Schedule

1. All talks in the Seminar Hall.
2. On Friday the talks are at 2pm and at 3pm.

R. Srinivasan, Toeplitz CAR flows.

$E_0$-semigroups are weakly continuous semigroups of endomorphisms on B(H), the algebra of all bounded operators on a separable Hilbert space. They are broadly divided into three types,  namely type I, II and III. R. T. Powers discovered the first example of type III $E_0-$semigroup. In 2000 Boris Tsirelson discovered an example of  uncountable family of mutually non cocycle conjugate of type III $E_0-$semigroups.  This is followed by few important works, but there is no activity relating to the original example of Powers. 

Toeplitz CAR flows are a class of $E_0$-semigroups including the first type III example, constructed by Powers. It is shown that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate $E_0$-semigroups of type III. A generalization of the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson) was proved. Consequently it was shown that Toeplitz CAR flows are always either of type I or type III.

This is a joint work with Masaki Izumi.

Krishna Hanumanthu, Syzygies and geometry of projective varieties.

Syzygies of certain vector bundles can shed important light on the intrinsic geometry of a projective variety. I will discuss some aspects of this phenomenon and a few new results. I will also list a number of open problems.

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Tue 2011-Sep-06

Shrihari Sridharan, SRB-measure leaks

In this talk, we shall study about the escaping rate of the Sinai-Ruelle-Bowen (SRB) measure through holes of positive measure constructed in the Julia set of hyperbolic rational maps (open dynamics). The dependence of this rate on the size and position of the hole shall be explained. For an easier and better understanding, the simple quadratic polynomial restricted on the unit circle will be analysed thoroughly.

Yashonidhi Pandey, Parahoric $G$-bundles on a compact Riemann surface for classical groups

Let $p: Y \rightarrow X$ be a Galois cover of smooth projective curves with Galois group $\pi$. A $\pi-G$-bundle $E$ is a principal $G$-bundle on $Y$ together with a $\pi$-linearisation. For the case of classical groups, we describe $\pi-G$-bundles intrinsically as objects defined of the base curve $X$. After the recent work of Balaji-Seshadri, they may be called parahoric $G$-bundles. We also give necessary and sufficient conditions for the non-emptiness of the moduli of (semi)-stable parahoric $G$-bundles on the projective line (for $G= O(n), Sp(2n)$).

This work is part of my post-doctoral work with Professors C.S.Seshadri and V.Balaji.

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Wed 2011-Sep-07

Pabitra Barik, Existence Of Fine Moduli Space of Stable Bundles over Compact Riemann Surface

Shiva Shankar, On the injective hull of spaces of periodic functions.

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Thu 2011-Sep-08

Micah Leamer, Asymptotic behavior of the dimensions of syzygies.

Let R be a commutative Noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M eventually stabilize to the depth of R. In this talk, I will give conditions under which a similar statement can be made regarding dimension. In particular, if R is equidimensional and the Betti numbers of M are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of M coincides with the dimension of R.

This is joint work with Kristen Beck at the University of Arizona.

Senthamarai Kannan, Torus quotients of homogeneous spaces.

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Fri 2011-Sep-09

Manoj Kummini, Multilinear free resolutions from higher tensors
(2pm)

We describe a construction of free resolutions from higher tensors. It provides a unifying view on a wide variety of complexes including the Eagon--Northcott, Buchsbaum--Rim and similar complexes, and the Eisenbud--Schreyer pure resolutions. As motivating examples, we will look at detereminantal varieties and multilinear maps from product of projective spaces. This is joint work with C. Berkesch, D. Erman and S. Sam.

Pramath Sastry, The Deligne-Verdier approach to Grothendieck Duality
(3pm)

Deligne's appendix to Hartshorne's "Residues and Duality" was regarded as a stunning shortening of Grothendieck's duality theory, for the twisted inverse image lax- functor ("upper shriek") was obtained by Deligne from general considerations. In contrast Grothendieck needed the prop of Cousin complexes and dualizing complexes. Later Verdier showed a way of using Deligne's results to concretely realise the "upper shriek" in the smooth case via differential forms. However, experts in the field soon realised that the shortening and the concreteness (even in the smooth case) were somewhat illusory. A full understanding in concrete terms needs an understanding of the trace map which in turn needs a concrete theory of residues. Conrad pointed out that even for as simple a case of projective n-space over a field, Verdier's approach does not give the trace or residues, at least not with the technology known when Conrad's book was written (2002). A little later I showed that the Deligne-Verdier approach to duality is compatible with arbitrary base change (not just flat base change) for Cohen-Macaulay maps, and this opened the road to realising the full potential of the Deligne-Verdier approach.

This is joint work with Suresh Nayak.

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