Talks

Numerical methods are powerful tools for advancing our understanding of quantum gravity. In this talk, I will introduce two complementary numerical approaches. The first focuses on solving nonlinear partial differential equations that arise in Loop Quantum Gravity (LQG)-inspired effective models. This framework enables us to investigate the formation and evolution of shock waves in spherically symmetric gravitational collapse. The second approach involves the use of complex critical points, Lefschetz thimble techniques, and the Metropolis Monte Carlo algorithm to study the Lorentzian path integral in Spinfoam models and Quantum Regge Calculus. These methods offer new insights into quantum cosmology and black-to-white hole transitions.

K-theoretical Coulomb branches are expected to have cluster structure. Cautis and Williams categorified this expectation. In particular, they conjecture (and prove in type A) that the category of perverse coherent sheaves on the affine Grassmannian is a cluster monoidal categorification. We discuss recent progress on this conjecture. In particular, we construct cluster short exact sequences of certain perverse coherent sheaves. We do that by constructing a bridge, relating this (geometric) category to the (algebraic) category of finite dimensional modules over the quantum affine group. This is done by relating both categories to the notion of Feigin--Loktev fusion product.

I will discuss the phenomenology of superconductors hosting both order parameter vortices and fractionally charged anyon excitations. I will demonstrate that in such systems superconductivity and topological order are intertwined under applied magnetic fields, leading to surprising observable consequences departing from traditional superconductivity from electronic pairing. In particular,  I will show that vortices nucleated by perpendicular magnetic fields must trap anyons in their cores. However, because only some vortices can trap an integer number of anyons, this places a constraint on the vortex phase winding. In general, rather than the expected hc/2e quantization of superconducting vortices, we find instead the enhanced flux quantum of hc/e, which I will argue should affect a wide range of observables. I will further develop a general Landau-Ginzburg theory describing vortex fluctuations and discuss the phase diagram as perpendicular magnetic field is increased, showing that condensation of the intertwined vortices leads to exotic insulating phases hosting neutral anyons and a nonvanishing thermal Hall effect.

I will present a unified framework for understanding the statistics and anomalies of excitations—ranging from particles to higher-dimensional objects—in quantum lattice systems. We introduce a general method to compute the quantized statistics of Abelian excitations in arbitrary dimensions via Berry phases of locality-preserving symmetry operations, uncovering novel statistics for membrane excitations. These statistics correspond to quantum anomalies of generalized global symmetries and imply obstructions to gauging, enforcing long-range entanglement. In particular, we show that anomalous higher-form symmetries enforce intrinsic long-range entanglement, meaning that fidelity with any SRE states must exhibit exponential decay, unlike ordinary (0-form) symmetry anomalies. As an application, we identify a new example of (3+1)D mixed-state topological order with fermionic loop excitations, characterized by a breakdown of remote detectability linked to higher-form symmetry anomalies.

In this talk, we will look at a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime.

Based on an upcoming joint work with Sourya Roy, University of Iowa.

Expanders are well-connected graphs that have been extensively studied and have numerous applications in computer science, including error-correcting codes. High-dimensional expanders (HDXs) generalize expanders to hypergraphs and have the powerful local-to-global property. Roughly speaking, this property states that the expansion of an HDX can be certified by the expansion of certain local structures. This property has made HDXs crucial in the recent breakthrough on locally testable codes (LTCs) [Dinur et al.'22]. These LTCs simultaneously achieve constant rate, constant relative distance, and constant query complexity. However, despite these desirable properties, these LTCs have yet to find applications in proof systems, as they lack the crucial multiplication property present in widely used polynomial codes. A major open question is: Do there exist LTCs with the multiplication property that achieve the same rate, distance, and query complexity as those constructed by Dinur et al.?

In this talk, I will provide intuition behind the connection between HDXs and LTCs, explain why the LTCs by Dinur et al. lack the multiplication property, and discuss my recent and ongoing work on constructing LTCs with the multiplication property. This talk is based on joint work with Irit Dinur, Huy Tuan Pham, and Rachel Zhang.