Course Information and Syllabus

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Course title

Gaussian Random holomorphic fields: Topics in Random Complex Geometry

Course number

Meeting times/Location

110.760

MW 12 - 1: 15  Krieger 302

Professors:

Office Hours

Bernard Shiffman and Steve Zelditch

M  2  - 3PM  in Krieger 406 and by appointment.

Course description

Field = function, vector field, tensor field, section of a line bundle etc. Gaussian random = we define  a Gaussian measure on a vector space of such fields.

Think of a field F =  \sum_{j = 1}^d  c_j F_j  where {F_j} is an orthonormal basis of functions in a given Hilbert space and c_j are independent Gaussian random variables.

It is a random field since the coefficients are random. We will study the zeros, critical points, and  `landscape'  of such random fields.

For instance if we assume the fields have norm one, then the graph of $|F|^2$ of a random F is a kind of random mountain chain. Imagine taking a level or height and considering

the graph of |F|^2 above that height; think of the part of a mountain above sea level. If the level is too high, the mountain is surely underwater. As the level is lowered,

one first sees one mountain peak. As the level is lowered, we see several, then we see ridges and so on. What is the structure of these `excursion sets'? Or the local maxima at a high level?

When do ridges form between the mountain peaks (i.e. when does the landscape percolate?) This kind of question is important in physics, ranging from astrophysics to string theory to water waves.

Much of geometry is concerned with extremal objects: most singular varieties, optimal forms...Random fields is concerned with the opposite regime of most likely objects. They are

often very rough.  To get an idea of what a random landscape looks like, take a look at these

 IMAGES

Compared to traditional probability courses, our approach to Gaussian random fields is unusual:   we define them  by first putting   Hermitian inner products of the form

 \int_M |f(z)|^2 e^{- N \phi(z}}  d\nu(z) on spaces of the functions or sections . The two point function or covariance is the Szego

kernel of this weighted Hilbert space of holomorphic fields. Usually in probability theory one assumes the covariance function has certain properties.

But in geometry it is more natural to start with the  inner product and then see what consequences

the choice of inner product has on the behavior of the szego kernel and Gaussian random fields. A basic type of result is that zeros of Gaussian random holomorphic fields tend

to be distributed like the equilibrium measure \mu^{eq}_{K, \phi}  defined by the weight $\phi$ and the set K where $d\nu$ is supported.

We will start with one complex dimension, where the zero set of a random holomorphic polynomial of degree N is a random set of N points. Thus the zero set is a random point process.

Later we will discuss higher dimensional generalizations, e.g. random divisors, random subvarieties of a variety, random holomorphic maps to projective space....

There are real variable analogues where one uses eigenfunctions of the Laplacian instead of holomorphic polynomials. A much studied random field in this setting is the Gaussian

free field. Time permitting, we may study the recent work of Duplantier-Sheffield proving rigorously the so-called KPZ relation from Liouville quantum gravity.

Pre-requisites:                                                              Some knowledge of complex analysis and geometry, e.g. line bundles and their holomorphic sections. We will cover the basics. The first Chapter of Griffiths-Harris is enough.

We will assign exercises to familiarize interested students with the definitions. There are many open research level problems in this area.

Textbooks

We are writing a book in this area, and will lecture on early drafts of the chapters.

Some other recommended texts are on the right.

2. Cramér, Harald; Leadbetter, M. R. Stationary and related stochastic processes. Sample function properties and their applications. Reprint of the 1967 original. Dover Publications, Inc., Mineola, NY, 2004.3.

3. Janson, Svante Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129. Cambridge University Press, Cambridge, 1997