Georgia Institute of Technology College of Sciences

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

This talk is a primer on solving certain kinds of counting problems through regular languages, finite automata and transfer matrices. Example problems: count the number of binary strings that contain "0110", count the number of binary strings that contain 0, 1, 2,... copies of "0110," a derivation of the negative binomial distribution function.

The only requirements for this talk is a basic familiarity with directed graphs, matrices and generating functions.

Teams Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1643050072413?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%22dc6c6c03-84d2-497a-95c0-d85af9cbcf28%22%7d

The Schinzel-Zassenhaus conjecture describes the narrowest collar width around the unit circle that contains a full set of conjugate algebraic integers of a given degree, at least one of which lies off the unit circle. I will explain what this conjecture precisely says and how it is proved. The method involved in this solution turns out to yield some other new results whose ideas I will describe, including to the closest interlacing of Frobenius eigenvalues for abelian varieties over finite fields, the closest separation of Salem numbers in a fixed interval, and the distribution of the short Kobayashi geodesics in the Siegel modular variety.

https://bluejeans.com/476147254/8544