09:00 to 09:50 Registration 09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) 10:00 to 11:00 On non-smooth functions of Wiener-Hopf operators We discuss trace formulae for the operator \begin{equation*} f(PAP) - Pf(A)P, \end{equation*} where $A$ is a pseudo-differential operator on $L^2(\mathbb R^d)$ with a smooth or discontinuous symbol, and $P$ is a multiplication by the indicator of a piece-wise smooth domain in $\mathbb R^d$. The function $f$ is not supposed to be smooth. The obtained formulae generalise results obtained by H. Widom in the 80's. These results are used to study the entanglement entropy of free fermions at positive temperature both in the low and high temperature limits. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 On the two-dimensional random walk in an isotropic random environment This is joint work with Erich Baur (Lyon) and Ofer Zeitouni (Weizmann Institute). We report on work in progress on the standard model of a random walk in random environment in the critical dimension 2. We investigate exit distributions from large sets which are supposed to be essentially the same as those for ordinary random walks. Random walks in random environment are well understood in dimension one, and for small disorder in dimensions above 2, but the two-dimensional case is largely open. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 From the mesoscopic to microscopic scale in random matrix theory Co-authors: Laszlo Erdos (IST), Horng Tzer Yau (Harvard), Jun Yin (Wisconsin Madison) Eugene Wigner has envisioned that the distributions of the eigenvalues of large Gaussian random matrices are new paradigms for universal statistics of large correlated quantum systems. These random matrix eigenvalues statistics supposedly occur together with delocalized eigenstates. I will explain recent developments proving this paradigm for eigenvalues and eigenvectors of random matrices. This is achieved by bootstrap on scales, from mesoscopic to microscopic. Random walks in random environments, homogenization and the coupling method play a key role. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 On fluctuations of eigenvalues of random band matrices We consider the fluctuation of linear eigenvalue statistics of random band $n$ dimensional matrices whose bandwidth $b$ is assumed to grow with n in such a way that $b/n$ tends to zero. Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we remove the main technical restriction $n>>b>>n^{1/2}$ of all the papers, in which band matrices were studied before. Moreover, the developed method allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc. INI 1 16:00 to 17:00 Primitive Pythagorean triples and "near" quasicrystals Counting things is a great favorite of children, and mathematicians as well, whatever the things are. In this talk, I discuss an old counting problem on primitive Pythagorean triples in view of modern theory of quasicrystals. INI 1 17:00 to 18:00 Welcome Wine Reception
 10:00 to 11:00 Pure point spectrum in the regime of zero Lyapunov exponents Co-author: Anton Gorodetski (UC Irvine) In the study of ergodic Schrodinger operators, a central role is played by the Lyapunov exponent of the associated Schrodinger cocycle. We discuss a construction showing that the regime of zero Lyapunov exponents can contain pure point spectrum. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Invariance of IDS under Darboux transformation and its application The integrated density of states (IDS) is a crucial quantity for studying spectral properties of ergodic Schroedinger opearators. Especially in one dimension it determines most of the spectral properties. On the other hand, in relation to completely integrable systems, Darboux transformation has been investigated from various points of views. In this talk the invariance of IDS under Darboux transformation will be shown, and as a byproduct the invariance of IDS under KdV flow will be remarked. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Behavior of the spectrum of the periodic Schrodinger operators near the edges of the gaps Co-author: Leonid Parnovski (UCL) It is a common belief that generically all edges of the spectrum of periodic Schrodinger operators are non-degenerate, i.e. are attained by a single band function at finitely many points of quasi-momentum and represent a non-degenerate quadratic minimum or maximum. We present the construction which shows that all degenerate edges of the spectrum can be made non-degenerate under arbitrary small perturbation. The corresponding perturbation is found in the class of potentials with larger (but proportional) periods; thus the final operator is still periodic but the lattice of periods changes. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 $L_1$-Estimates for Eigenfunctions of the Dirichlet Laplacian Co-authors: Michiel van den Berg (U Bristol), J\"urgen Voigt (TU Dresden) For $d \in {\bf N}$ and $\Omega \ne \emptyset$ an open set in ${\bf R}^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. We do {\it not} require $\Omega$ to be of finite volume. % If $\Phi$ is associated with an eigenvalue below the essential spectrum of $-\Delta_\Omega$, we provide estimates for the $L_1$-norm of $\Phi$ in terms of the $L_2$-norm of $\Phi$ and suitable spectral data of $-\Delta_\Omega$. The main idea in obtaining such estimates consists in finding a---sufficiently small---subset $\Omega' \subset \Omega$ where $\Phi$ is localized in the sense that $\Phi$ decays exponentially as one moves away from $\Omega'$. These $L_1$-estimates are then used in the comparison of the heat content of $\Omega$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. \vskip.5em This is joint work with Michiel van den Berg (Bristol) and J\"urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden). INI 1 19:30 to 22:00 Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co.