Abstract: This project is motivated by the generalized Dao line associated with a cyclic quadrilateral and a diameter of its circumcircle (Dao Thanh Oai, Advanced Euclidean Geometry, message 1781, September 20, 2014). Given a cyclic quadrilateral ABCD and a diameter l of its circumcircle, we will show that the generalized Dao line coincides with the orthopole line of ABCD with respect to l. Consequently, at least ``eleven'' notable points associated with the cyclic quadrilateral lie on this common line. This is a joint project with Dr. Fisher.

Title: An adaptive framework for first order gradient methods

Abstract: Gradient methods are widely used in optimization problems. In practice, while the smoothness parameter can be estimated utilizing techniques such as backtracking, estimating the strong convexity parameter remains a challenge; moreover, even with the optimal parameter choice, convergence can be slow. In this work, we propose a framework for dynamically adapting the step size and momentum parameters in first-order gradient methods for the optimization problem, without prior knowledge of the strong convexity parameter. The main idea is to use the geometric average of the ratios of successive residual norms as an empirical estimate of the upper bound on the convergence rate, which in turn allows us to adaptively update the algorithm parameters. The resulting algorithms are simple to implement, yet efficient in practice, requiring only a few additional computations on existing information. The proposed adaptive gradient methods are shown to converge at least as fast as gradient descent for quadratic optimization problems. Numerical experiments on both quadratic and nonlinear problems validate the effectiveness of the proposed adaptive algorithms. The results show that the adaptive algorithms are comparable to their counterparts using optimal parameters, and in some cases, they capture local information and exhibit improved performance.

Title: Projective Surfaces, Hypergeometric Functions, And Differential Geometry of 2-plane distributions in 5-dimensions

Abstract: There is a rich classical connection between the differential geometry of surfaces in projective 3-space and differential equations. For each such surface, there is a linear system of PDEs of rank-4 in local coordinates on the surface that determines the data of an immersion into projective space, and which depends inherently on the differential geometry of the surface. Conversely, any such rank-4 linear system in two variables determines the data of an immersed surface in projective 3-space and the local geometry. Thought of as a flat connection on a rank-4 vector bundle over the surface, there is a compatible parabolic structure group that encodes information on the local geometry of the immersed surface. The Laplace transform is a natural way of producing a new surface from the original in such a way that the two are suitably mutually tangent to certain families of tangent lines on each. It also generates a new rank-4 linear system from the original in a straightforward way. 

In this talk, geometric aspects of both will be explored. It will be shown that there is a natural compatible rank-3 sub-bundle with connection that completely determines the local geometry. To understand the geometry of the connection, it is natural to lift the tangent bundle of the surface to a 2-plane distribution on the 5-dimensional total space of the vector bundle and study the behavior off the complement of the zero section. We will explain the precise relationship between the two. Then, we will apply these results to rank-4 linear systems that are associated to Appell hypergeometric functions - important bivariate generalizations of the classical Gauss hypergeometric function. As time permits, we will discuss how the Laplace transform impacts the discussion above. Based off arXiv:2510.25918 with B. Ashley and arXiv:2602.12644 with M. Ryan.