Colloquium. Academic Year 21-22
Thursdays 4:10pm on Zoom, unless otherwise noted
Colloquium Chair (2021-2022): Doug Hardin
Colloquium- Simple groups of dynamical origin SC5211
Volodymyr Nekrashevych, Texas A&M
Meeting ID: 998 6775 5871
Pass code: 527745
Historically, the first examples of simple groups (found by Galois) were alternating groups. We will discuss their infinite generalizations associated with discrete dynamical systems on Cantor sets. Properties of these groups are intimately related to classical properties of dynamical systems. Such conditions as simplicity and finite generation of the groups are equivalent to standard conditions for dynamical systems: minimality and expansivity. This class of groups is also a source of examples of simple groups with prescribed properties: amenability, torsion, intermediate growth, and others, all of which are proved by analyzing the underlying dynamical system.
Host: Alexander Olshanskiy
Black holes: The inside story of gravitational collapse
Maxime Van De Moortel, Princeton University
Meeting ID: 984 6256 8562
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What does the interior of a black hole look like? Beyond the astrophysical motivation, it turns out that answering this question is at the heart of profound conjectures in General Relativity. One of them is the celebrated Penrose’s Strong Cosmic Censorship Conjecture supporting the deterministic character of the theory of gravitation. I will present recent advances on this topic, based on modern techniques in hyperbolic PDEs and Differential Geometry, and describe outstanding problems.
Gapped Ground State Phases of Quantum Lattice Models
Amanda Young, TU Munich
Meeting ID: 920 3292 5113
Quantum spin systems are many-body physical models where particles are bound to the sites of a lattice. These are widely used throughout condensed matter physics and quantum information theory, and are of particular interest in the classification of quantum phases of matter. By pinning down the properties of new exotic phases of matter, researchers have opened the door to developing new quantum technologies. One of the fundamental quantities for this classification is whether or not the Hamiltonian has a spectral gap above its ground state energy in the thermodynamic limit. Mathematically, the Hamiltonian is a self-adjoint operator and the set of possible energies is given by its spectrum, which is bounded from below. While the importance of the spectral gap is well known, very few methods exist for establishing if a model is gapped, and the majority of known results are for one-dimensional systems. Moreover, the existence of a non-vanishing gap is generically undecidable which makes it necessary to develop new techniques for estimating spectral gaps. In this talk, I will discuss my work proving non-vanishing spectral gaps for key quantum spin models, and developing new techniques for producing lower bound estimates on the gap. Two important models with longstanding spectral gap questions that I recently contributed progress to are the AKLT model on the hexagonal lattice, and Haldane’s pseudo-potentials for the fractional quantum Hall effect. Once a gap has been proved, a natural next question is whether it is typical of a gapped phase. This can be positively answered by showing that the gap is robust in the presence of perturbations. Ensuring the gap remains open in the presence of perturbations is also of interest, e.g., for the development of robust quantum memory.
A second topic I will discuss is my research studying spectral gap stability.