BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Divya Aggarwal (Indraprastha Institute of Information Technology\, Delhi) DTSTART:20210212T133000Z DTEND:20210212T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/1 DESCRIPTION:Title: Enumeration of matrices and splitting subspaces over finite field s\nby Divya Aggarwal (Indraprastha Institute of Information Technology \, Delhi) as part of Online Weekly Research Seminar for Early Career Mathe maticians from India\n\nLecture held in Zoom.\n\nAbstract\nWe will outline some enumeration techniques and discuss their applications in counting va rious kinds of matrices. We will then introduce the notion of T-splitting subspaces and discuss their connections with other areas. Enumeration of s plitting subspaces for an arbitrary operator T is an open problem. We will describe some recent progress on it and some future directions for resear ch.\n\nMeeting ID 926 1140 2828 and the Passcode is the smallest positive integer that can be written as the sum of two cubes in two different ways. \n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Ritu Dutta (Dibrugarh University) DTSTART:20210219T133000Z DTEND:20210219T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/2 DESCRIPTION:Title: Voting rules: an introduction\nby Ritu Dutta (Dibrugarh Unive rsity) as part of Online Weekly Research Seminar for Early Career Mathemat icians from India\n\nLecture held in Zoom.\n\nAbstract\nIn this talk\, we talk about different voting methods. Their merits and demerits\, how votin g rule(s) shaped our democracy\, what are the possibilities to improve our democratic institutions? If time permits\, we will discuss some recent vo ting reforms across the globe.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Gauranga K. Baishya (Tezpur University) DTSTART:20210226T133000Z DTEND:20210226T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/3 DESCRIPTION:Title: Combinatorial proof of a beautiful Euler-type Identity\nby Ga uranga K. Baishya (Tezpur University) as part of Online Weekly Research Se minar 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 the set o f even parts have exactly one element\, b(n) be the difference between 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 wh ich exactly\none part is repeated. One can show that a(n) = b(n) and b(n)= c(n) separately. We will prove combinatorially\, the beautiful identity th at 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 str ings. This is an expository talk.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Mohan Swaminathan (Princeton University\, USA) DTSTART:20210305T133000Z DTEND:20210305T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/4 DESCRIPTION:Title: Gromov's compactness theorem for (pseudo)holomorphic curves\n by Mohan Swaminathan (Princeton University\, USA) as part of Online Weekly Research Seminar for Early Career Mathematicians from India\n\nLecture he ld in Zoom.\n\nAbstract\nI will provide some motivation and an introductio n to the titular theorem which roughly states (in a special case) that the space of smooth projective curves (of a given genus and given area) in a closed Kahler manifold can be compactified by adding certain special types of singular curves. Assuming standard estimates on solutions of the Cauch y-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://researchseminars.org/talk/EarlyCareerIndia/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Azizul Hoque (Rangapara College\, Assam) DTSTART:20210312T133000Z DTEND:20210312T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/5 DESCRIPTION:Title: Parametrized Families of Quadratic Fields with Large n-rank\n by Azizul Hoque (Rangapara College\, Assam) as part of Online Weekly Resea rch Seminar for Early Career Mathematicians from India\n\nLecture held 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 that we b elieve such examples to be very rare. There is a conjecture (folklore) tha t the n-rank of k is unbounded when k runs through the quadratic fields. I t 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. We will show that for every large positive real number x\, there exists a sufficie ntly large positive constant c such that the number of quadratic fields wi th 3-rank at least 3 and absolute discriminant ≤ x is > $cx^{1/3}$. If t ime permits\, we will construct a parametric family of real (resp. imagina ry) quadratic fields with n-rank at least 2.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Bidisha Roy (Polish Academy of Sciences\, Warsaw\, Poland) DTSTART:20210409T133000Z DTEND:20210409T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/6 DESCRIPTION:Title: On some properties of consecutive Lehmer numbers modulo a prime\nby Bidisha Roy (Polish Academy of Sciences\, Warsaw\, Poland) as part of Online Weekly Research Seminar for Early Career Mathematicians from Ind ia\n\nLecture held in Zoom.