The Southeast Geometry Seminar (SGS) is a semiannual series of one day event sponsored by:
The square peg problem is deceptively simple: Given a continuous simple closed curve in the plane, prove that there are four points on the curve that form a perfect square. It was first posed in 1909, and remains open despite a long history of papers on the subject. We will discuss a number of new results in this area, including joint work with Denne and McCleary where we show that there are an odd number of squares in any simple closed curve which is differentiable.
It is a classical problem whether a closed 2-dimensional smooth Riemannian manifold admits a smooth isometric embedding in the Euclidean 3-space. A classical problem raised by Weyl asks whether the unit sphere with positive Gauss curvature admits an isometric embedding in the Euclidean 3-space. This was solved affirmatively by Nirenberg and Pogorelov independently. In this talk, we discuss the isometric embedding of torus. We will provide a necessary and sufficient condition to embed the region where the Gauss curvature is positive.
I will discuss the construction of a new example with positive sectional curvature on a 7-dimensional manifold homeomorphic to the unit tangent bundle of the 4-sphere. The metric is of Kaluza Klein type on an orbifold principle bundle over the 4-sphere. It is closely related to the geometry of self dual Einstein and 3-Sasakian metrics.
In classical capillarity theory one assumes a rigid "support" surface (e.g., capillary tube) that is fixed in space, and one seeks to describe the surface of an adjacent liquid (into which one dips the tube), in accordance with governing physical laws. For the related problem of a body floating at a fluid interface, the relevant physical laws are the same, however neither the liquid surface nor the rigid support is fixed in space, and each is subject to different kinds of constraints. Thus a new level of complexity appears, and new procedures are needed to obtain useful information. The present work offers an initial step toward characterizing the configurations that can occur, in accordance with classical energy principles. Several specific problems are addressed, notably that of determining conditions under which a body whose density exceeds that of the ambient liquid will float or sink. The floating configurations correspond to local energy minima that are not global, as the energy can be made negatively unbounded by submerging the body to increasing depth. Criteria are provided in a general case; for particular situations of interest, such as a long thin needle of general section, the criteria become explicit and directly applicable, although presumably not best possible.
Complex projective structures on surfaces can be understood in two ways. An analytic tradition, having much in common with univalent function theory, parametrizes the space of complex projective structures over a Riemann surface via Schwarzian derivatives. A more synthetic description, due to Thurston, proceeds through the operation of grafting, which in its simplest form, glues portions of projective structures together along circular boundaries. We explain the descriptions, their contexts, and the relationships between the two perspectives.
We will discuss some recent progress on the Li-Yau-Hamilton differential Harnack inequalities for positive solutions of heat equation on manifolds. Our focus will be on manifolds with negative lower bounds on the Ricci curvature. The interesting manifolds can be complete or compact with or without boundary.