Talk by A. Lerario

[Ugo Bruzzo]

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SISSA GEOMETRY SEMINARS
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Antonio Lerario, Institut Camille Jordan (Lyon), will give a talk on

"Random matrix theory and applications to geometry and random topology"

on Monday, March 10th, at 2:30 pm in Room 136 (SISSA building A).


Abstract:

Betti numbers b_2i(C) of a complete intersection C in complex projective space distributes in the range 0,...,dim(C) in a "delta shaped" function:
they are all "ones" except in the middle dimension (i=dim(C)/2) where all the topology is concentrated
(the sum of all Betti numbers is represented by the "integral" of this function).

Looking at the real analogue of this picture the only thing we can say is that the sum ("integral") of all the Betti numbers of the real part R is bounded by that of its complex counterpart C.

Despite this high level of freedom over the reals, we can still ask what is the typical behavior of this picture. In other words, picking a random real complete intersection, what do we expect its Betti numbers to be? And how do we expect them to distribute in the range 0,...,dim(R)?

In this talk I will combine techniques from algebraic topology (spectral sequences) and probability theory (random matrices) to give an answer to these questions in the case of an intersection of random quadrics. I will show that, in a sense, the real picture is on average the "square root" of the complex one.

(this is joint work with E. Lundberg)