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Quantum groups from character varieties
Abstract: The moduli spaces of local systems on marked surfaces enjoy many nice properties. In particular, it was shown by Fock and Goncharov that they form examples of cluster varieties, which means that they are Poisson varieties with a positive atlas of toric charts, and thus admit canonical quantizations. I will describe joint work with A. Shapiro in which we embed the quantized enveloping algebra U_q(sl_n) into the quantum character variety associated to a punctured disk with two marked points on its boundary. The construction is closely related to the (quantized) multiplicative Grothendieck-Springer resolution for SL_n. I will also explain how the R-matrix of U_q(sl_n) arises naturally in this topological setup as a (half) Dehn twist. Time permitting, I will describe some potential applications to the study of positive representations of the split real quantum group U_q(sl_n,R)

It is a common expectation in quantum gravity that the fundamental nature of space-time would be radically different from the smooth continuum of classical general relativity. In this talk it shall be shown that a quantum modification from loop quantum gravity crucial for singularity resolution is also responsible for deforming the underlying space-time in a manner which cannot be realized using classical geometric structures. With the minimum requirement that the quantum theory satisfies a well-defined notion of covariance explicit midisuperspace models manifesting such non-Riemannian space-times shall be shown to exhibit the physical phenomenon of non-singular signature change. Robustness and implications of this effect shall also be briefly discussed.

A central question in quantum computation is to identify which problems can be solved faster on a quantum computer. A Holy Grail of the field would be to have a theory of quantum speed-ups that delineates the physical mechanisms sustaining quantum speed-ups and helps in the design of new quantum algorithms. In this talk, we present such a toy theory for the study of a class of quantum algorithms for algebraic problems, including Shor’s celebrated factoring algorithm. Our theory is an extension of Gottesman’s stabilizer formalism based on elements of group and hypergroup theory. Using our methods, we develop classical simulation algorithms for Clifford-like circuits containing quantum Fourier transforms as well as new quantum algorithms for hidden symmetries and hyper-symmertry problems. During the talk, we will discuss the role of resources such as entanglement, interference and contextuality within our formalism and connect quantum speed-ups therein to the presence of precise algebraic structures.

Based on the following works:

[1] https://arxiv.org/abs/1210.3637
[2] https://arxiv.org/abs/1409.3208
[3] https://arxiv.org/abs/1409.4800
[4] https://arxiv.org/abs/1509.05806

Tensor networks offer an efficient representation of many-body wave-functions in an exponentially large Hilbert space by exploiting the area law of ground state quantum entanglement. I will start with a gentle introduction to the tensor network formalism. Then I will describe its application to realizing Wilson's renormalization group directly on quantum lattice models (e.g. quantum spin chains), with emphasis on the RG fixed points corresponding to conformal field theories. We will see how to define both global and local scale transformations on the lattice in such a way that, intriguingly, conformal invariance can be tested (and conformal data extracted) right at the UV cut-off scale.