. This is an expository talk.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohan Swaminathan (Princeton University\, USA)
DTSTART:20210305T133000Z
DTEND:20210305T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/4
DESCRIPTION:Title: Gromov's compactness theorem for (pseudo)holomorphic curve
s\nby Mohan Swaminathan (Princeton University\, USA) as part of Online
Weekly Research Seminar for Early Career Mathematicians from India\n\nLec
ture held in Zoom.\n\nAbstract\nI will provide some motivation and an intr
oduction to the titular theorem which roughly states (in a special case) t
hat the space of smooth projective curves (of a given genus and given area
) in a closed Kahler manifold can be compactified by adding certain specia
l types of singular curves. Assuming standard estimates on solutions of th
e Cauchy-Riemann equation as a black box\, I will then sketch the proof of
Gromov's theorem as an application of the Arzela-Ascoli theorem.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Azizul Hoque (Rangapara College\, Assam)
DTSTART:20210312T133000Z
DTEND:20210312T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/5
DESCRIPTION:Title: Parametrized Families of Quadratic Fields with Large n-ran
k\nby Azizul Hoque (Rangapara College\, Assam) as part of Online Weekl
y Research Seminar for Early Career Mathematicians from India\n\nLecture h
eld in Zoom.\n\nAbstract\nConstructing number fields with large n-rank has
proved to be a challenging practical problem\, due in part to the fact th
at we believe such examples to be very rare. There is a conjecture (folklo
re) that the n-rank of k is unbounded when k runs through the quadratic fi
elds. It was Quer who constructed 3 imaginary quadratic fields with 3-rank
equal to 6\, and this result still stands as the current record. We will
discuss two methods for constructing quadratic fields with large n-rank. W
e will show that for every large positive real number x\, there exists a s
ufficiently large positive constant c such that the number of quadratic fi
elds with 3-rank at least 3 and absolute discriminant ≤ x is > $cx^{1/3}
$. If time permits\, we will construct a parametric family of real (resp.
imaginary) quadratic fields with n-rank at least 2.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bidisha Roy (Polish Academy of Sciences\, Warsaw\, Poland)
DTSTART:20210409T133000Z
DTEND:20210409T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/6
DESCRIPTION:Title: On some properties of consecutive Lehmer numbers modulo a
prime\nby Bidisha Roy (Polish Academy of Sciences\, Warsaw\, Poland) a
s part of Online Weekly Research Seminar for Early Career Mathematicians f
rom India\n\nLecture held in Zoom.\n\nAbstract\nA Lehmer number modulo a p
rime $p$ is an integer $a$ with\n$1 \\leq a \\leq p - 1$ whose inverse $\\
bar{a}$ within the same range has\nopposite parity. A Lehmer number which
is also a primitive root modulo $p$ is called an Lehmer primitive root (L
PR).\n\nLet $N$ be a positive integer and $p$ be an odd prime number. In t
his talk\, we will discuss about existence of $N$-consecutive Lehmer numbe
rs and $N$- consecutive Lehmer primitive roots in $\\left(\\mathbb{Z}/ p\
\mathbb{Z} \\right)^*$\, where $p$ depends on $N$. In the second part\, w
e will discuss on non-primitive Lehmer numbers and their properties.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uday Bhaskar Sharma (Tata Institute of Fundamental Research\, Mumb
ai)
DTSTART:20210430T133000Z
DTEND:20210430T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/8
DESCRIPTION:Title: Commuting Tuples and Commuting Probability\nby Uday Bh
askar Sharma (Tata Institute of Fundamental Research\, Mumbai) as part of
Online Weekly Research Seminar for Early Career Mathematicians from India\
n\nLecture held in Zoom.\nAbstract: TBA\n\nIn this talk\, I will speak abo
ut simultaneous similarity classes of commuting tuples of elements of an a
lgebra and a group\, and explain its connection with commuting probability
.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumya Dey (The Institute of Mathematical Sciences\, Chennai)
DTSTART:20210507T133000Z
DTEND:20210507T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/9
DESCRIPTION:Title: Orbits of zipping maps of surfaces of infinite type\nb
y Soumya Dey (The Institute of Mathematical Sciences\, Chennai) as part of
Online Weekly Research Seminar for Early Career Mathematicians from India
