The use of different perspectives on a problem is a very powerful principle in statistical physics, and has been especially important in mathematical physics. I will illustrate this theme with old and recent applications. These include the interpretation of QFTs at imaginary time as statistical fields, the relation of statistical fields to…

In the Thouless-Anderson-Palmer approach to mean-field spin glasses, the free energy is presented as the infimum of a functional which TAP defined over the space of all possible magnetization vectors, subject to a convergence condition. Its self-averaging over exponentially many solutions at low temperature seems to be taken for granted, though…

Although lattice Yang-Mills theory is easy to rigorously define, the construction of a satisfactory continuum theory is a major open problem in dimension d ≥ 3. Such a theory should assign a Wilson loop expectation to each suitable collection L of loops in d-dimensional space. One classical approach is to try to represent this expectation as a…

The Adler-Bardeen non-renormalization is a basic property of anomalies with important physical implications, ranging from particle physics to condensed matter. We prove its validity in lattice models at a non-perturbative level, focusing in particular on fermion-vector boson models in 3+1 and 1+1 dimensions. The proof relies on regularity…

I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function on the critical line, especially in short intervals. The problems have interesting connections to statistical mechanics of disordered systems, both in their interpretations…

In this talk we explain recent results relating the six-vertex model and the Kardar-Parisi-Zhang (KPZ) universality class. In particular, we describe how the six-vertex model can be used to analyze stochastic interacting particle systems, such as asymmetric exclusion processes, and how infinite-volume pure…

In this talk I will give an overview of recent progress regarding mathematical questions and proofs of the spectral properties of Haldane pseudo-potentials. These are short range interactions projected onto the lowest Landau level and are tailored to model properties of fractional Hall fluids and their…

Twistronics is the study of how the angle (the twist) between layers of two-dimensional materials can change their electronic structure. When two sheets of graphene are twisted by those angles the resulting material exhibits flat bands which, as argued in the physics literature, is related to…

It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning. For the latter, we extend the recent proof by Lammers of the existence of a…

I will explore a toy model for our universe in which spontaneous symmetry breaking – acting on the level of operators (not states) - can produce the interacting physics we see about us from the simpler, single particle, quantum mechanics we study as undergraduates. Based on joint work with Modj Shokrian Zini, see arXiv:2011.05917 and arXiv:2108…

Finite volume (or area) models for topological insulators are closer to experiment than in_nite volume models. However, they are only indirectly connected to Brillouin zone and so we need to…

After reviewing recent progress in random Hermitian and non-Hermitian matrix theory, we prove that eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis (ETH), which is a strong form of Quantum Unique ergodicity (QUE) with optimal speed of convergence. This requires proving certain…

In the thin cylinder regime Haldane’s pseudo-potential corresponding to one-third filling results in a frustration-free fermionic lattice Hamiltonian which is dipole-conserving with an added electrostatic interaction. Its zero-energy eigenspace is exponentially large. Nevertheless, it admits a a rather simple, full description in terms of a…

We chose a $N\times N$ Hermitian matrix randomly picked from one of the random Gaussian matrix ensembles $(\beta =1,2,4)$ - the reference matrix. Perturbing it with a sequence of rank $t$ matrices, with $t$ taking the values $1\le t \le N$, we study the expected difference between the spectra of the perturbed and the reference matrices as a…

Quantum spin systems are many-body models which are of wide interest in modern physics and at the same time amenable to rigorous mathematical analysis. A central question about a quantum spin system is whether its Hamiltonian exhibits a spectral gap above the ground state. The existence of such a spectral gap has far-reaching consequences, e.g…

I will discuss some new results about an effective theory introduced by Lieb in 1963 to approximate the ground state energy of interacting Bosons at low density. In this regime, it agrees with the predictions of Bogolyubov. At high densities, Hartree theory provides a good approximation. In this talk, I will show that the '63 effective theory…

Born's probabilistic interpretation of Schroedinger's wave function is shown to lead to a semi-relativistic quantum mechanics of atoms, molecules, etc., coupled with electromagnetic radiation. No second quantization is invoked, yet the photon naturally shows up in this formulation.

The continuum scaling limit of the Ising model in d dimensions at the critical temperature whose magnetic field properly scales to zero with lattice spacing is (or should be) a non-Gaussian generalized random field Phi for d = 2 (and d = 3). This field is (or should be) related to arelativistic quantum field theory with one time and d-1 space…

Fyodorov, Hiary and Keating have conjectured that the maximum of the characteristic polynomial of random matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the size of local maxima of L-function along the critical axis. I will explain the origins of this conjecture and…

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered p-dimensional ground state sector. The index is fractional with the denominator given by p. In particular, this yields a new short proof of the…

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