Let be a finite set, and let be its projections to the corresponding two-planes. Denoting by the cardinality of a set , we have the following inequality:
This generalizes to sets in higher dimensions and projections to subspaces of possibly different dimensions. As long as each coordinate appears at least times on the right, the size of is bounded by the product of the projections.
One proof of the above inequality is via a mixed version of Cauchy-Schwartz or Holder-type inequalities. For functions of two variables, we have
Taking to be indicators of projections of to the -planes, the bound follows. However, proving the inequality for requires some work.
There is a different inequality, involving entropy of random variables, whose proof (and generalizations) are much more conceptual. Namely, if are random variables and denotes entropy, we have
Below the fold are the definitions, a proof, and an application. Continue reading
In this post, I will record some properties of reduced root systems, following Bourbaki. The reference is their “Lie groups and algebras, Ch. 4-6”. There will be no proofs, just a summary of the main properties. The post consists mainly of definitions and lists their properties. Continue reading
In this post, I will describe a special case of a theorem about torsion points on Jacobians and Teichmuller dynamics which is proved in this paper. Namely, I will describe a different proof of a theorem of Moeller that for Teichmuller curves, zeroes of the holomorphic -form must land in the torsion of (a factor of) the Jacobian. Some familiarity with Teichmuller dynamics will be assumed, as in the survey of Zorich.
In this post, I would like to discuss some of the ideas from my paper Semisimplicity and Rigidity of the Kontsevich-Zorich cocycle. It contains a number of somewhat disparate methods – Hodge theory, random walks and a bit of homogeneous and smooth dynamics. I will try to explain how these concepts interact and what the essential aspects are. Some familiarity with flat surfaces, as explained for example in the survey by Anton Zorich, will be assumed.
Below the fold I will discuss some of the analytic ideas from variations of Hodge structures. Then I’ll try to connect these to Teichmuller dynamics and explain how this leads to various semisimplicity results. For applications, I’ll discuss how this implies that measurable -invariant bundles have to be real-analytic. I’ll also mention how real multiplication on factors of the Jacobians arises.
Grothendieck famously wrote a paper called Hodge’s general conjecture is false for trivial reasons. In this post, I would like to record, for my own benefit, some of the observations made in it.
Consider the following question. If is continuous in each variable separately, need it be continuous in both? Although Cauchy (incorrectly) said it should, note it is not even obvious that it should be measurable.
A classical example is
Defining , this function is real-analytic in each variable separately, but it is not even continuous at the origin.
By contrast, in the complex-analytic situation we have the following result.
Theorem 1 (Hartogs) Suppose is holomorphic in each variable separately. Then is holomorphic on .
Below the fold, I will state some more results of this type and prove some of them. Continue reading
In this note, I would like to record mainly for my own benefit some basic things about how to descend algebraic data along Galois extensions of fields. I have benefited from some notes of K. Conrad available online. The exposition below is a shorter version of those.