\n\nLecture held in Zoom.\n\nAbstract\nWe shall introduce the mapping clas
s groups of surfaces of infinite type\, which are known as 'big' mapping c
lass groups\, and the associated Teichmüller spaces. In the second half o
f the talk we shall briefly discuss about an ongoing work with Dr. Gianluc
a Faraco\, which concerns some interesting mapping classes which we call '
zipping maps'\, and the orbits of their action on the Teichmüller space.\
n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eshita Mazumdar (Indian Statistical Institute\, Bengaluru)
DTSTART:20210521T133000Z
DTEND:20210521T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/10
DESCRIPTION:Title: A Stroll Through Davenport Constant\nby Eshita Mazumd
ar (Indian Statistical Institute\, Bengaluru) as part of Online Weekly Res
earch Seminar for Early Career Mathematicians from India\n\nLecture held i
n Zoom.\n\nAbstract\nFor a finite abelian group G\, the Davenport Constant
D(G) is defined to be the least positive integer k such that any sequence
S with length k over G has a non-trivial zerosum subsequence. The origina
l motivation for introducing Davenport Constant was to study the problem o
f non-unique factorization domain over number fields. The precise value of
this group invariant for any finite abelian group is still unknown. In my
talk I am going to present my most recent research works related to Daven
port Constant. In first half of my talk\, I will present an Extremal Probl
em related to Weighted Davenport Constant\, where we introduce and discuss
several exciting combinatorial results for finite abelian group. In secon
d half of my talk\, I will talk about my current project\, where my main a
im is to discuss the perfect power of a polynomial $f(x)\\in \\mathbb{Z}[x
]$ for integral values of x: While doing so we developed a new group invar
iant whichis a natural generalization of Davenport Constant.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gianluca Faraco (Indian Institute of Science\, Bengaluru)
DTSTART:20210326T133000Z
DTEND:20210326T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/11
DESCRIPTION:Title: Translation surfaces with poles and meromorphic different
ials\nby Gianluca Faraco (Indian Institute of Science\, Bengaluru) as
part of Online Weekly Research Seminar for Early Career Mathematicians fro
m India\n\nLecture held in Zoom.\n\nAbstract\nLet $S$ be an oriented surfa
ce of genus $g$ and $n$ punctures. The periods of any meromorphic differen
tial on $S$\, with respect to a choice of complex structure\, determine a
representation $\\chi:\\Gamma_{g\,n} \\to\\mathbb C$ where $\\Gamma_{g\,
n}$ is the first homology group of $S$. We characterize the representatio
ns that thus arise\, that is\, lie in the image of the period map $\\tex
tsf{Per}:\\Omega\\mathcal{M}_{g\,n}\\to \\textsf{Hom}(\\Gamma_{g\,n}\,\\Bb
b C)$. This generalizes a classical result of Haupt in the holomorphic cas
e. This is a joint work with S. Chenakkod and S. Gupta.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tushar Kanta Naik (Indian Institute of Science Education and Resea
rch\, Mohali)
DTSTART:20210319T113000Z
DTEND:20210319T124500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/12
DESCRIPTION:Title: Finite Groups with Exactly Two Conjugacy Class Size and t
he Analogous Study in Lie Algebra\nby Tushar Kanta Naik (Indian Instit
ute of Science Education and Research\, Mohali) as part of Online Weekly R
esearch Seminar for Early Career Mathematicians from India\n\nLecture held
in Zoom.\n\nAbstract\nThe classification of the finite simple groups is o
ne of the most celebrated achievements of the last century. On the other h
and\, finite p-groups of order $p^n$ for $n\\leq 4$ were classified early
in the history of group theory\, and modern work has extended these classi
fications to groups\, up to the order $p^7$. The number of p-groups grows
so quickly that further classifications along these lines seem a near-impo
ssible task. To reduce the difficulty\, it is of practice to study finite
p-groups with added conditions. One such condition is the sizes of conjuga
cy classes. In this talk\, we shall discuss the known results and remainin
g problems in the classification of finite p-groups with exactly two conju
gacy class sizes. In the end\, we shall throw some lights on the analogous
