No abstract uploaded!

In the moduli space $$ \mathcal{M} $$ _{$g$}of genus-$g$Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field$F$of degree$g$. If$g$= 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$ _{$g$}and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ . Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field$F$satisfying a strong geometry of numbers type restriction.

No abstract uploaded!

We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using L2 function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.

To understand the new physics and richness of quantum many-body system phenomena is one of the stimuli driving the condensed matter community forward. Importantly, the new insights and solutions for condensed matter theory sometimes come from the developed and developing knowledge of high energy theory, mathematical and particle physics, which is also true the other way around: Condensed matter physics has been providing crucial hints and playgrounds for the fundamental laws of high energy physics. In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states (SPTs) and topologically ordered states (TOs). We first explore the ground state structures of SPTs and TOs: the former can be symmetry twisted and the latter has robust degeneracy. The Berry phases generated by transporting and overlapping ground state sectors potentially provide universal topological invariants that fully characterize the SPTs and TOs. This framework provides us the aspects of symmetry and topology. We establish a field theory representation of SPT invariants in any dimension to uncover group cohomology classification and beyond — the former for SPTs with gapless boundary gauge anomalies, the latter for SPTs with mixed gauge-gravity anomalies. We study topological orders in 3+1 dimensions such as Dijkgraaf-Witten models, which support multi-string braiding statistics; the resulting patterns may be analyzed by the mathematical theory of knots and links. We explore the aspects of surface anomalies of bulk gapped states from the bulk-edge correspondence: The gauge anomalies of SPTs shed light on the construction of bosonic anomalies including Goldstone-Wilczek type, and also guide us to design a non-perturbative lattice model regularizing the low-energy chiral fermion/gauge theory towards the Standard Model while overcoming the Nielsen-Ninomiya fermion-doubling problem without relying on Ginsparg-Wilson fermions. We conclude by utilizing aspects of both quantum mechanical topology and spacetime topology to derive new formulas analogous to Verlinde’s via geometric-topology surgery. This provides new insights for higher dimensional topological states of matter.