\n\nAbstract\nA Lehmer number modulo a prime $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 als o a primitive root modulo $p$ is called an Lehmer primitive root (LPR).\n\ nLet $N$ be a positive integer and $p$ be an odd prime number. In this tal k\, we will discuss about existence of $N$-consecutive Lehmer numbers and $N$- consecutive Lehmer primitive roots in $\\left(\\mathbb{Z}/ p\\mathbb {Z} \\right)^*$\, where $p$ depends on $N$. In the second part\, we will discuss on non-primitive Lehmer numbers and their properties.\n LOCATION:https://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:20260422T225925Z UID:EarlyCareerIndia/8 DESCRIPTION:Title: Commuting Tuples and Commuting Probability\nby Uday Bhaskar S harma (Tata Institute of Fundamental Research\, Mumbai) as part of Online Weekly Research Seminar for Early Career Mathematicians from India\n\nLect ure held in Zoom.\nAbstract: TBA\n\nIn this talk\, I will speak about simu ltaneous similarity classes of commuting tuples of elements of an algebra and a group\, and explain its connection with commuting probability.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Soumya Dey (The Institute of Mathematical Sciences\, Chennai) DTSTART:20210507T133000Z DTEND:20210507T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/9 DESCRIPTION:Title: Orbits of zipping maps of surfaces of infinite type\nby Soumy a Dey (The Institute of Mathematical Sciences\, Chennai) as part of Online Weekly Research Seminar for Early Career Mathematicians from India\n\nLec ture held in Zoom.\n\nAbstract\nWe shall introduce the mapping class group s of surfaces of infinite type\, which are known as 'big' mapping class gr oups\, and the associated Teichmüller spaces. In the second half of the t alk we shall briefly discuss about an ongoing work with Dr. Gianluca Farac o\, which concerns some interesting mapping classes which we call 'zipping maps'\, and the orbits of their action on the Teichmüller space.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Eshita Mazumdar (Indian Statistical Institute\, Bengaluru) DTSTART:20210521T133000Z DTEND:20210521T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/10 DESCRIPTION:Title: A Stroll Through Davenport Constant\nby Eshita Mazumdar (Ind ian Statistical Institute\, Bengaluru) as part of Online Weekly Research S eminar for Early Career Mathematicians from India\n\nLecture held in Zoom. \n\nAbstract\nFor a finite abelian group G\, the Davenport Constant D(G) i s 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 original motiv ation for introducing Davenport Constant was to study the problem of non-u nique factorization domain over number fields. The precise value of this g roup invariant for any finite abelian group is still unknown. In my talk I am going to present my most recent research works related to Davenport Co nstant. In first half of my talk\, I will present an Extremal Problem rela ted to Weighted Davenport Constant\, where we introduce and discuss severa l exciting combinatorial results for finite abelian group. In second half of my talk\, I will talk about my current project\, where my main aim is t o 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 invariant wh ichis a natural generalization of Davenport Constant.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Gianluca Faraco (Indian Institute of Science\, Bengaluru) DTSTART:20210326T133000Z DTEND:20210326T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/11 DESCRIPTION:Title: Translation surfaces with poles and meromorphic differentials\nby Gianluca Faraco (Indian Institute of Science\, Bengaluru) as part of Online Weekly Research Seminar for Early Career Mathematicians from India \n\nLecture held in Zoom.\n\nAbstract\nLet $S$ be an oriented surface of g enus $g$ and $n$ punctures. The periods of any meromorphic differential on $S$\, with respect to a choice of complex structure\, determine a represe ntation $\\chi:\\Gamma_{g\,n} \\to\\mathbb C$ where $\\Gamma_{g\,n}$ is the first homology group of $S$. We characterize the representations that thus arise\, that is\, lie in the image of the period map $\\textsf{Per }:\\Omega\\mathcal{M}_{g\,n}\\to \\textsf{Hom}(\\Gamma_{g\,n}\,\\Bbb C)$. This generalizes a classical result of Haupt in the holomorphic case. This is a joint work with S. Chenakkod and S. Gupta.\n LOCATION:https://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:20260422T225925Z UID:EarlyCareerIndia/12 DESCRIPTION:Title: Finite Groups with Exactly Two Conjugacy Class Size and the Anal ogous Study in Lie Algebra\nby Tushar Kanta Naik (Indian Institute of Science Education and Research\, Mohali) as part of Online Weekly Research Seminar for Early Career Mathematicians from India\n\nLecture held in Zoo m.