study in Lie algebra.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaitra Chattopadhyay (Indian Institute of Technology Guwahati)
DTSTART:20210514T133000Z
DTEND:20210514T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/13
DESCRIPTION:Title: Simultaneous divisibility and indivisibility properties o
f class numbers of quadratic fields\nby Jaitra Chattopadhyay (Indian I
nstitute of Technology Guwahati) as part of Online Weekly Research Seminar
for Early Career Mathematicians from India\n\nLecture held in Zoom.\n\nAb
stract\nThe ideal class group and class number are important algebraic obj
ects associated to a number field. The famous "Class number $1$ conjecture
"\, due to Gauss\, motivates number theorists to have a closer look into t
he distribution of class numbers of quadratic fields. In particular\, the
divisibility properties of class numbers turn out to be useful to understa
nd the ideal class groups of quadratic fields. In this talk\, we shall bri
efly recall the divisibility results in the literature and touch upon the
topic of simultaneous divisibility of class numbers of triples of imaginar
y quadratic fields\, which is a joint work with M. Subramani. We conclude
with a recent result on the simultaneous indivisibility of pairs of real q
uadratic fields\, which is a joint work with A. Saikia.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rahul Kumar (Indian Institute of Technology Gandhinagar)
DTSTART:20210528T133000Z
DTEND:20210528T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/14
DESCRIPTION:Title: A generalized modified Bessel function and explicit trans
formations of certain Lambert series\nby Rahul Kumar (Indian Institute
of Technology Gandhinagar) as part of Online Weekly Research Seminar for
Early Career Mathematicians from India\n\nLecture held in Zoom.\n\nAbstrac
t\nAn exact transformation\, which we call a \\emph{master identity}\, is
obtained for the series $\\sum_{n=1}^{\\infty}\\sigma_{a}(n)e^{-ny}$ for $
a\\in\\mathbb{C}$ and Re$(y)>0$. As corollaries when $a$ is an odd integer
\, we derive the well-known transformations of the Eisenstein series on $\
\textup{SL}_{2}\\left(\\mathbb{Z}\\right)$\, that of the Dedekind eta func
tion as well as Ramanujan's famous formula for $\\zeta(2m+1)$. Correspondi
ng new transformations when $a$ is a non-zero even integer are also obtain
ed as special cases of the master identity. These include a novel companio
n to Ramanujan's formula for $\\zeta(2m+1)$. Although not modular\, it is
surprising that such explicit transformations exist. The Wigert-Bellman id
entity arising from the $a=0$ case of the master identity is derived too.
The latter identity itself is derived using Guinand's version of the Voron
o\\"{\\dotlessi} summation formula and an integral evaluation of N.~S.~Kos
hliakov involving a generalization of the modified Bessel function $K_{\\n
u}(z)$. Koshliakov's integral evaluation is proved for the first time. It
is then generalized using a well-known kernel of Watson to obtain an inter
esting two-variable generalization of the modified Bessel function. This g
eneralization allows us to obtain a new transformation involving the sums-
of-squares function $r_k(n)$. This is joint work with Atul Dixit and Aashi
ta Kesarwani.\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Yaqubi (Ferdowsi University of Mashhad\, Iran)
DTSTART:20210625T133000Z
DTEND:20210625T144500Z
DTSTAMP:20260404T111410Z
UID:EarlyCareerIndia/15
DESCRIPTION:Title: Enumeration of direct animals with lattice paths\n
by Daniel Yaqubi (Ferdowsi University of Mashhad\, Iran) as part of Online
Weekly Research Seminar for Early Career Mathematicians from India\n\nLec
ture held in Zoom.\n\nAbstract\nThe aim of this talk is the enumeration of
direct animals with lattice paths. Lattice paths have been studied
for a very long time\, explicitly at least since the second half of
the 19th century. A typical problem in lattice paths is the enumeration
of all $\\mathcal{S}$-lattice paths (lattice paths with respect to the se
t $\\mathcal{S}$). A non-trivial simple case is the problem of findi
ng the number of lattice paths starting from the origin $(0\,0)$ and endin
g at a point $(m\,n)$ using only right step $(1\,0)$ and up step $(0\,1)$
(i.e.\, $\\mathcal{S}=\\{(1\,0)\,(0\,1)\\}$).\n
LOCATION:https://stable.researchseminars.org/talk/EarlyCareerIndia/15/
END:VEVENT
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