**Date & Time:** Friday, 2nd September at 4.30 pm

**Venue:** Ramanujan Hall

**Title: **: Universal property of the Chern character form of the universal
connection

**Speaker:**Prof. Mahuya Datta,
Indian Statistical Institute, Kolkata

**Abstract: **(in LATEX format):
Gromov proved that given any closed 2-form $\omega$ on a manifold $M$,
there is a smooth immersion $f:M\to\C P^n$ which pulls back the standard
symplectic form $\sigma$ on the complex projective space $\C P^n$ onto
$\omega$, provided $n$ is sufficiently large and certain necessary
cohomological condition is satisfied. This means that the symplectic manifolds
$(\C P^n,\sigma)$ are the universal objects in the `category' of pairs
$(M,\omega)$, where $\omega$ is a closed 2-form on $M$.

We generalise this result for any even degree form $2k$. We identify certain $2k$-forms $ch_k$ on the complex Grassmannians $Gr_n(\C^q)$. These forms arise as the coefficients of the Chern character form of the universal connection on the Stiefel bundle over $Gr_n(\C^q)$. We then show that any closed $2k$-form $\omega$ on a smooth manifold $M$ can be realised as the pull back of $ch_k$ by some map $f: M\lgra Gr_n(\C^q)$ provided $q$ and $n$ are sufficiently large, and some necessary cohomology conditions are satisfied.