Title: Error Correction and List Decoding for Reed Solomon Codes
Abstract:
In this talk, we will have a look at three results, starting with the following.
[Berlekamp-Welch] Given a univariate polynomial function over F_q
with data corruption at t < q/2 points, we can recover the function
completely if the degree of the function is sufficiently low.
A generalization of the above is as follows, where instead of
'recovering' the function, we find all its 'close approximates'.
[Madhu Sudan] Given data points (x_i,y_i), i \in [n], and parameters
k and t, we can list all polynomials with degree at most k, which
satisfy at least t data points.
This result can further be generalized as follows.
[Madhu Sudan] Given data points (x_i,y_i) with weights w_i, i \in
[n], and parameters k and W, we can list all polynomials with degree
at most k such that the sum of weights of data points satisfied by the
polynomial is at least W.
The last two results provide list-decoding of Reed-Solomon codes.
Time:
4:00pm-5:00pm
Location:
Room 113
Description:
Title: Asymptotics of the number of points of symplectic lattices in subsets of Euclidean spaces
Abstract: It is well known that a "good" large subset of the Euclidean space contains approximately as many lattice points as its volume. This need not hold for a general subset. On the other hand, a classical theorem of Siegel asserts that for any subset of positive measure, the "average" number of points (in an appropriate sense) of a general unimodular lattice contained in it, equals the measure of the set. In place of the average over the entire space of lattices one may also ask for analogous results for smaller subclasses. In a recent work with Jayadev Athreya, we explored this issue, with some modifications that place the problem in perspective, for the case of symplectic lattices, viz. lattices (in even-dimensional spaces) obtained from the standard lattice under symplectic transformations. In this talk I shall describe the overall asymptotics in this case, together with the historical background of the results and techniques involved.
Time:
3:30pm-5:00pm
Location:
Ramanujan Hall
Description:
Title: A new proof of Zariski's Theorem about Complete ideals in two-dimensional regular local rings.
Abstract: Zariski's first paper in algebra written in 1938 proved among many other results that product
of complete ideals is complete in the polynomial ring $K[X,Y]$ where $K$ is an algebraically
closed field of characteristic zero. This was generalised to two-dimensional regular local rings
in Appendix 5 of Zariski-Samuel's classic "Commutative Algebra". We will present a new proof
of this theorem using a formula of Hoskin-Deligne about co-length of a zero-dimensional
complete ideal in a two-dimensional regular local ring in terms of quadratic transforms of