Computational Arithmetic - Geometry for Algebraic Curves

By Prof. Nitin Saxena | IIT Kanpur

Learners enrolled: 560 | Exam registration: 18

ABOUT THE COURSE:
We want to count the number of roots of an algebraic system (over finite fields). This is a very difficult question in general(eg. #P-hard). However, there are fast algorithms known for special cases. In this course we will focus on the "2-variable" case, i.e. curves. This case already demands significant theory and has an amazing list of applications in computer science. We will cover some important aspects of this 20th century mathematics in a self-contained way, and see as many applications as time permits.

INTENDED AUDIENCE: Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.

PREREQUISITES: A very good grasp in at least one of these: Algebra, Number theory, Geometry, Calculus, Analysis, or Topology,.

INDUSTRY SUPPORT: Discrete Optimization, Cryptography/ Cyber Security, Coding theory, Computer Algebra, Symbolic Computing Software, Learning Software

Summary

Course Status :	Completed
Course Type :	Elective
Language for course content :	English
Duration :	12 weeks
Category :	Computer Science and Engineering Foundations of Computing
Credit Points :	3
Level :	Undergraduate/Postgraduate
Start Date :	22 Jul 2024
End Date :	11 Oct 2024
Enrollment Ends :	05 Aug 2024
Exam Registration Ends :	16 Aug 2024
Exam Date :	26 Oct 2024 IST

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1: Algebraic-Geometry via Zariski Topology

Week 2: Variety—affine or projective. Rational Function and Germs.

Week 3: Morphism, Function and Birationality. Tangent Space and Smoothness.

Week 4: Resolving Singularity via Valuation. Discrete Valuation Ring (dvr).

Week 5: Abstract Curve and its Smooth Model.

Week 6: Approximate Valuation Theorem.

Week 7: Divisor Group. L-sheaf or Line Bundle.

Week 8: Principal Divisor and Degree. Class Group and Genus.

Week 9: Adele, Differential and the Riemann-Roch Theorem.

Week 10: Hasse-Weil Local Zeta Function. Functional Equation.

Week 11: Galois Cover and proving Weil’s Riemann Hypothesis.

Week 12: Character Sum. Torsion Point. Cohomology in the zeta function.

Books and references

https://www.cse.iitk.ac.in/users/nitin/courses/CS688-2022-23-II/
Text Book:
Algebraic Curves over Finite Fields, Carlos Moreno

Instructor bio

Prof. Nitin Saxena

IIT Kanpur

Prof. Nitin Saxena did B.Tech in CSE from the Indian Institute of Technology, Kanpur in 2002 and Ph.D. in the same department, under Manindra Agrawal, in 2006. He is broadly interested in Algebraic Complexity, Computational Algebra, Geometry and Number Theory. He was a visiting graduate student in Princeton University (2003-2004) and National University of Singapore (2004-2005); a postdoc at CWI, Amsterdam (2006-2008) and a Bonn Junior Fellow (W2 Professor) at Hausdorff Center for Mathematics, Bonn (2008-2013). Since April 2013, he is a faculty in CSE, IIT Kanpur. He is the coordinator of the Center for Developing Intelligent Systems (CDIS). He has two of the most prestigious honors in the field of theoretical computer science and discrete mathematics, Gödel Prize (2006) and Fulkerson Prize (2006). In fact, he is the youngest Gödel Prize winner ever. He is a recipient of numerous other prestigious awards including the Distinguished Alumnus Award (2003, IIT Kanpur) for his exceptional academic brilliance, the INSA Young Scientist Medal (2015), the SwarnaJayanti Fellowship (2013-14), and the Shanti Swarup Bhatnagar Award in Mathematical Sciences (2018). He is a Fellow of IASc, NASI, and INAE. In 2022, he was profiled by DST (Ministry of S&T, India) in the list of “75 Scientists under 50 shaping today’s India”. In 2009, he was voted one of the top 50 alumni by the IIT Kanpur Alumni Association to celebrate 50 years of the institute.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 26 October 2024 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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