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Series: Math Physics Seminar

Series: Math Physics Seminar

Existence of ballistic transport for Schr ̈odinger operator with a quasi-
periodic potential in dimension two is discussed. Considerations are based on the
following properties of the operator: the spectrum of the operator contains a semiaxis
of absolutely continuous spectrum and there are generalized eigenfunctions being close
to plane waves ei⟨⃗k,⃗x⟩ (as |⃗k| → ∞) at every point of this semiaxis. The isoenergetic
curves in the space of momenta ⃗k corresponding to these eigenfunctions have a form
of slightly distorted circles with holes (Cantor type structure).

Series: Math Physics Seminar

During the last few years there has been a systematic pursuit for sharp estimates of the energy components of atomic systems in terms of their single particle density. The common feature of these estimates is that they include corrections that depend on the gradient of the density. In this talk I will review these results. The most recent result is the sharp estimate of P.T. Nam on the kinetic energy. Towards the end of my talk I will present some recent results concerning geometric estimates for generalized Poincaré inequalities obtained in collaboration with C. Vallejos and H. Van Den Bosch. These geometric estimates are a useful tool to estimate the numerical value of the constant of Nam's gradient correction term.

Series: Math Physics Seminar

We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. For this purpose, we first introduce the full BCS functional and the translation invariant BCS functional. Our main result states that theminimizers of the full BCS functional coincide with the minimizers of the translation invariant BCS functional for temperatures in the aforementioned interval. In the case of vanishing angular momentum our results translate to the three dimensional case. Finally, we will explain the strategy and main ideas of the proof. This is joint work with Andreas Deuchert, Christian Hainzl and Michael Loss.

Series: Math Physics Seminar

We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with ﬁnite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers.(Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)

Series: Math Physics Seminar

The discrete truncated moment problem considers the question whether
given a discrete subsets $K \subset \mathbb{R}$ and a sequence of real
numbers one can find a measure supported on $K$ whose (power) moments
are exactly these numbers. The truncated
moment is a challenging problem. We derive a minimal set of necessary
and sufficient conditions. This simple problem is surprisingly hard and
not treatable with known techniques. Applications to the truncated
moment problem for point processes, the so-called
relizability or representability problem are given. The relevance of
this problem for statistical mechanics in particular the theory of
classic liquids, is explained. This is a joint work with M. Infusino, J.
Lebowitz and E. Speer.

Series: Math Physics Seminar

In this talk we will consider a few different mathematical
models of gas-like systems of particles, which interact through
binary collisions that conserve momentum and mass.
The aim of the talk will be to present how one can employ ideas from
dynamical systems theory to derive macroscopic properties of such models.

Series: Math Physics Seminar

We consider the motion of a particle on the two-dimensional hexagonal lattice
whose sites are occupied by flipping rotators, which scatter the particle
according to a deterministic rule. We find that the particle's trajectory
is a self-avoiding walk between returns to its initial position. We
show that this behavior is a consequence of the deterministic scattering rule
and the particular class of initial scatterer configurations we consider. Since
self-avoiding walks are one of the main tools used to model the growth of
crystals and polymers, the particle's motion in this class of systems is potentially
important for the study of these processes.

Series: Math Physics Seminar

Two-point symmetrizations are simple rearrangementsthat have been used to prove isoperimetric inequalitieson the sphere. For each unit vector u, there is atwo-point symmetrization that pushes mass towardsu across the normal hyperplane.How can full rotational symmetry be recovered from partialinformation? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d);in particular, a continuous function that is symmetric under thesereflections must be radial. How many two-point symmetrizationsare needed to verify that a function which increases under thesesymmetrizations is radial? I will show that d+1 such symmetrizationssuffice, and will discuss the ergodicity of the randomwalk generated by the corresponding folding maps on the sphere.(Joint work with G. R. Chambers and Anne Dranovski).

Series: Math Physics Seminar

We build a family of
spectral triples for a discrete aperiodic tiling space, and derive the
associated Connes distances. (These are non commutative geometry
generalisations of Riemannian structures, and associated geodesic
distances.) We show how their metric properties lead to a characterisation
of high aperiodic order of the tiling. This is based on joint works with
J. Kellendonk and D. Lenz.