Faculty Mentor:
Chris Connell 

Description:
There are a number open conjectures about the geometry of curves and surfaces in Euclidean space. Some of these conjectures also involve the relationship between extremal curves and surfaces with certain curvature properties. Among the special classes of surfaces, the algebraic ones stand out in importance both classically and in modern applications. There exist surprising constraints on negatively curved algebraic surfaces. There are also good candidates of what appear to be invariants of nonpositively curved cubic surfaces computed from the portion of the surface living in the "projective plane at infinity". In this project, I would like to explore this topic and see if we can identify more generally these algebraic invariants of negatively or nonpositively curved algebraic surfaces and establish their relationship with the algebraic degree of the surface. In a similar vein, there is an open question to classify extremal surfaces built from specific constructions that start with an arbitrary curve in 3-space. Thus second type of problem generalizes the isoperimetric theorem.

Prerequisites:
Vector calculus and a keen interest in geometry. An introduction to differential geometry (e.g. a course on curves and surfaces in Euclidean space) would be very desirable, but not absolutely necessary. Experience with a computer algebra system such as Mathematica or Maple is also a plus, but not in itself necessary.

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A mechanical linkage is a collection of points joined by line segments. Some endpoints might be anchored while others can be moved. From steam engines to the robotic arms of today, linkages have always been important in engineering. But the purpose of this project is to investigate linkages from a purely mathematical viewpoint. In particular, the geometry and topology of the space of configurations of a linkage will be studied. For example, can the space of configurations of a particular linkage be naturally decomposed into convex polyhedra? Among the tools useful for studying such questions are programs like Geometer's Sketchpad.

Prerequisites:
A penchant for visualization. Some familiarity with topology would be useful though not absolutely necessary.

Faculty Mentor:
Matvei Libine

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Homotopy Type Theory is a new development that establishes surprising connections between logic, algebra, geometry, topology, computer science, and physics. This project will involve formalization of mathematical results in homotopy type theory. The exact topics will depend on the student's interest and background.

Prerequisites:
Good background in abstract algebra and logic; excellent programming skills; knowledge of some topology is a plus.

The summer of 2014 will see the Laboratory for Mathematical Psychology further its research on face perception as well as memory and visual display search. This work involves experimentation as well as mathematical modeling of human perception and cognition. The mathematical modeling uses both analytic (e.g., theorem proving) as well as computational (e.g., calculations from parameterized models) and simulations (e.g., of real-time stochastic processes). Experimentation on face cognition focuses on perception of human identity, race, gender, and emotional expression. Some of our work includes collaboration with neuroscientists to perform neuroimaging during our human behavior experimentation. Algebra is a must and calculus, probability, and stochastic processes will be employed but can be explained to participating students.

Prerequisites:

None.

For negatively curved manifolds, the universal cover naturally gives a boundary at infinity that distinguishes asymptotically distinct orbits (up to a time shift) for the unit tangent bundle. The question is whether this boundary structure can be replicated in other dynamical settings. This project proposes to begin with hyperbolic toral automorphisms and see whether the related constructed boundaries can be used to classify measures on the torus that that are constructed from measures on the boundaries. 

Prerequisites:
A clever and inquisitive mind. It would also be useful to have some familiarity with differential geometry and measure theory. The first toral automorphism to be looked at are matrices viewed as acting on the torus, hence no prior knowledge is needed.