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Mathematics Lecturer Discusses Quintic Surfaces

A visual representation of the Togliatti quantic surface (American Mathematical Society) A visual representation of the Togliatti quantic surface (American Mathematical Society)

Visiting mathematics professor Julie Rana from the University of Minnesota School of Mathematics Center for Educational Programs presented a lecture as a part of a “job talk” that professorial candidates are invited to host. Rana’s lecture focused on the nature of quintic surfaces, the subject in which she received her master’s degree. This is a concept included under the broad umbrella of algebraic geometry.

Part of a series of talks organized by the mathematics department, the event was open to the public. The lecture drew in students whose interest in mathematics extends outside of the classroom as well as professors seeking to further their knowledge of the addressed content.

The talk focused on the “interplay between geometric objects we get and algebraic equations we work with” said Rana in an incredibly simple introduction to the complex topic that she would further address. Rana began by establishing the significance of the degrees within a polynomial equation, providing a foundation to build upon until reaching the focus of her lecture.

After going through the graphical depictions of functions with degrees of one through four, Rana began to discuss her area of expertise, quintic surfaces. A quintic surface is the result of a polynomial function consisting of an exponential degree of five, aptly named quintic. A quintic surface occupies a forty-dimensional space and is the lowest degree of surface which can be classified as a “general type.”

To demonstrate the definition of a general type, Rana displayed various convex polygons and revealed the general types by drawing a line segment connecting two points within the shape; if the line remained in the polygon, it conformed with one of the requirements in the categorization of a “general type.”

She elaborated upon general type, including the description of invariants, a geometric shape whose total angle must be a multiple of 180 degrees regardless of the stretching and manipulation of the shape as a whole.

Rana concluded her lecture by reaffirming that the quintic surface is established by the polynomial function which meets the standards set by the general type and exists within a forty-dimensional space.

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Mathematics Lecturer Discusses Quintic Surfaces