CACAAG Seminar

Date: Monday, 24 July, 5 pm

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Host: Madhusudan Manjunath

Venue: Ramanujan Hall

Speaker: Aryaman Jal

Affiliation: KTH, Stockhom

Title: Polyhedral geometry of bisectors and bisection fans

 Abstract: Every symmetric convex body induces a norm on its affine hull.
The object of our study is the bisector of two points with respect to this
norm. A topological description of bisectors is known in the 2 and
3-dimensional cases and recent work of Criado, Joswig and Santos (2021)
expanded this to a deeper study of the geometric, combinatorial and
topological properties of the bisector. A key object introduced was the
bisection fan of a polytope which they were able to explicitly describe in
the case of the tropical norm. We discuss the bisector as a polyhedral
complex, introduce the notion of bisection cones and describe the
bisection fan corresponding to other polyhedral norms. This is joint work

Algebraic Groups seminar

Tuesday, 25 July 2023, 4 pm

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Venue: Room 114

Host: Shripad Garge

Speaker: Dibyendu Biswas

Affiliation: IIT Bombay

Title: Commutative unipotent groups - I

Abstract: We study additive functions with the aim of understanding commutative unipotent groups.

Ph.D. defense seminar

Date:  Wednesday, 26th July, 3 pm

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Host: Debanjana Mitra

Venue: Ramanujan Hall

Speaker: Wasim Akram

Title: Feedback stabilization of certain evolution equations and its numerical study

Mathematics Colloquium

Wednesday, 26 July, 4 pm

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Host: J. K. Verma

Venue: Ramanujan Hall

Speaker: Vaibhav Pandey

Affiliation: Purdue University, West Lafayette, IN, USA

Title: Classical invariant theory

Abstract: Invariant theory has played a key role in the development of modern commutative algebra. The classical invariant theory has a rich history going back at least to the seminal contributions of David Hilbert and Emmy Noether. The beauty of the subject lies in its strong insistence on computations and on the far-reaching consequences of these computations in modern algebra.

In this talk, we will focus on the classical invariant rings---they are the fixed subrings of the natural actions of linear algebraic groups on polynomial rings. Exploring key properties of these rings over rational numbers was a major research program in the late eighteenth century. We will talk about these properties and prove the corresponding ring theoretic properties of these invariant rings over fields of positive characteristic. This is joint work with Mel Hochster, Jack Jeffries, and Anurag Singh.

Commutative algebra seminar

Speaker: Vaibhav Pandey
Affiliation: Purdue University
Venue: Ramanujan Hall
Date: Thursday, 27 July 2023
Title: On the natural null cones of the classical invariant rings.

Abstract: The generic determinantal ring can be seen as a subring of a polynomial ring in a natural manner. When the field is infinite, this subring is, in fact, the invariant ring of the natural action of the general linear group on the ambient polynomial ring. Over a field of characteristic zero, the general linear group is linearly reductive, so the invariant ring splits from the polynomial ring. This immediately implies a wealth of nice properties for the generic determinantal rings including the Cohen-Macaulay property and rational singularities.

When the field has positive characteristic, the general linear group is typically not linearly reductive. It is natural to ask if the above embedding continues to split over an infinite field of positive characteristic. We show that strikingly, this embedding does NOT split in ANY positive characteristic if the general linear group is NOT linearly reductive.

Broadly speaking, this non-splitting is due to the Cohen-Macaulay property of the irreducible components of the null cone of this representation. We analyze the irreducible components of the null cones of the general linear group and also the null cone of the symplectic group in greater depth. We show that they are all F-regular in positive characteristic and calculate their divisor class groups independent of characteristic. This talk is based on two separate projects: One with Mel Hochster, Jack Jeffries, and Anurag Singh and another with Uli Walther and Yevgeniya Tarasova. 
 

Ph.D. Defence Seminar 

Friday, 28 July 2023, 2.30 pm

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Venue: Ramanujan Hall

Host: Rekha Santhanam

Speaker: Soumyadip Thandar 

Affiliation: IIT Bombay

Title: Classifying Equivariant Rational Homotopy Type up to Isomorphic Cohomology Diagrams