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 |
| Date | Speaker | Title | Abstract |
|---|---|---|---|
| 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. |
| Apr. 23rd 2025 | Colleen Delaney (Perdue) | 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. |
| Apr. 30th 2025 | Svala Sverrisdóttir (UC Berkeley) | Algebraic Varieties arising from second qunatization | 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. |
| 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 $\sigma$-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. |
| May. 21st 2025 | Jane Panangaden (Pitzer College) | Reformulations of Furstenberg's $\times 2 \times 3$ conjecture. | Furstenberg’s $\times 2 \times 3$ 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 $\times 2$ or $\times 3$ 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 $C^\ast$ algebra. |
| May. 21st 2025 | Matthew Heydeman (Harvard) | TBA | TBA |
| June. 4th 2025 | Yulia Alexandr (UCLA) | Maximum information divergence from linear and toric models | I will revisit the problem of maximizing information divergence from a new perspective using logarithmic Voronoi polytopes. We will see that for linear models, the maximum is always achieved at the boundary of the probability simplex. For toric models, I will describe an algorithm that combines the combinatorics of the chamber complex with numerical algebraic geometry. I will pay special attention to reducible models and models of maximum likelihood degree one, with many colorful examples. This talk is based on joint work with Serkan Hoşten. |
| June. 18th 2025 | Leo Shaposhnik (Freie Universität Berlin) | TBA | TBA |