In Fall 2020 and Spring 2021, Masha Gordina and I orgaznied a learning seminar on Mathematical Physics including (but not limited to) Quantum Field Theory, Yang-Mills Theory, General Relativity, and so on. The purpose of the seminar is to gather together grad students, postdocs, and faculty who want to learn more about connections between mathematics and physics. Talks will be given on Zoom (10am eastern).
This is a learning seminar, and hence most of the talks will be expository. No previous mathematical physics background is assumed, and the talks will be self-contained.
Date: September 19 2020
Speaker: Benny Wegener,(University of Rome Tor-Vergata)
Title: Operator Algebras and Quantum Field Theory
Abstract: The goal of this talk is to give an intuition why Operator Algebras are suitable objects to describe QFT phenomena. Along the way I will give basic introductions to the mathematical formulation of Quantum Mechanics and Causality. The main result is the GNS theorem, on which the axioms of Algebraic QFT are built. If time allows, I want to introduce the Weyl C*-algebra. With help of this algebra the conceptual differences between Classical Physics, Quantum Mechanics and Quantum Field Theories can be explained.
Date: September 25 2020
Speaker: Marco Carfagnini (UConn)
Title: A Probabilistic Lagrangian
Abstract: Onsager and Machlup published a series of papers in 1953 about fluctuations of irreversible stochastic processes, defying what is nowadays known as the Onsager-Machlup functional (or Probabilistic Lagrangian). In this talk we first describe the physics behind this problem, and then we will formalize it in a more mathematical way. Some essential probabilistic notions such as the Wiener measure on the path space are introduced, and then we will provide connections with other fields of mathematics such as differential geometry and functional analysis.
Date: October 2 2020
Speaker: Zhongshan An, (UConn)
Title: Initial boundary value problem for vacuum Einstein equations
Abstract: In general relativity, spacetime metrics are solutions to the Einstein equations, which are wave equations moduli gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. On contrast, the initial boundary value problem has been much less understood. In this talk we will first talk about spacetimes, Einstein equations and the Cauchy problem. Then we will discuss obstacles to establishing a well-defined initial boundary value problem and new results on it.
Date: October 10 2020
Speaker: Tommaso Cornelis Rosati, (ICL)
Title: The continuum Anderson Hamiltonian and population dynamics
Abstract: In this talk we will introduce a random operator that arises from physical and biological models. The construction of the operator, in higher dimensions, relies on solution theories for singular stochastic PDEs. After a concise introduction to such theories we will explain how the operator arises as the scaling limit of branching particle systems in a random environment. Finally, we study the fluctuations of such scaling limit.
Date: October 16 2020
Speaker: Ambar Sengupta, (UConn)
Title: The Foundations of Quantum Mechanics: Logic and Hilbert Spaces
Abstract: This is a look back at an old theory: von Neumann’s formulation of Quantum Mechanics through the logic and geometry of subspaces of a Hilbert space. This includes viewing observables as self-adjoint operators and general states as trace class operators on a Hilbert space. Symmetries are described by representations of groups, and composite systems are described by tensor products. We will also take a quick look at the Hilbert spaces of most interest in quantum computing.
Date: October 23 2020
Speaker: Olga Chekeres, (UConn)
Title: Path Integrals in Quantum Mechanics
Abstract: A quantum mechanical particle, in contrast to a classical one, does not follow a defined trajectory, but rather a superposition of trajectories. This property was translated into mathematical equations by Richard Feynman who developed a path integral formulation of quantum mechanics. In this talk we will briefly review the canonical setting and then derive a path integral propagator in quantum mechanics. We will see how the classical trajectory arises in the classical limit of the theory. We will also take a look at some examples and calculation techniques.
Date: October 30 2020
Speaker: Xiaodong Yan (UConn)
Title: An approximate smectic liquid crystal model: compactness and sharp lower bound
Abstract: Smectic liquid crystals are remarkable examples of geometrically frustrated multi-layer system of soft matters. The sutble interplay between the geometry of the layers and equal spacing imposes theoretical complications and understanding the layer structures is a challenging task. In this talk, we shall discuss an approximate nonlinear smectic liquid crystal model in 2 dimensions. Our main results are a compactness theorem for a sequence with bounded energy and a sharp lower bound on the energy. Our analysis indicates that for the 2d model we studied, the defect energy of asymptotically minimal configurations corresponds to the energy of a 1D ansatz, which confirms equipartition between the bending and compression terms in the energy functional is optimal. This is joint work with Michael Novack.
Date: November 13 2020
Speaker: Gerald Dunne (UConn)
Title: Path Integrals: At the Interface of Physics and Mathematics
Abstract: Functional integrals form the basis of quantum field theory, and lead to physical predictions that have been confirmed to remarkable precision. Attempts to make them better defined mathematically have also led to novel insights, conjectures and results in mathematics. I will discuss some examples, and also some open problems, coming from the physics side.
