NCMW - Seshadri Constants (2019)

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Venue: NISER,  Bhubaneswar, Odisha(Orissa)
Date:  16th, Dec 2019 to 21st, Dec 2019

 

School Convener(s)

Name Prof. Indranil Biswas Dr. Krishna Hanumanthu Dr. Ritwik Mukherjee
Mailing Address Professor,
School of Mathematical Sciences, TIFR.
Associate Professor,
CMI
Reader-F,
School of Mathematical Sciences, NISER

 


 Speakers with their affiliations:

Name of the speaker Position Affiliation
D. S. Nagaraj Professor Indian Institute of Science Education and Research, Tirupati
Krishna Hanumanthu Associate Professor Chennai Mathematical Institute, Chennai
Omprokash Das Reader Tata Institute of Fundamental Research, Mumbai
Ritwik Mukherjee Reader-F National Institute of Science Education and Research Bhubaneswar
Anantadulal Paul Research Scholar National Institute of Science Education and Research Bhubaneswar
Nilakantha Das Research Scholar National Institute of Science Education and Research Bhubaneswar
Rahul Kumar Singh Post-doctoral Fellow Tata Institute of Fundamental Research, Mumbai
Praveen Kumar Roy Post-doctoral Fellow Tata Institute of Fundamental Research, Mumbai

 

 List of Tutors

Name of the tutor Position Affiliation
Snejajit Misra Post-doctoral Fellow Tata Institute of Fundamental Research, Mumbai
Nabanita Ray Post-doctoral Fellow Tata Institute of Fundamental Research, Mumbai
Aditya Subramaniam Research Scholar Chennai Mathematical Institute, Chennai

 

 


 Syllabus:

Name of the Speaker : Detailed Syllabus
D.S. Nagaraj Curve and surface theory. Helectured on Riemann-Roch theorem for curves and surfaces, blow ups of nonsingular surfaces at points, adjunction formula, intersection theory on surfaces, Hodge index theorem. The main reference used was relevant portions of R. Hartshorne's Algebraic geometry.
Krishna Hanumanthu Basics of Seshadri constants. After giving basic definitions, examples, he covered the motivation for defining them by discussing Fujita conjecture and jet separation. Then the connections to linear systems of plane curves and the Nagata and SHGH conjectures was discussed. The questions of irrationality of Seshadri constants was also discussed. The main references were the chapter on Seshadri constants in R. Lazarsfeld's Positivity in Algebraic Geometry I and some expository lecture notes.
Omprokash Das Positive characteristic analogues of Seshadri constants called the Frobenius Seshadri constants. Using a Frobenius analogue of jet separation, M. Musta\c{t}\u{a} and K. Schwede defined Frobenius Seshadri constants and proved many interesting properties and connections to the usual Seshadri constants. These lectures covered most of their results and the mainreference was their original paper: A Frobenius variant of Seshadri constants, Mathematische Annalen 2014.
Ritwik Mukherjee Gave an introduction to Enumerative Geometry of singular curves. He explained the idea of viewing Enumerative Numbers as zero sets of sections of Vector Bundles and interpreting these numbers as intersection of Cycles. The two crucial things in this subject is to show transversality and compute degenerate contributions to the intersection number. The main references were the papers by Ritwik Mukherjee and Somnath Basu on enumeration of singular curves.
Anantadulal Paul Continued with the setup of Ritwik Mukherjee’s lecture and explained how to enumerate curves in a surface that are tangent to a smooth divisor. As before, these numbers can also be viewed as intersection of cycles (or equivalently zeros of a section of an appropriate bundle). The main reference was Anantadulal Paul’s paper on enumerating curves with tangencies.
Nilakantha Das Continued with the setup of Ritwik Mukherjee’s lecture and explained how to enumerate curves in a family of surfaces. He focused on the example of enumerating singular curves in P^3, whose image lies inside a P^2. The main reference was the paper by Nilkantha Das and Ritwik Mukherjee on enumerating singular planar curves in P^3.
Rahul Kumar Singh Moduli space of stable maps. He explained how Kontsevich and Manin obtained a formula for the characteristic number of rational curves in P^2. He then went on to explain how one can generalize this to a family version (in particular, enumerating rational curves in P^3, whose image lies in a P^2). Finally, he spoke about generalizing this result to higher singularity (cusps). The main reference for this talk were the papers by Anantadulal, Ritwik Mukherjee and Rahul Singh on counting rational curves in a moving family of P^2 and the paper by Ritwik Mukherjee and Rahul Singh on counting cuspidal curves in a moving family of P^2.
Praveen Kumar Roy A research talk on some recent results on Seshadri constants for line bundles on hyperelliptic surfaces and surfaces of general type.

