These notes are based on a series of lectures given first at the University of Warwick in spring 2008 and then at the Courant Institute in spring 2009. It is an attempt to give a reasonably self-contained presentation of the basic theory of stochastic partial differential equations, taking for granted basic measure theory, functional analysis and probability theory, but nothing else. 
The approach taken in these notes is to focus on semilinear parabolic problems driven by additive noise. These can be treated as stochastic evolution equations in some infinite-dimensional Banach or Hilbert space that usually have nice regularising properties and they already form a very rich class of problems with many interesting properties. Furthermore, this class of problems has the advantage of allowing to completely pass under silence many subtle problems arising from stochastic integration in infinite-dimensional spaces.

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An Introduction to Stochastic PDEs

We construct a hereditarily indecomposable Banach space with dual space isomorphic to l1. Every bounded linear operator on this space is expressible as λI + K, with λ a scalar and K compact.

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A hereditarily indecomposable {\mathcal{L}_{\infty}} -space that solves the scalar-plus-compact problem

We prove the edge universality of the beta ensembles for any ${\beta \ge 1}$, provided that the limiting spectrum is supported on a single interval, and the external potential is ${\fancyscript{C}^4}$ and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class ${\fancyscript{C}^4}$.

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Edge Universality of Beta Ensembles

Publisher Summary The theory of Banach spaces has its foundation on set-theoretic topology. Thus, the classical work of Banach rests heavily on the deeper properties of topology: completeness of the reals gives the Hahn–Banach theorem. Baire's category theorem proves the open mapping theorem and the uniform boundedness principle, while Tychonoff's compactness theorem proves the Alaoglou theorem. The interaction between set-theoretic topology and Banach spaces is not only internal, that is, in the sense of using analytic methods in the study of topological spaces and conversely of using topological arguments in the study of Banach spaces. The interaction is to a large extent also because of the fact that both branches use the same set-theoretic methods Set-theoretic, topological, and Banach spaces problems and methods coalesce and interact in problems concerning calibers, existence of independent families, and some isomorphic embeddings into Banach spaces. This chapter further discusses infinitary combinatorics. The ordinals are defined in such a way that an ordinal is the set of smaller ordinals.

Banach Spaces and Topology

We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to obtain local laws for matrix ensembles that are \emph{anisotropic} in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity. For definiteness, we present the method for sample covariance matrices of the form $Q := T X X^* T^*$, where $T$ is deterministic and $X$ is random with independent entries. We prove that with high probability the resolvent of $Q$ is close to a deterministic matrix, with an optimal error bound and down to optimal spectral scales. 
As an application, we prove the edge universality of $Q$ by establishing the Tracy-Widom-Airy statistics of the eigenvalues of $Q$ near the soft edges. This result applies in the single-cut and multi-cut cases. Further applications include the distribution of the eigenvectors and an analysis of the outliers and BBP-type phase transitions in finite-rank deformations; they will appear elsewhere. 
We also apply our method to Wigner matrices whose entries have arbitrary expectation, i.e. we consider $W+A$ where $W$ is a Wigner matrix and $A$ a Hermitian deterministic matrix. We prove the anisotropic local law for $W+A$ and use it to establish edge universality.

Anisotropic local laws for random matrices

This book introduces graduate students and resarchers to the study of the geometry of Banach spaces using combinatorial methods. The combinatorial, and in particular the Ramsey-theoretic, approach to Banach space theory is not new, it can be traced back as early as the 1970s. Its full appreciation, however, came only during the last decade or so, after some of the most important problems in Banach space theory were solved, such as, for example, the distortion problem, the unconditional basic sequence problem, and the homogeneous space problem. The book covers most of these advances, but one of its primary purposes is to discuss some of the recent advances that are not present in survey articles of these areas. We show, for example, how to introduce a conditional structure to a given Banach space under construction that allows us to essentially prescribe the corresponding space of non-strictly singular operators. We also apply the Nash-Williams theory of fronts and barriers in the study of Cezaro summability and unconditionality present in basic sequences inside a given Banach space. We further provide a detailed exposition of the block-Ramsey theory and its recent deep adjustments relevant to the Banach space theory due to Gowers.

Ramsey Methods in Analysis

Genericity and amalgamation of classes of Banach spaces

We investigate in this paper the complementation of copies of .This yields a new characterization of a class of injective Banach spaces.2000 Mathematical Subject Classification: 46B20, 46B26.

Spiros A. Argyros

Papers

A hereditarily indecomposable {\mathcal{L}_{\infty}} -space that solves the scalar-plus-compact problem

Ramsey Methods in Analysis

Genericity and amalgamation of classes of Banach spaces

Complementation and Embeddings of c0(I) in Banach Spaces

A class of banach spaces with few non strictly singular operators