Georgia Institute of Technology College of Sciences

Georgia Tech is the site of the 2017 SIAM Conference on Applied Algebraic Geometry (July 31 to August 4). This biennial meeting is an activity of the Activity Group in Applied Geometry of SIAM, the Society for Industrial and Applied Mathematics. The SIAM Activity Group in Algebraic Geometry aims to bring together researchers who use algebraic geometry in industrial and applied mathematics. "Algebraic geometry" is interpreted broadly to include at least algebraic geometry, commutative algebra, noncommutative algebra, symbolic and numeric computation, algebraic and geometric combinatorics, representation theory, and algebraic topology. These methods have already seen applications in biology, coding theory, cryptography, combustion, computational geometry, computer graphics, quantum computing, control theory, geometric design, complexity theory, machine learning, nonlinear partial differential equations, optimization, robotics, and statistics. School of Mathematics professors Greg Blekherman, Anton Leykin, and Josephine Yu lead the local organizing committee.

When perturbed with a small periodic forcing, two (or more) coupledconservative oscillators can exhibit instabilities: trajectories thatbecome unstable while accumulating ``unbounded'' energy from thesource. This is known as Arnold diffusion, and has been traditionallyapplied to celestial mechanics, for example to study the stability ofthe solar system or to explain the Kirkwood gaps in the asteroid belt.However, such phenomenon could be extremely useful in energyharvesting systems as well, whose aim is precisely to capture as muchenergy as possible from a source.In this talk we will show a first step towards the application ofArnold diffusion theory in energy harvesting systems. We will consideran energy harvesting system based on two piezoelectric oscillators.When forced to oscillate, for instance when driven by a small periodicvibration, such oscillators create an electrical current which chargesan accumulator (a capacitor or a battery). Unfortunately, suchoscillators are not conservative, as they are not perfectly elastic(they exhibit damping).We will discuss the persistence of normally hyperbolic invariantmanifolds, which play a crucial role in the diffusing mechanisms. Bymeans of the parameterization method, we will compute such manifoldsand their associated stable and unstable manifolds. We will alsodiscuss the Melnikov method to obtain sufficient conditions for theexistence of homoclinic intersections.

In this thesis, we introduce multilinear dyadic paraproducts and Haar multipliers, and discuss boundedness properties of these operators and their commutators with locally integrable functions in various settings. We also present pointwise domination of these operators by multilinear sparse operators, which we use to prove multilinear Bloom’s inequality for the commutators of multilinear Haar multipliers. Along the way, we provide several characterizations of dyadic BMO functions.

I will talk about homomorphisms between surface braid groups. Firstly, we will see that any surjective homomorphism from PB_n(S) to PB_m(S) factors through a forgetful map. Secondly, we will compute the automorphism group of PB_n(S). It turns out to be the mapping class group when n>1.