Date: November 20 2020
Speaker: Alessandro Pizza (University of Rome Tor-Vergata)
Title: Local Lie-Schwinger conjugations and gapped quantum chains.
Abstract: We consider quantum chains whose Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we prove that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant. Under the same hypothesis, we prove that the ground state energy is analytic for values of the coupling constant in a fixed interval, uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain, that can be also applied to complex Hamiltonians obtained by considering complex values of the coupling constant. We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved, in contrast to most approaches.
Date: December 4 2020
Speaker: Anna Paola Todino (University of Bochum)
Title: The Geometry of Spherical Random Eigenfunctions
Abstract: Recently, considerable interest has been drawn by the analysis of geometric functionals (Lipschitz-Killing curvatures, hereafter LKCs) for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In dimension 2, LKCs correspond to the area, half of the boundary length and the Euler-Poincaré characteristic. The asymptotic behavior of their expected values and variances have been investigated and quantitative central limit theorems have been established in the high energy limits, after exploiting Wiener chaos expansions and Stein-Malliavin techniques. These results have been then extended to local behavior; more precisely Nodal Lengths in shrinking domains and excursion area in a spherical cap were considered. These studies are strongly motivated by cosmological applications, in particular in connection to the Cosmic Microwave Background.
Date: December 11 2020
Speaker: Giorgio Cipolloni (IAST)
Title: Fluctuation in the spectrum of non-Hermitian i.i.d. random matrices
Abstract: We consider a large non-Hermitian i.i.d. matrix X with real or complex entries, and we analyse its spectrum using Girko’s formula. We will then show some applications of this approach proving: (i) edge universality (the non-Hermitian analogue of Tracy-Widom); (ii) Gaussian fluctuation of linear statistics of the eigenvalues for test function having 2+\epsilon derivatives. The proof relies on three main ingredients: (i) Rigorous SUSY; (ii) local law for product of resolvents of the Hermitisation of X at two different spectral parameters, (iii) coupling of several dependent Dyson Brownian motions.
Date: Febuary 5 2021
Speaker: Sylvie Vega-Molino (UConn)
Title: On de Sitter Space; an Investigation of the Relativity of Inertia and the Cosmological Constant
Abstract: In this talk we examine two famous papers of Willem de Sitter, “On the Relativity of Inertia” (1917) and “On the Curvature of Space” (1918) in which an examination of the consequences of Einstein’s introduction of the cosmological constant into his field equations of general relativity is carried out. In particular, de Sitter discusses the notion of the relativity of inertia (in the Machian sense) and proposes the now famous de Sitter space as a resolution for concerns regarding the behavior of the metric at infinity. In the following decades this notion gained traction in light of the measured accelerated growth of the observable universe, which is consistent with de Sitter’s model. The talk will include some background on the theory and history of General Relativity and will still be well-suited for those not particularly familiar with the topic.
Date: Febuary 19 2021
Speaker: Filip Dul, (UMass Amherst)
Title: A Bird’s-Eye View of Anomalies
Abstract: In quantum field theory, the failure of a symmetry of a classical system to be lifted to the quantized system is called an “anomaly”. I’ll introduce an intuitive notion of this concept before crossing the bridge to geometry, where the Atiyah-Singer Index Theorem supplies a rigorous and purely mathematical way to quantify anomalies as topological invariants. The purpose of the talk will be to appreciate the beauty and breadth of the Index Theorem while keeping prerequisites as simple as possible.
Date: April 12 2021
Speaker: Erik Wendt (UConn)
Title: An introduction to ergodic theory and thermodynamic formalism I
Abstract: In these talks we will look into some of the main theorems in ergodic theory and thermodynamic formalism, an offshoot of ergodic theory of contemporary interest. We cover the physical origins of ergodic theory and its role as a bridge between analysis and probability, and introduce some of its most surprising (and powerful) results. After this we dive into thermodynamic formalism, a branch of ergodic theory which has had many recent applications to various areas of mathematics including number theory, functional analysis, and geometry. We end with a discussion of the variational principle for Gurevich pressure, which gives a surprising relation between the entropy, pressure, and equilibrium measures for a topological Markov shift.
Date: April 19 2021
Speaker: Erik Wendt (UConn)
Title: An introduction to ergodic theory and thermodynamic formalism II
Abstract: In these talks we will look into some of the main theorems in ergodic theory and thermodynamic formalism, an offshoot of ergodic theory of contemporary interest. We cover the physical origins of ergodic theory and its role as a bridge between analysis and probability, and introduce some of its most surprising (and powerful) results. After this we dive into thermodynamic formalism, a branch of ergodic theory which has had many recent applications to various areas of mathematics including number theory, functional analysis, and geometry. We end with a discussion of the variational principle for Gurevich pressure, which gives a surprising relation between the entropy, pressure, and equilibrium measures for a topological Markov shift.