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The prototypical example of a symmetric tensor category is Rep(G) for G a compact group. The other main source of symmetric tensor categories are Deligne's interpolation categories (S_t, GL_t, and O_t) which extend a family of group representation categories defined at positive integers to all complex numbers. Harman and Snowden have developed a theory of symmetric tensor categories coming from Oligomorphic permutation groups together with a well-behaved measure on G-sets. This includes Deligne's S_t which comes from the infinite symmetric group together with a measure where the usual permutation G-set has volume t. The simplest new example of their theory is the group of order-preserving bijections of the real line with the measure given by Euler characteristic. In joint work with Harman and Snowden (arXiv:2211.15392) we give a detailed description of this new symmetric tensor category, which has a number of novel properties. We call this category the Delannoy category because the dimensions of Hom spaces are given by Delannoy numbers. In this talk I'll outline our main results, including the classification of simple objects, the tensor product rules, and a combinatorial model for the category using Delannoy paths.

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Significant advancements in our understanding of the physical world have been driven by increasingly precise atomic spectroscopy. The level of accessible precision entered a new realm with the advent of laser cooling and trapping. Now we can extend the ultrahigh spectroscopic precision, or atomic clock technology, to more complex quantum particles like diatomic molecules. The ability to quantify molecular degrees of freedom, such as nuclear vibrations, with nearly atomic-clock precision illuminates their previously hidden properties. Moreover, it suggests possibilities to leverage this precision for probing fundamental aspects of physical interactions, including enhanced tests of Newtonian gravity at the nanometer scale.

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The single-channel Kondo impurity problem is a classic example of strongly coupled physics. In the Kondo problem, a single magnetic impurity is placed in a metal — the resulting system exhibits interesting properties such as a resistance minimum as a function of temperature. The problem was solved by Wilson’s numerical renormalization group and later by the Bethe ansatz technique. The Bethe ansatz exactly diagonalizes the Kondo hamiltonian for arbitrary impurity spin $s$ and numerically computes the impurity free energy for all temperatures. In this talk, I’ll present an alternate analytic solution for the Kondo problem at large $s$ that builds on recent results in boundary conformal field theory. This solution allows us to access analytically intermediate scales of the Kondo problem at large $s$; our results in this regime agree with the numeric results of the Bethe ansatz.

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Hamiltonian simulation of lattice gauge theories (LGTs) is a non-perturbative method of numerically solving gauge theories that offers novel avenues for studying scattering processes in gauge theories. With the advent of quantum computers, Hamiltonian simulation of LGTs has become a reality. Simulating scattering on quantum computers requires the preparation of initial scattering states in the interacting theory on the quantum hardware. Current state preparation methods involve bridging the scattering states in the free theory to the ones in the interacting theory adiabatically. Such quantum algorithms have limitations when applied to LGTs, and they tend to be computational resource intensive, rendering their implementation a challenge on the noisy intermediate-scale quantum (NISQ) era devices. In this work, we propose a wave packet state preparation algorithm for a 1+1D Z2 LGT coupled to dynamical matter. We show how this algorithm circumvents the adiabatic process by building and implementing the wave packet creation operators directly in the interacting theory using an optimized ansatz consisting of hadronic degrees of freedom in the confined Z2 LGT. Moreover, we numerically confirm the validity of this ansatz for a U(1) LGT in 1+1D. Finally, we demonstrate the viability of our algorithm for NISQ devices by comparing the classical simulation with the results obtained using the Quantinuum H1-1 quantum computer upon a simple symmetry-based noise mitigation technique.

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Recently, there has been a growing interest in the relations between homotopy theory in mathematics and theoretical physics. Homotopy theory is used to classify and study physical systems. Also, physically motivated conjectures have led to many interesting developments in homotopy theory. I have been studying this subject as a mathematician. 

My recent works have been motivated by the Segal-Stolz-Teichner program, which is one of the most deep and important subjects relating homotopy theory and physics. They propose a geometric model, in terms of supersymmetric quantum field theories, of a homotopy-theoretic object "Topological Modular Forms". Based on this, we show the absence of anomaly in heterotic string theory (joint work with Yuji Tachikawa), and found a new physical and geometric understanding of duality (with Y.Tachikawa) and periodicity (with Theo Jonhson-Freyd) in homotopy theory. These works lead us to further interesting conjectures to explore. I would like to illustrate this exciting interplay between mathematics and physics. 

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