Events for 10/05/2022 from all calendars

Noncommutative Geometry Seminar

Time: 2:00PM - 3:00PM

Location: BLOC 302

Speaker: Ryo Toyota, TAMU

Title: Controlled K-theory and K-homology

Abstract: I will introduce a new perspective of K-homology of spaces. This work is motivated by a paper of Guoliang Yu, where he showed that the K-theory of the localization algebra is isomorphic to K-homology for finite simplicial complexes. The localization algebra consists of functions from [1,\infty) to Roe algebra whose propagations go to 0. "The reason" we get K-homology is that by focusing operators whose propagation is small, we can recover some local information on spaces we lost by taking Roe algebras. Here we discuss how we can recover K-homology by focusing on operators whose propagation is smaller than a certain threshold r instead of thinking of operator valued functions. I will report what we can prove and what should be true.

Numerical Analysis Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Andrea Bonito, Texas A&M University

Title: Paper Folding and Curved Origami: Modeling, Analysis and Simulation

Abstract: The unfolding of a ladybird's wings, the trapping mechanism used by a flytrap, the design of self-deployable space shades, and the constructions of curved origami are diverse examples where strategically placed material defects are leveraged to generate large and robust deformations. With these applications in mind, we derive plate models incorporating the possibly of curved folds as the limit of thin three-dimensional hyper-elastic materials with defects. This results in a fourth order geometric partial differential equation for the plate deformations further restricted to be isometries. The latter nonconvex constraint encodes the plates inability to undergo shear nor stretch and is critical to justify large deformations.

We explore the rigidity of the folding process by taking advantage of the natural moving frames induced by piecewise isometries along the creases. We then deduce relations between the crease geodesic curvature, normal curvature, torsion, and folding angle.

Regarding the numerical approximation, we propose a locally discontinuous Galerkin method. The second order derivatives present in the energy are replaced by weakly converging discrete reconstructions. Furthermore, the isometry constraint is linearized and incorporated within a gradient flow. We show that the sequence of resulting equilibrium deformations converges to a minimizer of the exact energy (and, in particular, to an isometry) as the discretization parameters tend to infinity. This theory does not require additional smoothness on the plate deformations besides having a finite energy. The capabilities and efficiency of the proposed algorithm is documented throughout the presentation by illustrating the behavior of the model on relevant examples.

Groups and Dynamics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 506a

Speaker: Volodymyr Nekrashevych, Texas A&M University

Title: Dimensions of self-similar groups

Abstract: Every contracting self-similar group defines the associated limit space with a natural class of metrics on it. If the group is the iterated monodromy group of a locally expanding covering map of a compact metric space X, then the limit space is canonically homeomorphic to X. For example, the Julia set of a sub-hyperbolic rational function is the limit space of its iterated monodromy group (where the usual metric on the Riemann sphere belongs to the above mentioned natural class of metrics on the limit space). We will discuss how the topological dimension of the limit space can be interpreted in terms of the action of the group on the rooted tree. We will also discuss possible applications of the Ahlfors-regular dimension of the limit space (for the natural class of metrics).