Student Seminar, Fall 2017

 

Upcoming Student Seminar Talks

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NEW SCHEDULE for student seminars — we will be meeting during common hours on Thursdays or Fridays in Bailey 207, with light lunch served at 12:30pm in Bailey 204, unless otherwise noted.

 

Fall term 2017

 

Mathematics of Gerrymandering

Professor Jeff Jauregui, Union College
Friday, October 20th, 1:00pm, Bailey 207

Gerrymandering is the manipulation of political boundary lines for the benefit (or to the detriment) of a particular group, a practice nearly as old as the United States. In a case that is currently before the U.S. Supreme Court, the legality of partisan gerrymandering is being challenged. One of the key elements for the plaintiffs is the “efficiency gap,” a purported mathematical measure of gerrymandering that was developed by Stephanopoulos and McGhee in 2015. We will discuss the efficiency gap, including its value and its limitations, and other aspects of how math can be applied to help combat gerrymandering.

 

 

Past talks

“Another world” for mathematics

Professor Kimmo Rosenthal, Union College

Thursday, October 12th, 1:00pm, Bailey 207

“There is another world, but it is in this one.” One possible interpretation of this enigmatic quote by Paul Eluard is that “another world” is the invisible world of the mind. Mathematics does involve entering another world. In Math 199 students enter a “world” where they acquire the foundation on which to build their mathematical knowledge through the means of classical logic and the theory of sets and functions. But, is there “another world” for mathematics?

This raises other questions: Is there still a place for “math for math’s sake”? Does lack of broad general interest make something have less value? The quote below from Murnane suggests that what truly matters is the act of (intellectual) exploration itself. Perhaps abstract mathematics should be viewed more as a rigorous and aesthetic intellectual art form, to be valued for its beauty alone without paying heed to the modern-day shibboleths of “relevance” and “applicability”.

We will briefly discuss intuitionistic logic (as opposed to classical logic) and then some very basic ideas behind topos theory (topos is a Greek word meaning “place”); hence a topos is a “place” for doing mathematics. Finally, we will consider the idea of infinitesimally small numbers in this context. Does “not non-zero” have to mean zero? Could we have a notion of a number d with d≠0 and yet d2=0? We will end with a very short proof (without using limits) of the product rule for derivatives, showing why there is no f’g’ term (a proof that is valid, but only in “another world”).

“An explorer’s task is to postulate the existence of a land beyond the known land. Whether or not he finds that land and brings back news of it is unimportant. He may choose to lose himself in it forever and add one more to the sum of unexplored lands.” Gerald Murnane, The Plains

 

What Can Symmetries Do for You? Shapes of Spaces

Professor Megan M. Kerr, Wellesley College

Thursday, September 28th, 1:00pm, Bailey 207

Differential Geometry is the study of shapes.  In practice, it is about the interplay of multivariable calculus and linear algebra, with applications to a broad array of problems, from understanding the shape of the universe to understanding the shape of red blood cells.  The curvature of a surface measures the shape; for example, the curvature of a small round sphere is greater than that of a big round sphere — a sphere with a very large radius looks flat (zero curvature).  There are infinitely many ways to bend and stretch a surface without making holes or creases, and so, a (topological) space can take on infinitely many shapes.

We will consider a special class of spaces with a high degree of symmetry.  Happily, these symmetries arise naturally. We will explore what happens when we vary the shape of a given manifold, controlling the variations so that the symmetries — or most of them — remain.  A little Linear Algebra makes the geometry a piece of cake, and likewise a little geometry provides a fresh perspective on the algebra.

 

Partially Ordered Sets and Noetherian Rings

Cory Colbert, Gaius Charles Bolin Fellow in Mathematics at Williams College
Thursday, September 21st, 1:00pm, Bailey 207

A partially ordered set is a set with order relations. If R is a commutative ring, we can look at its set of prime ideals and form a partially ordered set with respect to subset inclusion. We are concerned with the reverse direction: if X is a partially ordered set, when can we build a commutative ring R such that its set of prime ideals equals X? We’ll start from the beginning, so this talk will be accessible to everyone.

The Isoperimetric Inequality

Professor Christina Tonnesen-Friedman, Union College
Friday, September 15th, 12:55pm, Bailey 207

Among all closed non-self-intersecting planar curves of a fixed length, the circle encloses the maximum area. This statement seems rather obvious, but the proof is not quite as simple as one might think.

We will go through two proofs; a very intuitive geometric one (with a flaw) and a calculus proof.

The talk assumes a little bit of multivariable calculus, but is otherwise self-contained.

Please view a list of seminars from previous years here.

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