Speaker: Dr. Shubham Rastogi
Date and time: January 13, 11:30 am
Venue: Ramanujan Hall

Title: On *-regular isometric dilations

Abstract: Every contraction on a Hilbert space has an isometric dilation.
And\^o extended this result to the pairs of commuting contractions. S.
Parrott showed that this dilation result does not extend to an $n$-tuple
of commuting contractions, in general for $n\geq 3.$ However, provided
that the $n$-tuple satisfies Brehmer's positivity condition, the dilation
exists. In fact, S. Brehmer proved that an $n$-tuple  of commuting
contractions satisfies Brehmer's positivity if and only if it admits a
minimal $*$-regular isometric dilation.  Moreover, D. Gasper and N. Suciu
showed that the minimal $*$-regular isometric dilation comprises doubly
commuting isometries. In this talk, we shall see an extension of this
result to a sequence of commuting contractions.

An $n$-tuple of doubly commuting pure isometries can be modeled by the
tuple of multiplication by the co-ordinate functions on a vector-valued
Hardy space over the polydisc. A similar result does not hold true for a
sequence of doubly commuting pure isometries. This brings us to the
question of characterizing a sequence that has the sequence of
multiplication by the co-ordinate functions on a vector-valued Hardy space
over the Hilbert multidisc, as its minimal $*$-regular isometric dilation.
We will address this question in the talk. The talk is based on a work in
progress with B. K. Das.

Partial Differential Equations seminar
Speaker: Mikko Salo (University of Jyvaskyla, Finland)
Title: Harmonic functions and their analogues in inverse problems
Time, day and date: 2:00:00 PM - 3:00:00 PM, Tuesday, January 14
Venue: Online mode

Commutative Algebra seminar
Speaker: R, V, Gurjar (TIFR Mumbai (Retd))
Title: Automorphisms of Algebraic Varieties.
Time, day and date: 4:00:00 PM - 5:00:00 PM, Tuesday, January 14
Venue: Room 215

Abstract

In the first lecture we will discuss general results about diffeomorphisms (resp. biholomorphisms) of compact differentiable (resp. compact complex) manifolds, and general algebraic varieties.
In the second talk we will indicate a proof of an important result.
Let X be a smooth projective variety of general type. Then the automorphism group of X is finite.
This applies to compact Riemann surfaces of genus > 1. Using this I will indicate the proof of Hurwitz's theorem:
Theorem. Let C be a compact Riemann surface of genus g > 1. Then Aut(C) has order at most 84(g-1). This bound is best possible.

Colloquium:
Speaker: Suman Kumar Sahoo (Department of Mathematics, IIT Bombay)
Title: Propagation of Singularities and Inverse Problems
Time, day and date: 4:00:00 PM - 5:30:00 PM, Wednesday, January 15
Venue: Ramanujan Hall

Abstracts:

The propagation of singularities in partial differential equations describes how the singularities of solutions evolve over time or space. In this talk, we explore how these singularities can be leveraged to recover geometric information about the coefficients of certain PDEs, such as the scattering relation.