Events
Department of Mathematics and Statistics
Texas Tech University
According to the Ahlfors-Gehring theorem, a simply connected domain $\Omega$ in the extended complex plain is a quasidisk if and only if there exists a sufficient condition for the injectivity of a holomorphic function in $\Omega$ in relation to the growth of its Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also give sufficient conditions for the existence of homeomorphic and quasiconformal extensions to $\overline{\mathbb{C}}$ for harmonic mappings defined on quasidisks.
The Ahlfors-Gehring theorem has been extended to finitely connected domains $\Omega$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $\Omega$ if and only if the components of $\partial\Omega$ are either points or quasicircles. We generalize this theorem to harmonic mappings.
 | Tuesday
Oct. 6
3:30 PM
MATH 017
|
| Real-Algebraic Geometry
Spec I Again
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
|
In this talk, I will discuss an equivalence between the category of etale groupoids internal to locales and a certain subcategory of inverse semigroups. This generalizes the well-known equivalence of pseudogroups and effective etale Lie groupoids, as well as the correspondence between etale groupoids and quantales, due to Pedro Resende and Lawson–Lenz.Motivated by the long-standing question of how precisely classical mechanics relates to quantum mechanics (QM), I show how equations discovered in 1927 by Erwin Madelung mathematically relate to the Schrödinger equation and Newtonian mechanics. Historically, Madelung's equations were constitutive for the de-Broglie-Bohm interpretation of QM, considered its first major `hidden variable' formulation. However, even if one takes the less stringent philosophical position of viewing QM as a statistical theory describing ensembles of particles, Madelung's equations provide a deep insight into quantum dynamics, the origin of `(first) quantization' and complex numbers in QM, as well as the classical limit. This provides the basis for viewing the Madelung equations as physically more fundamental than the Schrödinger equation.
Furthermore, if one compares the expectation values of the fundamental observables of position, momentum, energy, as well as angular momentum with the expectation values of the respective, naturally chosen functions in the framework of Kolmogovian probability theory, one is naturally lead to the question whether von Neumann's functional-analytic axiomatization of QM was a consequence of the historical development of QM - rather than empirical necessity - so that an axiomatization in terms of Kolmogorov's framework and differential-geometric evolution equations - such as Madelung's - would be more appropriate. By potentially depriving quantum probability theory of its empirical basis, the question is thus of foundational importance for mathematical probability theory in general. Finally, I will argue for a stochastic interpretation of the Schrödinger theory, as pioneered by Fenyes, Nelson, Bohm and Vigier, placing it in the same class of physical theories as the theory of diffusion.
This talk is primarily based on one of my articles, published in Foundations of Physics 47, 1317–1367 (2017).
Watch online via this Zoom link.
In local algebra, the quotient of the bounded derived category by the subcategory of perfect complexes is often referred to as the
singularity category. The quotient is trivial for a regular ring, and for a Gorenstein ring it is triangulated equivalent the
stable category of maximal Cohen--Macaulay modules or, from our point of view, to the homotopy category of totally acyclic complexes of
finitely generated projective modules. I will give an overview of a few recent papers in which we use the flat--cotorsion theory to extend these ideas to schemes.
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We considered the generalization of the Einstein model of Brownian motion when the key parameter of the time interval of free jump degenerates via a solution and its gradient. This phenomena manifests in two scenarios: a) when fluid is highly dispersing like a non-dense gas and b) when distance of the flow w.r.t source is so big that velocity and the gradient of pressure are not subject to the linear Darcy equation. In this work we jointly investigate the question of what feature will exhibit particle flows if the time interval of free jump inverse is proportional to the density of the fluids and its gradient. It was shown that in this scenario, the flow exhibits a localization feature, namely: if at some moment of time t0 in the region gradient of pressure or pressure itself is equal to zero then for some T during time interval [t0, t0 + T] there is no flow as well. This directly links to Barenblatt's finite speed of propagation property for degenerate equations. Method of proof is very different and based on the application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique. PDF available.
Refining the fundamental ∞-groupoid functor Π: Top → ∞Grpd to the context of topological ∞-groupoids Sh∞(Top), we introduce an abstract shape operation ∫: Sh∞(Top) → ∞Grpd which exists in many ∞-toposes, in particular those known as cohesive, where this shape operation has particular left and right adjoints (respectively sharp # and flat ♭), and preserves finite products. We illustrate the use of these adjoints again in the exemplary context of topological ∞-groupoids/topological stacks, in particular to define the “points-to-pieces” transformation. In the axiomatic setting of ∞-toposes, we explain how these operations specify (co)reflective subuniverses, and provide geometric interpretations of this fact. The shape and flat (co)modalities preserve group objects and their deloopings, as well as group object homotopy-quotients, which results in a formulation of differential cohomology internal to any cohesive ∞-topos. For example, given objects X, A in a cohesive ∞-topos, we explain how a morphism X →♭A represents a A-local system on X, i.e., a cocycle in (nonabelian) cohomology with A-coefficients.