Toric Structures on Symplectic Bundles of Projective Spaces

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces \bC P^r\times \bC P^s with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product \bC P^r \times \bC P^1 can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $\omega$. In this paper we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,\omega)$ with $\dim H^2(M)=2$. In particular, all such manifolds are \bC P^r bundles over \bC P^s for some $r,s$. We show that there is a unique toric structure if $r<s$, and also that if $r,s\geq 2$ then $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,\omega)$ has a unique toric structure, such as the case where $(M,\omega)$ is monotone.Comment: 26 pages, one figur

Properties of Hamiltonian Torus Actions on Closed Symplectic Manifolds

In this thesis, we will study the properties of certain Hamiltonian torus actions on closed symplectic manifolds. First, we will consider counting Hamiltonian torus actions on closed, symplectic manifolds M with 2-dimensional second cohomology. In particular, all such manifolds are bundles with fiber and base equal to projective spaces. We use cohomological techniques to show that there is a unique toric structure if the fiber has a smaller dimension than the base. Furthermore, if the fiber and base are both at least 2-dimensional projective spaces, we show that there is a finite number of toric structures on M that are compatible with some symplectic structure on M. Additionally, we show there is uniqueness in certain other cases, such as the case where M is a monotone symplectic manifold. Finally, we will be interested in the existence of symplectic, non-Hamiltonian circle actions on closed symplectic 6-manifolds. In particular, we will use J-holomorphic curve techniques to show that there are no such actions that satisfy certain fixed point conditions. This lends support to the conjecture that there are no such actions with a non-empty set of isolated fixed points

Incorporating “Just in Time” Teaching to Enhance the Lecture/Recitation Format in Calculus

The Department of Mathematics and Statistics at Georgia State University has recently switched MATH 2211/2212 to three hours of lecture and one hour of recitation instead of four hours of lecture weekly. This has left Calculus instructors with some difficulty adapting to less lecture time each week. This project was supported by STEM funds to prepare a major overhaul of the Calculus sequence at GSU to better fit the newly changed lecture/recitation format. The key objectives of the proposed work are: To incorporate “Just in Time” teaching methods to make more efficient use of the reduced lecture time. By preparing in advance a comprehensive set of review materials and pre-quizzes, we attempted to encourage students to do more preparation ahead of time making lectures more helpful and efficient. To enhance the use of technology in our instruction of Calculus by developing supplemental demonstrations using CAS. We focused on the more challenging topics, so that less class time was required to gain understanding of these difficult concepts. We made these available in advance to supplement other “Just in Time” materials. At all stages, we collected both qualitative and quantitative data on students’ learning and attitudes

Adapting to Trig: Using the ALEKS Adaptive Technology to Improve Students’ Learning and Retention in A College Trigonometry Course

The Department of Mathematics and Statistics at Georgia State University has recently introduced the MATH 1112 “College Trigonometry” course designed to prepare students in several STEM majors for Calculus courses. This project received STEM funds to introduce the paradigm of individualized adaptive learning, offered by the ALEKS online platform, into teaching of College Trigonometry at GSU. The key objectives of the proposed work are: To incorporate the ALEKS adaptive learning technology into the College Trigonometry course. The project team designed a comprehensive course program that included both trigonometry topics and review topics necessary for students’ success in the current and future courses. By its inherent design, ALEKS software automatically generated individualized learning paths for each student, reintroducing earlier topics and review topics as necessary. To facilitate self directed learning initiatives by capitalizing on the adaptive nature of ALEKS’s individualized learning paths, which places more responsibility on students to plan their work outside of class. We offered students additional guidance and tutoring support by making a lab space with a dedicated GTA available to them. To enhance the use of technology in the course by developing supplemental demonstrations using CAS. At all stages, we collected both qualitative and quantitative data on students’ learning and attitudes