\n\nAbstract\nThe classification of the finite simple groups is one of t he most celebrated achievements of the last century. On the other hand\, f inite 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 classificatio ns to groups\, up to the order $p^7$. The number of p-groups grows so quic kly that further classifications along these lines seem a near-impossible task. To reduce the difficulty\, it is of practice to study finite p-group s with added conditions. One such condition is the sizes of conjugacy clas ses. In this talk\, we shall discuss the known results and remaining probl ems in the classification of finite p-groups with exactly two conjugacy cl ass sizes. In the end\, we shall throw some lights on the analogous study in Lie algebra.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Jaitra Chattopadhyay (Indian Institute of Technology Guwahati) DTSTART:20210514T133000Z DTEND:20210514T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/13 DESCRIPTION:Title: Simultaneous divisibility and indivisibility properties of class numbers of quadratic fields\nby Jaitra Chattopadhyay (Indian Institut e of Technology Guwahati) as part of Online Weekly Research Seminar for Ea rly Career Mathematicians from India\n\nLecture held in Zoom.\n\nAbstract\ nThe ideal class group and class number are important algebraic objects as sociated to a number field. The famous "Class number $1$ conjecture"\, due to Gauss\, motivates number theorists to have a closer look into the dist ribution of class numbers of quadratic fields. In particular\, the divisib ility properties of class numbers turn out to be useful to understand the ideal class groups of quadratic fields. In this talk\, we shall briefly re call the divisibility results in the literature and touch upon the topic o f simultaneous divisibility of class numbers of triples of imaginary quadr atic fields\, which is a joint work with M. Subramani. We conclude with a recent result on the simultaneous indivisibility of pairs of real quadrati c fields\, which is a joint work with A. Saikia.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Rahul Kumar (Indian Institute of Technology Gandhinagar) DTSTART:20210528T133000Z DTEND:20210528T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/14 DESCRIPTION:Title: A generalized modified Bessel function and explicit transformati ons of certain Lambert series\nby Rahul Kumar (Indian Institute of Tec hnology Gandhinagar) as part of Online Weekly Research Seminar for Early C areer Mathematicians from India\n\nLecture held in Zoom.\n\nAbstract\nAn e xact transformation\, which we call a \\emph{master identity}\, is obtaine d 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 d erive the well-known transformations of the Eisenstein series on $\\textup {SL}_{2}\\left(\\mathbb{Z}\\right)$\, that of the Dedekind eta function as well as Ramanujan's famous formula for $\\zeta(2m+1)$. Corresponding new transformations when $a$ is a non-zero even integer are also obtained as s pecial cases of the master identity. These include a novel companion to Ra manujan's formula for $\\zeta(2m+1)$. Although not modular\, it is surpris ing that such explicit transformations exist. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. The lat ter identity itself is derived using Guinand's version of the Vorono\\"{\\ dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function $K_{\\nu}(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generali zation allows us to obtain a new transformation involving the sums-of-squa res function $r_k(n)$. This is joint work with Atul Dixit and Aashita Kesa rwani.\n LOCATION:https://researchseminars.org/talk/EarlyCareerIndia/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Yaqubi (Ferdowsi University of Mashhad\, Iran) DTSTART:20210625T133000Z DTEND:20210625T144500Z DTSTAMP:20260422T225925Z UID:EarlyCareerIndia/15 DESCRIPTION:Title: Enumeration of direct animals with lattice paths‎\nby Dani el Yaqubi (Ferdowsi University of Mashhad\, Iran) as part of Online Weekly Research Seminar for Early Career Mathematicians from India\n\nLecture he ld 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 v ery long time‎\, ‎explicitly at least since the second half of the 19t h century‎. A typical problem in lattice paths is the enumeration of all $\\mathcal{S}$-lattice paths (lattice paths with respect to the set $\\ma thcal{S}$)‎. ‎A non-trivial simple case is the problem of finding the number of lattice paths starting from the origin $(0\,0)$ and ending 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://researchseminars.org/talk/EarlyCareerIndia/15/ END:VEVENT END:VCALENDAR