**Course description:**Mathematics 219 is a Core Course for the Integrated Maths Ph.D. and Undergraduate Math Major programs in IISc Mathematics. For more details (including a list of topics), click on the link with text 'Mathematics 219' above.**Meetings:**Tue Thu 10 – 11:30 am, in Lecture Hall LH-1 (Maths Dept.).**Please join the Microsoft Team "**(Note: Eighteen UG-2020 members have already been added.)__MA219 - IISc Math - Fall 2022__", from a valid IISc email account. If this does not work for some reason, then please email me (khare@iisc) from your IISc email account.**Pre-requisites:**For undergraduates: UM 102.**Handout/syllabus:**Click on this link.

Homework 1 (due by Thursday August 11, 2022, in TA's office hours or earlier in class)

Homework 2 (due by Thursday August 18, 2022, in TA's office hours or earlier in class)

Homework 3 (due by Thursday August 25, 2022, in TA's office hours or earlier in class)

Homework 4 (due by Thursday September 8, 2022, in TA's office hours or earlier in class)

Homework 5 (due by Friday September 16, 2022, in TA's office between 5 and 5:30 pm, else in TA's office hours on Thursday, or earlier in class)

Homework 6 (to try on your own if you like; not for submitting)

Homework 7 (due by Thursday October 6, 2022, in TA's office hours or earlier in class)

Homework 8 (due by Friday October 14, 2022, in TA's office by 9:30 am, else in TA's office hours on Thursday, or earlier in class)

Homework 9 (due by Thursday October 20, 2022, in TA's office hours or earlier in class)

Homework 10 (due by Friday October 28, 2022, in TA's office hours by 6 pm, else in TA's office hours on Thursday, or earlier in class)

Homework 11 (due by Friday November 11, 2022, in TA's office hours by 5 pm, else in TA's office hours on Thursday, or earlier in class)

Homework 12 (due by Thursday November 17, 2022, in TA's office hours or earlier in class)

Homework 13 (due by Tuesday November 29, 2022, in TA's office hours)

**Course description:**Mathematics 315 is an Elective Course on (semisimple) Lie algebras and their representations.**Meetings:**Tue Thu 3:30 – 5 pm, in Lecture Hall LH-4 (Maths Dept.).- Please join the
**Microsoft Team**"__MA315 - IISc Math - Fall 2022__", from a valid IISc email account. If this does not work for some reason, then please email me (khare@iisc) from your IISc email account.

**Course description:**Mathematics 219 is a Core Course for the Integrated Maths Ph.D. and Undergraduate Math Major programs in IISc Mathematics. For more details (including a list of topics), click on the link with text 'Mathematics 219' above.**Meetings:**Tue Thu 8:30 – 10 am, on Microsoft Teams. See this link, for which you first need to be logged into either VPN (remote access) or in your IISc Microsoft Outlook account in a separate tab in the same browser.- To join the
**Microsoft Team and mailing list**, check the link in the preceding point for which Team to join – from a valid IISc email account. If this does not work for some reason, then please email me (khare@iisc) from your IISc email account. (Note: UG-2019 members will be automatically added.) **Pre-requisites:**For undergraduates: UM 102.**Handout/syllabus:**Click on this link.

**Course description:**Mathematics 219 is a Core Course for the Integrated Maths Ph.D. and Undergraduate Math Major programs in IISc Mathematics. For more details (including a list of topics), click on the link with text 'Mathematics 219' above.**Meetings:**Tue Thu 10 – 11:30 am, Microsoft Teams. See this link, for which you first need to be logged into either VPN (remote access) or in your IISc Microsoft Outlook account in a separate tab in the same browser.**To join the Team, please email me (khare@iisc) from your IISc email account.**(Note: UG-2018 members are added.)**Pre-requisites:**For undergraduates: UM 102.**Handout/syllabus:**Click on this link.

