Seminars and short courses

Neural dynamical systems with stable attractor structures such as point attractors and continuous attractors are widely hypothesized to underlie meaningful temporal behavior that requires working memory. However, perhaps counterintuitively, having good working memory is not sufficient for supporting useful learning signals that are necessary to adapt to changes in the temporal structure of the environment. We show that in addition to the well-known continuous attractors, the periodic and quasi-periodic attractors are also fundamentally capable of supporting learning arbitrarily long temporal relationships. Due to the fine tuning problem of the continuous attractors and the lack of temporal fluctuations, we believe the less explored quasi-periodic attractors are uniquely qualified for learning to produce temporally structured behavior. Our theory has wide implications for the design of artificial learning systems, and makes predictions on the observable signatures of biological neural dynamics that can support temporal dependence learning. Based on our theory, we developed a new initialization scheme for artificial recurrent neural networks which outperforms standard methods for tasks that require learning temporal dynamics. Finally, we speculate on their biological implementations and make predictions on neuronal dynamics.

The principle of the holography of information states that in a theory of quantum gravity a copy of all the information available on a Cauchy slice is also available near the boundary of the Cauchy slice. This redundancy in the theory is already present at low energy. In the context of the AdS/CFT correspondence, this principle can be translated into a statement about the dual conformal field theory. We carry out this translation and demonstrate that the principle of the holography of information holds in bilocal holography.

Complete Calabi-Yau metrics provide singularity models for limits of Kahler-Einstein metrics. We study complete Calabi-Yau metrics with Euclidean volume growth and quadratic curvature decay. It is known that under these assumptions the metric is always asymptotic to a unique cone at infinity. Previous work of Donaldson-S. gives a 2-step degeneration to the cone in the algebro-geometric sense, via a possible intermediate object (a K-semistable cone). We will show that such intermediate K-semistable cone does not occur. This is in sharp contrast to the case of local singularities. This result together with the work of Conlon-Hein also give a complete algebro-geometric classification of these metrics, which in particular confirms Yau’s compactification conjecture in this setting. I will explain the proof in this talk, and if time permits I will describe a conjectural picture in general when the curvature decay condition is removed. Based on joint work with Junsheng Zhang (UC Berkeley).

We study the Inclusion Process with vanishing diffusion coefficient, which is known to exhibit condensation and metastable dynamics for cluster locations. Here we focus on the dynamics of mass distribution rather than locations, and consider the process on the complete graph in the thermodynamic limit with fixed particle density. We describe the mass distribution for a given configuration by a measure on a suitably scaled mass space and derive a limiting measure-valued process. When the diffusion coefficient scales like the inverse of the system size, the scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative point of view for this well-established dynamics. Testing configurations with size-biased functions, our approach can be generalized to other scaling regimes. This leads to an interesting characterization of the limiting dynamics via duality and provides a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. This is joint work with Simon Gabriel and Paul Chleboun (both Warwick)

Error-correcting codes are known to define chiral 2d lattice CFTs where all the $U(1)$ symmetries are enhanced to $SU(2)$. In this paper, we extend this construction to a broader class of length-$n$ codes which define full (non-chiral) CFTs with $SU(2)^n$ symmetry, where $n=c+ \bar c$. We show that codes give a natural discrete ensemble of 2d theories in which one can compute averaged observables. The partition functions obtained from averaging over all codes weighted equally is found to be given by the sum over modular images of the vacuum character of the full extended symmetry group, and in this case the number of modular images is finite. This averaged partition function has a large gap, scaling linearly with $n$, in primaries of the full $SU(2)^n$ symmetry group. Using the sum over modular images, we conjecture the form of the genus-2 partition function. This exhibits the connected contributions to disconnected boundaries characteristic of wormhole solutions in a bulk dual.

I’ll describe two approaches to constructing a universal state sum. The first approach (arXiv:2104.02101) is more elementary and assumes semisimplicity. Special cases of this state sum include Turaev–Viro, Crane–Yetter, Douglas–Reutter, the Reshetikhin–Turaev Dehn surgery formula (thought of as a state sum), Brown–Arf for $\mathrm{Pin}_-$ 2-manifolds, and Dijkgraaf–Witten. The second approach (joint work with David Reutter) is more general and does not assume semisimplicity. If there’s time I’ll sketch a program to use the non-semisimple state sum to reproduce a cluster of non-semi-simple 3-manifold invariants due to many different authors (Lyubashenko, Kuperberg, Hennings, ... Geer, Gainutdinov, Patureau-Mirand, ... ).