Georgia Institute of Technology College of Sciences

Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.

Dynamical systems model the way that real-world systems evolve in time. While the time-asymptotic behavior of many systems can be characterized by “simple” dynamical features such as equilibria and periodic orbits, some systems evolve in a chaotic, seemingly random way. For such systems it is no longer meaningful to track one trajectory at a time individually- instead, a natural approach is to treat the initial condition as random and to observe how its probabilistic law evolves in time. This is the core idea of ergodic theory, the topic of this talk. I will not assume much beyond some basics of probability theory, e.g., random variables. 

I will talk about introduction to mathematical image processing, and cover how numerical PDE can be used in data understanding.  This talk will present some of variational/PDE-based methods for image processing, such as denoising, inpainting, colorization.  If time permits, I will introduce identification of differential equation from given noisy data.   

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric. 

Bluejeans link: https://bluejeans.com/893955256