January 30, 2017 - February 3, 2017
Public Access Category: All |
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- Time:
- 11:00am-1:00pm
- Location:
- Room No. 215
- Description:
- Title: Jannsen's theorem on semisimplicity - 3
Abstract: In these lectures we shall introduce motives and present
results in Jannsen's paper, which say that the "conjectural" category of
motives is semisimple abelian iff the adequate equivalence relation taken
is
- Time:
- 11:00am
- Location:
- Ramanujan Hall
- Description:
- TITLE: Progress in Error-Correction: A Survey
Speaker: Venkatesan Guruswami, Carnegie Mellon Univ.
ABSTRACT:
Error-correcting codes play a crucial role in safeguarding data against the
adverse effects of noise during communication and storage. They are also
powerful tools underlying several recent advances in theoretical computer
science and combinatorics. The central challenge in coding theory is to
construct codes with minimum possible redundancy for different error models
and requirements on the decoder, along with efficient algorithms for
error-correction using those codes. Much progress has been made toward this
quest in the nearly seven decades since the birth of coding theory. Several
fundamental problems, however, continue to challenge us, and exciting new
directions routinely emerge to address current technological demands as well
as applications in computational complexity and cryptography. This talk will
survey some of our recent works on error-correction in various models, such
as:
- worst-case errors, where we construct list decodable codes with redundancy
as small as the target error fraction;
- i.i.d. errors, where we show polar codes enable efficient error-correction
even as the redundancy approaches Shannon capacity;
- bit deletions, where we give codes that can correct the largest known
fraction of deletions;
- single symbol erasure, a model of substantial current interest for
tackling node failures in distributed storage, where we give novel repair
algorithms for Reed-Solomon codes as well as simple new codes with
low-bandwidth repair mechanisms.
- Time:
- 3:30pm-5:00pm
- Location:
- Room No. 216
- Description:
- Title: Values of quadratic forms at integer points II
Abstract: This will be a continuation of the overview from the last week. Some details will be briefly recalled from the last time, for continuity and the benefit of new audience if any.
- Time:
- 5:00pm-7:00pm
- Location:
- Room No. 215
- Description:
- Title: Some consequences of the Riemann hypothesis for varieties over
finite field
Abstract: We will talk about a result of M. Katz and W. Messing, which
says the following. From the Riemann hypothesis and the hard Lefschetz
theorem in l-adic cohomology, the corresponding facts for any Weil
cohomology follow.
- Time:
- 11:00am-12:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: The Capset bound of Croot-Lev-Pach and Ellenberg-Gijswijt
Abstract: A construction of Behrend from the 1940s shows that there are subsets of [N] of size N^{1-o(1)} that contain no 3-term APs (also called capsets). For a long time, it was open whether there is such a construction over F_3^n (i.e. a capset in F_3^n of size 3^{n-o(n)}). Recently, building on work of Croot, Lev and Pach, it was shown by Ellenberg and Gijswijit ( https://arxiv.org/abs/1605.09223 ) that such a construction does not exist: i.e. any capset in F_3^n can have size at most c^n for some c < 3. The construction has had several applications already in Combinatorics and Theoretical Computer Science. We will see a proof of the theorem of Ellenberg and Gijswijt
- Time:
- 11:30am
- Location:
- Room No. 216
- Description:
- Title. Rational Surface Singularities.
Title. We will prove a purely numerical criterion due to M. Artin to test
the rationality of a surface singularity. In practice this is the
criterion which is used when a rational surface singularity is being
considered.
- Time:
- 2:30pm-3:30pm
- Location:
- Ramanujan Hall
- Description:
- Title: Existence and regularity theory in weighted Sobolev spaces and
applications.
Abstract: My emphasis in this talk will be on functional analytical tools to
the solvability and uniqueness of solutions to the nonhomogeneous boundary
value problems, dealing with degenerate PDEs of elliptic type. My aim is to
consider possibly general class of weights. In particular, I consider the
$B_{p}$-class of weights, introduced by Kufner and Opic, which is much more
general class than the commonly studied Muckenhoupt $A_{p}$-class.
- Time:
- 4:00pm-5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Generalized Hamming weights of (projective) Reed-Muller codes.
Abstract: Reed-Muller codes are among the most elementary and most studied codes. Less studied, but equally elementary are their projective counterparts, the protective Reed-Muller codes. Many open questions remain about these codes. Mathematically, a very interesting question is the determination of the generalized Hamming weights. The determination of these weights is equivalent to the determination of the maximum number of common solutions to certain system of polynomial equations. In this talk, I will give an overview of recent work and developments on the theory of generalized Hamming weights of projective Reed-Muller codes. This work was carried out together with Mrinmoy Datta and Sudhir Ghorpade.
- Time:
- 3:30pm-5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Koszul Algebras IV