# Seminars & Events for Mathematical Physics Seminar

##### Quantum graphs and Neumann networks

Quantum graph models are extremely useful but they also have some drawbacks. One of them concerns the physical meaning of the vertex coupling. The self-adjointness requirement leaves a substantial freedom expressed through parameters appearing in the conditions matching the wave function at the graph vertices. It is a longstanding problem whether one can motivate their choice by approximating the graph Hamiltonian by operators on a family of networks, i.e. systems of tubular manifolds the transverse size of which tends to zero. It appears that the answer depends on the conditions imposed on tube boundaries.

##### The reasonable effectiveness of mathematical deformation theory in physics, especially quantum mechanics and maybe elementary particle symmetries

In 1960 Wigner marveled about ``the unreasonable effectiveness of mathematics in the natural sciences," referring mainly to physics. In that spirit we shall first explain how a posteriori relativity and quantum mechanics can be obtained from previously known theories using the mathematical theory of deformations. After a tachyonic overview of how the standard model of elementary particles arose from empirically guessed symmetries we indicate how these symmetries could (very reasonably) be obtained from those of relativity using deformations (including quantization). This poses difficult and interesting mathematical problems with potentially challenging applications to physics.

##### Schramm -- Loewner Evolution and Liouville Quantum Multifractality

We describe some recent advances in the study of the fundamental coupling of a canonical model of random paths, the Schramm--Loewner Evolution (SLE), to a canonical model of random surfaces, Liouville Quantum Gravity (LQG). The latter is expected to be the conformally invariant continuum limit of various models of random planar maps. Via the KPZ relation the multifractal spectra of planar SLE morph into natural quantum analogues in LQG. We make this explicit for extreme nesting in the Conformal Loop Ensemble (CLE) in the plane and on a random planar map, and for the SLE joint harmonic measure and winding spectrum. Based on joint work with G. Borot (MPI Bonn), J. Miller (Cambridge) and S. Sheffield (MIT).

##### Local eigenvalue statistics for random regular graphs

I will discuss results on local eigenvalue statistics for uniform random regular graphs. Under mild growth assumptions on the degree, we prove that the local semicircle law holds at the optimal scale, and that the bulk eigenvalue statistics (gap statistics and averaged energy correlation functions) are given by those of the GOE. Joint work with J. Huang, A. Knowles, and H.-T. Yau.

##### Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas

Understanding the various aspects, and even the qualitative structure of phase diagrams of interacting many-body systems in the presence of static disorder still poses a big challenge. In this context, and in view of the woefully short list of rigorous results on disordered systems with interaction, limiting or integrable model systems present a testing ground for numerical works, conjectures and ideas. The Tonks-Girardeau gas subject to a random external potential is such an example. In this talk I will give an overview of results obtained recently for this model. (Based on joint works with R. Sims and R. Seiringer).

##### Asymptotics of the eigenvalues of operators for mirror curves

Using the coherent state transform I will establish the asymptotical behaviour of the Riesz mean for functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. Furthermore, I will prove the Weyl law for the eigenvalue counting function of these operators, therefore implying that their inverses are trace class. Joint work with A. Laptev and L. A. Takhtajan.

##### A potential mechanism for a singular solution of the Euler Equation

I will describe a potential mechanism for a singular solution of the Euler equation. The mechanism involves the interaction of vortex filaments, but occurs sufficiently quickly and at a small enough scales that could have plausibly evaded experimental and computational detection. Joint work with Sahand Hormoz and Alain Pumir.

##### Localization in the disordered Holstein model

The Holstein model (in the one particle sector) describes a lattice particle interacting with independent Harmonic oscillators at each site of the lattice. We consider this model with on site disorder in the particle potential. This is proposed a simple model in which it may be possible to test some ideas regarding multi/many-body localization. Provided the oscillator frequency is not too small and the hopping is weak, we are able to prove localization for the eigenfunctions, in particle position and in oscillator Fock space. Some open problems regarding the character of high energy eigenstates will be discussed. This is joint work with Rajinder Mavi.

##### The Bogoliubov free energy functional

Einstein's classic treatment of Bose-Einstein condensation for non-interacting bosons predicts at what temperature the phase transition to BEC occurs, but how this critical temperature changes when the bosons start to interact weakly is an old problem, first posed by Feynman in 1953. In this talk, I will address this question in a variational model. We have called it the 'Bogoliubov free energy functional', as it is related to Bogoliubov's 1947 theory of weakly-interacting Bose gases. I will describe several aspects of the model: its derivation, the existence of minimizers, the phase diagram, and finally the expression for the critical temperature, which has the form predicted by Lee and Yang in 1958 and a constant that agrees with numerical simulations.

##### The lace expansion for $\phi^{4}$

The lace expansion provides a formula for the two point function which has been useful for critical percolation, self-avoiding walk and related problems in high dimensions. Recently Akira Sakai has shown that the Ising model and the one component $\phi^4$ model admit similar formulas. I will review the basic features of the lace expansion and describe recent work with Mark Holmes and Tyler Helmuth which extends the lace expansion to O(2) symmetric $\phi^4$.

##### Physical implications of the chiral anomaly - from condensed matter physics to cosmology

Starting with an analysis of chiral edge currents in 2D electron gases exhibiting the quantum Hall effect I will discuss the role of anomalous chiral edge currents and of anomaly inflow in 2D insulators with explicitly broken parity and time-reversal and in time-reversal-invariant 2D topological insulators exhibiting edge spin-currents. I will also briefly discuss the topological field theories that yield the correct response equations for the 2D bulk of such materials. After a short excursion into the theory of 3D topological insulators I will then discuss an analogue of the QHE in 4 dimensions. Among other things, this will lead us to a model of dark matter and dark energy involving an axion that couples to the Pontryagin density of a massive gauge field.

##### Sharp quasi-invariance for the 4th order nonlinear Schroedinger equation

Starting with the work of Lebowitz-Rose-Speer in 1988, there has been great interest and significant progress in establishing invariance of certain Gibbs-type measures with respect to dispersive equations in 1 and 2 dimensions. The invariant measures in question are absolutely continuous with respect to the free field, and thus concentrated on functions of low regularity. N. Tzvetkov recently inituated the study of the absolute continuity of the flow at positive times with respect to measures supported on Sobolev spaces of arbitrary regularity, of type "Z^-1 exp (- ||u||_s^2)". This delicate property is known as quasi-invariance. We will present a result establishing quasi-invariance of the 4th order cubic nonlinear Schroedinger equation, and explain why the situation is radically different in the absence dispersion.

##### Probability Distribution of the Time at Which an Ideal Detector Clicks

We consider a non-relativistic quantum particle surrounded by a detecting surface and ask how to compute, from the particle's initial wave function, the probability distribution of the time and place at which the particle gets detected. In principle, quantum mechanics makes a prediction for this distribution by solving the Schrodinger equation of the particle of interest together with the 10^23 (or more) particles of the detectors, but this is impractical to compute. Is there a simple rule for computing this distribution approximately for idealized detectors? I will argue in favor of a particular proposal of such a rule, the "absorbing boundary rule," which is based on a 1-particle Schrodinger equation with a certain "absorbing" boundary condition on the detecting surface.