This term's topology learning seminar will be on Ozsvath-Szabo homology.

About 7 years ago, Ozsvath and Szabo invented this construction (which they call "Heegaard Floer homology") in an attempt to give a different definition of Seiberg-Witten theory. Their theory has been incredibly successful in applications to low-dimensional topology.

In brief, they show how to associate a family of homology groups to a 3--manifold by choosing a Heegaard splitting and computing a suitable Lagrangian intersection Floer homology. Some of the most important features of the construction are:

1. 4--dimensional cobordisms induce maps between homology groups; the invariants of closed 4-manifolds are conjecturally equal to the Seiberg-Witten invariants.

2. There is a version of the homology for knots in $S^3$, which leads to an exact formula (not just a bound!) for the genus of knots. Consequently their homology distinguishes the unknot, and can be used to prove many old conjectures about surgery on knots.

3. The theory gives rise to powerful invariants of contact structures on 3--manifolds and can distinguish tight from overtwisted.

4. The homology for knots, unlike all earlier gauge-theoretic invariants, can actually be calculated by purely combinatorial means. There is a strong hope that this will eventually lead to a complete combinatorial calculation of the Ozsvath-Szabo/Seiberg-Witten/Donaldson invariants of 4-manifolds.

The first meeting will be Tuesday 10th April, in room 7218, at 10.30am.
I will give an introductory talk and then we will arrange the schedule of speakers for the rest of the term. Anyone is welcome to attend - attendance does not necessarily lead to being volunteered for a talk!

Sinc--interpolation is an infinitely smooth interpolation on the whole real
line based on a series of shifted and dilated sinus--cardinalis functions used as
Lagrange basis. It often converges very rapidly, so for example for functions
analytic in an open strip containing the real line and which decay fast enough at
infinity. This decay does not need to be very rapid, however, as in Runge's function
$1/(1+x^2)$. Then one must truncate the series, and this truncation error is much
larger than the discretisation error (it decreases algebraically while the latter
does it exponentially).

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In our talk we will give a formula for the error commited when merely using
function values from a finite interval symmetric about the origin.
The main part of the formula is a polynomial in the distance between the nodes
whose coefficients contain derivatives of the function at the extremities.

The concept of relative measure was fairly popular among Polish
mathematician of 1930 in Lvov. Steinhaus and Urbanik were working
on introducing a relative measure and relative measurability into
the area of random variables.
Recent work on signal processing and time series has led to
re-discovery of the 'old-school' theorems and application to
data generated by signals or time-series. Some fundamental work
was done by Garnder and continuation of this work was done
by Leskow and Napolitano.

A short informal introduction to nonstochastic approach
to time series inference via relative measurability will
be presented. Applications to signal forecasting will be
presented.

Let G be a group acting``nicely" on a space, $X$, with some
structure(topological, differentiable, algebraic, combinatorial.). A
basic problem is to find an effective way of determining if two points
$x,y$ are in the same orbit (or at least in an appropriate closure of an
orbit). In this lecture I will look at methods that can be used in
concert with computers to
approach such problems through the determination of "enough" computable
invariant functions. There will be several examples including measures of
quantum entanglement.

There will be a 10 minute organizational meeting for the schedule of talks this quarter, after which I will discuss a recent proof of the local Langlands conjecture for GSp(4). Joint with with Shuichiro Takeda.

A q-query Locally Decodable Code (LDC) is an error-correcting
code that encodes an n-bit message x as a codeword $C(x)$, such that one can
probabilistically recover any bit $x_i$ of the message by querying only $q$
bits
of the codeword $C(x)$, even after some constant fraction of codeword bits
has
been corrupted. The goal of LDC related research is to minimize the length
of
such codes.

A q-server private information retrieval (PIR) scheme is a cryptographic
protocol that allows a user to retrieve the $i-th$ bit of an $n-bit$ string $x$
replicated between $q$ servers while each server individually learns no
information about $i$. The goal of PIR related research is to minimize the
communication complexity of such schemes.

We present a novel algebraic approach to LDCs and PIRs and obtain vast
improvements upon the earlier work. Specifically, given any Mersenne prime
$p=2^t - 1$, we design three query LDCs of length $Exp(n^{1/t})$, for every
$n$. Based
on the largest known Mersenne prime, this translates to a length of less
than $Exp(n^{10^{-7}})$, compared to $Exp(n^{1/2})$ in the previous
constructions.
We also design 3-server PIR schemes with communication complexity of
$O(n^{10^{-7}})$ to access an n-bit database, compared to the previous best
scheme with complexity $O(n^{1/5.25})$.

It has often been conjectured that there are infinitely many Mersenne primes. Under this conjecture, our constructions yield three query locally
decodable codes of subexponential length and three server private
information
retrieval schemes with subpolynomial communication complexity.