Georgia Institute of Technology College of Sciences

Quantum Teichmüller space was first introduced by Chekhov and Fock as a version of 2+1d quantum gravity. The definition was translated over time into an algebra of curves on surfaces, which coincides with an extension of the Kauffman bracket skein algebra. In this talk, we will discuss the relation between the Teichmüller space and the Kauffman bracket, and time permitting, the quantized version of this correspondence.

Meeting URL: https://bluejeans.com/106460449/5822

In this talk, I will discuss recent advances and challenges in modelling complex dynamics of pedestrian-bridge interactions,  These challenges include a proper understanding of the biomechanics of walking on a moving structure and of the psychology of walking in crowds. I will explain the fundamental mechanism behind pedestrian-induced lateral instability of bridges due to some positive feedback from uncorrelated walkers whose foot forces do not cancel each other but create a bias. I will also present the results of our past and ongoing work that reveal the role of foot placement strategies and social force dynamics in initiating bridge instabilities. In particular, I will show that  (i)  paradoxically, depending on the human balance law (and the frequency of bridge motion), larger crowds can stabilize  bridge motions and (ii)  crowd heterogeneity can promote large vibrations of bridges.

Recording link:  https://bluejeans.com/s/h0TpdyBRatJ 

Let M be a complex-hyperbolic n-manifold, i.e. a quotient of the complex-hyperbolic n-space $\mathbb{H}^n_\mathbb{C}$ by a torsion-free discrete group of isometries, $\Gamma = \pi_1(M)$. Suppose that M is  convex-cocompact, i.e. the convex core of M is a nonempty compact subset. In this talk, we will discuss a sufficient condition on $\Gamma$ in terms of the growth-rate of its orbits in $\mathbb{H}^n_\mathbb{C}$ for which M is a Stein manifold. We will also talk about some interesting questions related to this result. This is a joint work with Misha Kapovich.

https://bluejeans.com/196544719/9518

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 quantitites 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.

URL for the talk: https://bluejeans.com/602513114/7767