Date and Time: Monday 10 February, 04:00 pm - 05:00 pm.
Venue: Room 216, Department of Mathematics.
Title: Hilbert 17th problem: classical proof and recent effectivity results.
Abstract: Hilbert 17th problem is asking whether a non negative polynomial
is always a sum of squares. We discuss Artinâ€™s (1927) positive answer to
this problem and explain why this answer did not provide an effective
method for constructing the sum of squares. We describe primitive
recursive effective results obtained by Kreisel and his students in the
fifties. Finally we explain the first elementary recursive degree bound we
obtain, a tower of five exponentials. A precise bound in terms of the
number and degree of the polynomials and their number of variables is
provided. This is a joint work with Henri Lombardi and Daniel Perrucci.
Time:
2:00pm-3:15pm
Location:
Room No. 114, Department of Mathematics
Description:
Partial Differential Equations seminar.
Speaker: M. Vanninathan.
Affiliation: IIT Bombay.
Date and Time: Tuesday 11 February, 02:00 pm - 03:15 pm.
Venue: Room 114, Department of Mathematics.
Title: Asymptotic solutions of Hyperbolic PDE II.
Abstract: We discuss several aspects of asymptotic solutions to some
models of Hyperbolic PDE with small wave lengths including their
construction and their justification. Necessary tools to carry out these
tasks will be introduced.
Time:
3:30pm-5:00pm
Location:
Room No. 215 Department of Mathematics
Description:
Commutative Algebra and Algebraic Geometry seminar.
Speaker: R.V. Gurjar.
Affiliation: IIT Bombay.
Date and Time: Tuesday 11 February, 03:30 pm - 05:00 pm.
Venue: Room 215, Department of Mathematics.
Title: Different differents.
Abstract: We will discuss Noether (also called homological) different,
Dedekind different, and Kahler different and their relationship with each
other.
Time:
11:00am
Location:
Room No. 114, Department of Mathematics
Description:
Combinatorics Seminar.
Speaker: Anuj Vora.
Affiliation: Systems and Control Dept., IIT Bombay.
Date and Time: Wednesday 12 February, 11:00 am - 12:30 pm.
Venue: Room 114, Department of Mathematics.
Title: Zero Error Strategic Communication.
Abstract: We consider a setting between a sender and a receiver, where the
receiver tries to exactly recover a source sequence privately known to the
sender. However, unlike the usual setting of communication, the sender
here aims to maximize its utility and may have an incentive to lie about
its true information. We show that the maximum number of sequences that
can be recovered by the receiver grows exponentially and is given by the
largest independent set of a graph defined on sequences. We then define a
notion of the strategic capacity of a graph and show that it is lower
bounded by the independence number of a suitably defined graph on the
alphabet. Moreover, the Shannon capacity of the graph is an upper bound on
the capacity. This talk will briefly discuss the Shannon's zero-error
capacity problem. We then proceed to derive bounds on the strategic
capacity and give exact values for perfect graphs. If time permits, we
will also discuss the case where the receiver aims for asymptotically
vanishing probability of error.
Time:
4:00pm-5:00pm
Location:
Room No. 105 Department of Mathematics
Description:
Probability seminar.
Speaker: Kartick Adhikari.
Affiliation: I.I.T Technion, Israel.
Date and Time: Thursday 13 February, 04:00 pm - 05:00 pm.
Venue: Room 105, Department of Mathematics.
Title: The Spectrum of Dense Random Geometric Graphs.
Abstract: We study the spectrum of Laplacian of a random geometric graph,
in a regime where the graph is dense and highly connected. As opposed to
other random graph models (e.g. the Erdos-Renyi random graph), even when
the graph is dense, not all the eigenvalues are concentrated around 1. In
the case where the vertices are generated uniformly in a unit
d-dimensional box, we show that for every $0\le k \le d$ there are
$\binom{d}{k}$ eigenvalues at $1-2^{-k}$. The rest of the eigenvalues are
indeed close to 1. The spectrum of the graph Laplacian plays a key role in
both theory and applications. Aside from the interesting mathematical
phenomenon we reveal here, the results of this paper can also be used to
analyze the homology of the random Vietoris-Rips complex via spectral
methods.
The talk will be based on a joint work with R. Adler, O. Bobrowski and R.
Rosenthal.
Time:
2:30pm-3:30pm
Location:
Room No. 114, Department of Mathematics
Description:
Partial Differential Equations seminar.
Speaker: Vivek Tewary.
Affiliation: IIT Bombay.
Date and Time: Friday 14 February, 02:30 pm - 03:30 pm.
Venue: Room 114, Department of Mathematics.
Title: Bloch wave approach to almost periodic homogenization.
Abstract: Bloch wave homogenization is a spectral method for obtaining
effective coefficients for periodically heterogeneous media. This method
hinges on the direct integral decomposition of periodic operators, which
is not available in a suitable form for almost periodic operators. In
particular, the notion of Bloch eigenvalues and eigenvectors does not
exist for almost periodic operators. However, we are able to recover the
homogenization result in this case, by employing a sequence of periodic
approximations to almost periodic operators. Another approach, that
employs periodic lifting of quasiperiodic operators is also discussed. We
also establish a rate of convergence for approximations of homogenized
tensors for a class of almost periodic media