**Course description:**This course explores matrix positivity and operations that preserve it. These involve fundamental questions that have been extensively studied over the past century, and are still being studied in the mathematics literature, including with additional motivation from modern applications to high-dimensional covariance estimation. The course will bring together techniques from different areas: analysis, linear algebra, combinatorics, and symmetric functions.**Meetings:**Tue Thu 3:30 – 5 pm, Mathematics Lecture Hall LH-3**Pre-requisites:**A course in linear algebra, and a course in calculus/real analysis.**Handout/syllabus:**See this link (most/all of the topics should get discussed, though not in the order in the link).**Course notes:**Click on this link for the course notes.**Homework assignments**

Homework 1 (due by Friday August 23, in TA's office hours or earlier in class)

Homework 2 (due by Friday September 6, in TA's office hours or earlier in class)

Homework 3 (due by 5pm on Monday September 16, in Instructor's office hours)

**Course description:**Mathematics 212 is a Core Course for the Integrated Maths Ph.D. and Undergraduate Math Major programs in IISc Mathematics. For more details (including a list of topics), click on the link with text 'Mathematics 212' above.**Meetings:**Tue Thu 11:30 am – 1 pm, Mathematics Lecture Hall LH-3**Pre-requisites:**For undergraduates: UM 203.**Handout/syllabus:**Click on this link.**Homework assignments**

Homework 1 (due by Friday August 16, in TA's office hours or earlier in class)

Homework 2 (due by Friday August 23, in TA's office hours or earlier in class)

Homework 3 (due by Friday August 30, in TA's office hours or earlier in class)

Homework 4 (due by Friday September 6, in TA's office hours or earlier in class)

Homework 5 (due by 5pm on Monday September 16, in Instructor's office hours)

Homework 6 (due by Friday October 4, in TA's office hours or earlier in class)

Homework 7 (due by Friday October 11, in TA's office hours or earlier in class)

Homework 8 (due by Friday November 8, in TA's office hours or earlier in class)

Homework 9 (due by Friday November 15, in TA's office hours or earlier in class)

Homework 10 (due by Monday November 25, in TA's office hours; now updated)

**Course description:**Click on the link with text 'Mathematics UM102' above.

**Course handout:**Click on this link.**Notes on uniqueness of determinant:**Click on this link.**Homework assignments**

Homework 1 (due by Wednesday January 9, in class)

Homework 2 (due by Wednesday January 16, in class)

Homework 3 (Quiz 1 on Monday January 28, in class)

Homework 4 (Quiz 2 on Monday February 4, in class)

Homework 5 (Quiz 3 on Monday February 11, in class)

Homework 6 (Not for a Quiz nor for HW submission, but relevant to mid-sem syllabus)

Homework 7 (Quiz 4 on Monday March 18, in class)

Homework 8 (Quiz 5 on Monday March 25, in class)

Homework 9 (Quiz 6 on Monday April 1 – not a joke! – in class)

Homework 10 (Quiz 7 on Monday April 8, in class)

Homework 11 (Quiz 8 on Monday April 15, in class)

Homework 12 (Not for a Quiz nor for HW submission, but relevant to Final syllabus)

**Course description:**This course explores matrix positivity and operations that preserve it. These involve fundamental questions that have been extensively studied over the past century, and are still being studied in the mathematics literature, including with additional motivation from modern applications to high-dimensional covariance estimation. The course will bring together techniques from different areas: analysis, linear algebra, combinatorics, and symmetric functions.**Meetings:**Wed Fri 11am – 12:30pm, Mathematics Lecture Hall LH-3**Midterm:**Friday February 23, in class.**Pre-requisites:**A course in linear algebra, and a course in calculus/real analysis.**List of topics:**See this link (all but one or two sub-topics were discussed, though not in the order in the link).**Course notes:**Click on this link for the course notes.

- Mathematics 221 is a Core Course in real analysis, for the Integrated
Ph.D. program in IISc Mathematics.

This semester, we will closely follow Rudin's*Principles of Mathematical Analysis*. **Course handout:**Click on this link.**Homework assignments**

Homework 1 (due by Friday August 25, in TA's office hours or earlier in class)

Homework 2 (due by Friday September 8, in TA's office hours or earlier in class)

Homework 3 (due by Friday September 29, in TA's office hours or earlier in class)

Homework 4 (due by Tuesday October 24, in class)

Homework 5 (due by Tuesday November 7, in class)

Homework 6 (due by Friday November 17, in TA's office hours or earlier in class)

Homework 7 (due by Wednesday December 6, in Instructor's office hours by 4pm, or earlier in class)