Information, Geometry & Physics Seminar

Together with Matilde Marcolli, I am co-organizing the Information, Geometry & Physics Seminar at Caltech.

Time	Winter 2025, 15:00-16:00 on Wednesdays
Location	310 Linde Hall

Schedule

Date	Speaker	Title	Abstract
June 18th 2025	Leo Shaposhnik (Freie Universität Berlin)	On Infinite Tensor Networks, Complementary Recovery and Type II Factors	Understanding the limit of infinitely large discrete systems is notoriously hard and one often resorts to methods of effective field theory that try to describe the large scale structure of the system by the introduction of an effective description that approximates "low energy/large distance" observables in terms of quantum field theories. While it is possible to perform phenomenological studies in such terms, a more microscopic and formal description of the infinitely large system is difficult to obtain and precisely how the quantum field theory emerges from it is often left implicit. In particular, it is unclear what properties of the discrete system can lead to relativistic invariance of the effective description. In the context of AdS/CFT two related ideas appeared: First, that one should think of the (from the boundary perspective) emergent radial direction as a renormalization group (RG) flow that encodes long distance physics of the boundary as degrees of freedom that live deep inside the bulk and building on this idea, that the mechanism by which the bulk is encoded in the boundary is in terms of an quantum error correcting code, where degrees of freedom that live deep inside the "entanglement wedge" (the region dual to a boundary region) are well protected against errors that take place in the complementary boundary region. A perspective on the renormalization group that grew out of these considerations is that one can think of the renormalization group as an approximate quantum error correcting code which encodes long distance physics in the Hilbert space of UV degrees of freedom and protects them against errors that arise from these UV degrees of freedom. In this talk, I will describe how one can combine these ideas using tensor networks that are believed to implement RG flows, to obtain an explicit and well defined description of the infinitely large network in terms a net of von Neumann algebras that describe subsystems of the limiting system. I will argue that from this point of view, there is little hope for the HaPPY code to give rise to a CFT as a limiting description and what qualitative difference networks implementing the MERA have that might allow for the possibility of an effective description in terms of a quantum field theory. If time permits I will comment on a relationship between the type classification of von Neumann algebras and how our work gives rise to the idea that, to prepare states that look like vacua of quantum field theories, it is necessary that they have a high degree of "magic", a quantum computational resource that makes these states hard to simulate by classical computers. This talk is based on arxiv:2504.00096
June 4th 2025	Yulia Alexandr (UCLA)	Maximum information divergence from linear and toric models (CANCELLED)	I will revisit the problem of maximizing information divergence using logarithmic Voronoi polytopes. For linear models, the maximum is at the boundary of the probability simplex. For toric models, I describe an algorithm using the chamber complex and numerical algebraic geometry, with focus on reducible models and ML degree one models. Joint work with Serkan Hoşten.
May 28th 2025	Matthew Heydeman (Harvard)	Quantum Black Hole Entropy from a Localization Integral	Feynman's recipe for quantum mechanics states that a physical quantity may be computed by integrating over "paths" that could contribute to that quantity. This approach is very successful in quantum field theory, but fails completely for General Relativity, where naively we must integrate over an uncontrollable set of spacetime manifolds. In this talk, we will describe a situation where, despite the non-linearity of General Relativity, the path integral may be done exactly in some limit. This situation is that of a very low temperature black hole, and the relevant degrees of freedom are the metric fluctuations near the horizon. The path integral which computes the black hole partition function may be rewritten in terms of a topological field theory, and ultimately the black hole entropy is computed by the Duistermaat–Heckman localization formula. This gives corrections to Hawking's famous area formula, and generalizing this result leads to a classification of black holes based on their preserved (super)-symmetries.
May 21st 2025	Jane Panangaden (Pitzer College)	Reformulations of Furstenberg's $\times 2 \times 3$ conjecture.	Furstenberg’s x2x3 conjecture is an important conjecture in ergodic theory with connections to number-theoretic dynamics. It concerns probability measures on the unit circle which have the property that integrating the x2 or x3 speedup of any function is equivalent to integrating the original function. The conjecture states that every such measure that is also ergodic is either the Lebesgue measure or finitely supported. In this talk, we present joint work with Peter Burton in which we reformulate Furstenberg’s conjecture in two alternate settings. The first is a complex analytic setting in which the measures correspond to Carathéodory functions and the second is an operator setting in which the measures correspond to tracial states on a certain group C* algebra.
May 14th 2025	Jonathan Beardsley (University of Reno)	A mysterious appearance of quantum probability in projective geometry	In joint work with S. Nakamura, I showed that the category of projective geometries and collineations can be fully and faithfully embedded into Connes and Consani's ΓSet model for modules over the field with one element. It follows that one can produce simplicial sets from projective geometries via G. Segal's "delooping" procedure, an extension of the standard classifying space construction for Abelian groups. Dynkin systems are seemingly unrelated structures generalizing σ-algebras in measure theory. Historically, as a result of Dynkin's πσ-Theorem, they have been useful in studying Markov processes. However, they have also been proposed as an alternative to σ-algebras in quantum probability and arise as sets of "precise" events in imprecise probability theory. In this talk, I will describe a recent computation of the delooping of the discrete projective geometry on n points. The resulting simplicial set is the n-fold wedge sum of a simplicial set of (pointed) Dynkin systems containing σ-algebras as a sub-simplicial set. In particular, the delooping of the one-point trivial geometry is the simplicial set of Dynkin systems on finite sets.
Apr. 30th 2025	Svala Sverrisdóttir (UC Berkeley)	Algebraic Varieties arising from second quantization	In second quantization, electronic systems are modeled by elements in the exterior algebra. The creation and annihilation operators of particles generate a Clifford algebra, known as the Fermi-Dirac algebra. We use tools from non-commutative algebra to study the Fermi-Dirac algebra and give an alternative proof of Wick's theorem, a foundational result in quantum chemistry. The central challenge in electronic structure theory is solving the electronic Schrödinger equation. In coupled cluster theory, this eigenvalue problem is approximated through a hierarchy of polynomial equations at various levels of truncation, called the coupled cluster equations. We show that the truncated eigenstates parameterize well-known algebraic varieties, including the Grassmannian, flag varieties, and spinor varieties.
Apr. 23rd 2025	Colleen Delaney (Purdue)	An "efficient" classical algorithm for some 3-manifold TQFT invariants	We will share some recent results that are instructive for approaching the classification of 3D TQFTs and topological order by computational complexity. We explain how the Turaev-Viro-Barrett-Westbury state-sum invariants that arise from Tambara-Yamagami categories are efficient to compute for 3-manifolds, provided there is a bound on their first Betti number. On the one hand, this is a pretty good algorithm given the fact that these TQFT invariants are #P-hard to compute for the smallest member of the family of Tambara-Yamagami categories (and should be hard to compute more generally). On the other hand, it isn't too surprising given that Tambara-Yamagami categories are only a slight generalization of the finite dimensional representation category of a finite abelian group, whose associated TVBW invariants are easy to compute. In any case, one can interpret our parametrized algorithm to mean that our inability to classically compute these quantum invariants in polynomial time is due to the fact that 3-manifolds can have a large first Betti number. This talk is based on joint work with Clément Maria and Eric Samperton.
Feb. 26th 2025	Terrence George (UCLA)	Electrical networks and Lagrangian Grassmannians	An electrical network is a planar graph with weights on its edges called conductances. Thomas Lam showed that electrical networks form a subset of the totally nonnegative part of the Grassmannian and described the subset as a linear slice. We show that this linear slice consists of points that are isotropic for a particular symplectic form. This is joint work with Sunita Chepuri and David Speyer.