 

 


 Time Table

Day Date Lecture 1 (9.30 – 11.00) Tea (time) Lecture 2 (11.30 – 1.00) Lunch (time) Lecture 3 (02.30 – 3.30) Tea (time) Discussio n/ Tutorial (04.00 – 5.00) Tea & snacks (time)
    name of the speaker (abbrevia tion)   name of the speaker (abbreviati on)   name of the speaker (abbrevia tion)   name of the speaker & tutors (abbrevia tion)  
Mo n 16/12/1 9 DSN   DSN   RM   DSN, NR,AS  
Tue s 17/12/1 9 DSN   DSN   RS   DSN, NR,AS  
We d 18/12/1 9 AP   ND   AP   ND  
Thu 19/12/1 9 KH   KH   KH, AS, SM   KH, AS, SM  
Fri 20/12/1 9 OD   OD   PRK   OD,SM NR,  
Sat 21/12/1 9 KH   OD   OD, NR, SM   OD,SM NR,  

 

Full forms for the abbreviations of speakers and tutors :

Speakers:
DSN: D. S. Nagaraj
RM: Ritwik Mukherjee
RS: Rahul Singh
AP: Anantadulal Paul
ND: Nilakantha Das
KH: Krishna Hanumanthu
OD: Omprokas Das
PRK: Praveen Kumar Roy
Tutors:
NR: Nabanita Ray
SM: Snehajit Misra
AS: Aditya Subramaniam


List of actual Participants

 

Sr. SID Full Name Gender Affiliation Position in College / University University/Institute M.Sc. / M.A. Year of Passing M.Sc. / M.A Ph.D. Deg. Date
1 28738 Mr. Aishik Chattopadhyay Male Chennai Mathematical Institute PhD Pondicherry University 2018  
2 29337 Mr Arghya Pramanik Male IIT bombay PHD University Of Hyderabad 2015  
3 29355 Dr. Saugata Purkayastha Male Assam Don Bosco University Assistant Professor Department of Mathematics, Gauhati University 2010 1 Dec 2015
4 29522 Mr. Pradeep Das Male Harish-Chandra Research Institute PhD Student IIT Kanpur 2011  
5 29534 Mr. Anoop Singh Male Harish-Chandra Research Institute PhD University of Allahabad 2013  
6 29613 Mr. Anindya Ganguly Male IIT Kharagpur MS(by Research) IIT Kharagpur 2016  
7 29627 Mr. Saikat Maity Male University of Calcutta B.Sc complete year student      
8 29691 Mr. Ritankar Nath Male Tata Institute of Fundamental Research, Mumbai Integrated PhD Tata Institute of Fundamental Research, Mumbai    
9 30174 Dr. Shameek Paul Male RKMVERI Assistant Professor University of Mumbai 2005 19 Jan 2012
10 30221 Ms. Ananya Chaturvedi Female Tata Institute of Fundamental Research Postdoctoral Fellow University of Houston 2012 15 Dec 2016
11 30282 Mr. Debabrota Mondal Male Indian institute of technology Bhubaneswar MSc Student IIT Bhubaneswar App / Awt Res.  
12 30359 Dr. Parangama Sarkar Female Chennai Mathematical Institute Postdoc IIT Madras 2011 3 Jun 2016
13 30482 Mr. Amit Kumar Singh Male IIT Madras PhD University of Allahabad 2011  
14 30483 Ms Arusha C Female IISER Bhopal PhD Student M.S University of Baroda 2015  
15 30507 Mr. Sujoy Chakraborty Male TIFR Mumbai Research Scholar CMI, chennai 2015  
16 30537 Mr. Areeb Shah Male Chennai Mathematical Institute B.Sc. Student      

 


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