Abstracts for Maths/Physics seminars


Seeking topological information in knot homologies

Yi Ni, Caltech

Knot homologies are generalizations of the polynomial invariants for knots and links. The most famous
knot homologies are  Khovanov homology and Knot Floer homology. I will discuss the topological
information one can find in knot homologies, including genus, 4-genus, fibration and triviality.

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Asymptotic Quantum Electrodynamics and Representation Theory

Giuseppe Dito, Universite de Bourgogne, France

It has been known for a long time that asymptotic fields in QED are not free
fields and that Lorentz invariance is broken on the physical space.
In this talk, I will present a construction of an asymptotic representation of the
Poincar\'e group defined on a Fock space $\mathcal F ^{\rho}$, $1/2 < \rho < 1$, with indefinite metric
built on a separable Hilbert space $E^\rho$. This Hilbert space appears in the scattering theory of
the classical Maxwell-Dirac equations and contains electromagnetic potentials decreasing as slowly as
$(1/|x|)^{3/2 - \rho + \epsilon}$ for some $\epsilon >0$. The translation generators of this asymptotic
representation are given by quartic expressions with the feature that the mass-square operator,
when restricted to the one-electron space, admits a sharp eigenvalue. However, the standard indefinite metric is an
 unbounded sesquilinear form in $E^\rho$ and, as a consequence,  Gupta-Bleuler methods
permit to construct a physical subspace $V_{GB}$ of $\mathcal F ^{\rho}$ which is only
invariant under $\mathbb{R}^4 \ltimes SU(2)$, but not under Lorentz transformations.

This is a work in progress with Erik Taflin.