Abstract: Let V be an irreducible algebraic subvariety of C^n X C^n of
If Schanuel Conjecture holds, under some natural conditions on V, we
show that, if V is defined over the rationals, there exists a in C^n
such that (a, exp(a)) is a generic point of V.
Ergodic theoretic methods in the context of homogeneous dynamics have been highly successful in number theoretic and other applications. A lacuna of these methods is that usually they do not give rates or effective estimates. Einseidler, Venkatesh and Margulis proved a rather remarkable quantitative equidistribution result for periodic orbits of semisimple groups in homogenous spaces that can be viewed as an effective version of a result of Mozes and Shah based on Ratner's measure classification theorem.
This is the second of two lectures on the paper Einseidler,, Margulis, Mohammadi and Venkatesh https://arxiv.org/abs/1503.05884. In this second lecture I will explain how the authors obtain using property tau (uniform spectral gap for arithmetic quotient) quantitaive equidistribution results for periodic orbits of maximal semisimple groups. Surprisingly, one can then use this theorem to establish property tau...
Contingency tables are matrices with fixed row and column sums. They are in natural correspondence with bipartite multi-graphs with fixed degrees and can also be viewed as integer points in transportation polytopes. Counting and random sampling of contingency tables is a fundamental problem in statistics which remains unresolved in full generality.
Open Gromov-Witten (OGW) invariants count pseudoholomorphic maps from a Riemann surface with boundary to a symplectic manifold, with constraints that make sure the moduli space of solutions is zero dimensional. In joint work with J. Solomon (2016-2017), we defined OGW invariants in genus zero under cohomological conditions. In this talk, also based on joint work with J. Solomon, I will describe a family of PDEs satisfied by the generating function of our invariants. We call this family the open WDVV equations.
In the past decades There has been considerable interest in the probability that two random elements of (finite or certain infinite)
I will describe new works (by myself and by others) on probabilistically nilpotent groups, namely groups in which the probability that [x_1,...,x_k]=1 is positive/bounded away from zero.
It turns out that, under some natural conditions,
these are exactly the groups which have a finite/bounded index
subgroup which is nilpotent of class < k.
The proofs have some combinatorial flavor.
Let X be the spherical building associated to the group G=GL(n,F) ,
where F is a finite field. We will survey some results on the homology of X with constant and twisted coefficients, and on the corresponding expansion properties.
In this talk we recall Conlon's random construction of sparse 2-dim simplicial complexes arising from Cayley graphs of F_2^t . We check what expansion properties this construction has (and doesn't have): Mixing of random walks, Spectral gap of the 1-skeleton, Spectral gap of the links, Co-systolic expansion and the geometric overlap property.
Locally testable codes are error-correcting codes that admit
super-efficient checking procedures. In the first part of the talk, we will
see why expander based codes are NOT locally testable. This is in contrast
to typical "good" error correcting properties which follow from expansion.
We will then see that despite this disconnect between expansion and
testability, all known construction of locally testable codes follow from
the high-dimensional expansion property of a related complex leaving open
an intriguing connection between local-testability and high-dimension
Computing R=P.Q ,the product of two mXm Boolean matrices [BMM] is an ingredient
of many combinatorial algorithms.
Many efforts were made to speed it beyond the standard m^3 steps, without using
the algebraic multiplication.
To divide the computation task, encoding of the rows and column indices were
used (1.1) j by (j1,j2) k by (k1,k2)
e.g. using integer p j2=j mod p ,j1=ceiling of j/p.
Clearly, the product of the ranges of the digits= m1.m2 - is approximately m.
L.Lee’s article reduced BMM to parsing substrings of a fixed string u with
In this talk we shall review a paper by Gromov and Guth, in which they introduced several ways to measure the geometric complexity of an embedding of simplicial complexes to Euclidean spaces.
One such measurement is strongly related to the notion of high dimensional expanders introduced by Gromov, and in fact, it is based on a paper of Kolmogorov and Barzadin from 1967, in which the notion of an expander graph appeared implicitly.
We shall show one application of bounded degree high dimensional expanders, and present many more open questions arising from the above mentioned paper.