Title: On a local version of the fifth Busemann-Petty Problem
Abstract:
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved. Their fifth problem asks the following.
Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let
C(K,x)=vol(K\cap H_x)dist (0, G).
Title: Nodal statistics of graph eigenfunctions
Abstract: Understanding statistical properties of zeros of Laplacian
eigenfunctions is a program which is attracting much attention from
mathematicians and physicists. We will discuss this program in the
setting of "quantum graphs", self-adjoint differential operators
acting on functions living on a metric graph.
Numerical studies of quantum graphs motivated a conjecture that the
distribution of nodal surplus (a suitably rescaled number of zeros of
Title: Perturbations of non-normal matrices
Abstract: Eigenvalues of Hermitian matrices are stable under perturbations in the sense that the $l_p$ norm of the difference between (ordered)eigenvalues is bounded by the Schatten norm of the perturbation. A similar control does not hold for non-Normal matrices. In the talk, I will discuss
Abstract: An inhomogeneous Markov chain X_n is a Markov chain whose state spaces and transition kernels change in time. A “local limit theorem” is an asymptotic formula for probabilities of th form
Prob[S_N-z_N\in (a,b)] , S_N=f_1(X_1,X_2)+....+f_N(X_N,X_{N+1})
in the limit N—>infinity. Here z_N is a “suitable” sequence of numbers.
I will describe general sufficient conditions for such results.
If time allows, I will explain why such results are needed for the study of certain problems related to irrational rotations.
Krieger’s generator theorem shows that any free invertible ergodic measure preserving action (Y,\mu, S) can be modelled by A^Z (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is A^Z) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which Z^d-dynamical systems are universal. These conditions are general enough to prove that
1) A self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo)
Title: Eigenvector correlation in the complex Ginibre ensemble
Abstract:
The complex Ginibre ensemble is a non-Hermitian random matrix on $\mathbb{C}^N$ with i.i.d. complex Gaussian entries normalized to have mean zero and variance $1/N$. Unlike the Gaussian unitary ensemble, for which the eigenvectors are orthogonal, the geometry
of the eigenbases of the Ginibre ensemble are not particularly well understood.
Title: Fuglede's spectral set conjecture for convex polytopes.
Abstract:
A set \Omega \subset \mathbb{R}^d is called spectral if the space L^2(\Omega) admits an orthogonal basis of exponential functions. Back in 1974, B. Fuglede conjectured that spectral sets could be characterized geometrically as sets which can tile the space by translations. This conjecture inspired extensive research over the years, but nevertheless, the precise connection between the notions of spectrality and tiling, is still a mystery.
