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Invited Paper Session Abstracts - Geometric Ideas and Where to Find Them

Friday, August 3, 1:30 p.m. - 4:20 p.m., Plaza Ballroom D, Plaza Building

Results from geometry have long captivated the attention of mathematicians because of the surprising beauty, wide utility, and intriguing proofs behind the results. Geometric concepts are often a thread connecting areas of mathematics as well as a link between mathematics and other fields. In this session, we focus on new ways of looking at geometric theorems as well as applications to various fields of mathematics, including linear algebra, complex analysis, and dynamics.

Organizers: Ulrich Daepp, Pamela Gorkin, and Karl Voss, Bucknell University

String Art and Calculus

1:30 p.m. - 1:50 p.m.
Greg Quenell, State University of New York, Plattsburgh

Draw lines connecting (1,0) to (0,9), (2,0) to (0,8), (3,0) to (0,7), and so on. A curve starts to appear in the first quadrant. It looks like a branch of a hyperbola, but it isn't. We consider this curve and other curves that show up in a similar situation: we drive two lines of nails into a board and stretch pieces of string between the nails in one line and those in the other. By varying the spacing of the nails or placing the nails along more general paths, we generate a whole gallery of "string art" curves.

From Benford's Law to Poncelet's Theorem

2:00 p.m. - 2:20 p.m.
Karl Voss, Bucknell University

Sometimes seemingly unrelated mathematical ideas do in fact have a connection, in this case through geometry. Benford’s law describes an unexpected distribution of first digits in many sets of data that we encounter in the world. As counter-intuitive as Benford’s Law may seem upon first examination, there is an unanticipated connection to Poncelet’s Theorem, a remarkable geometric result. The journey connecting these ideas links statistics, geometry, and analysis.

Ellipses ...

2:30 p.m. - 2:50 p.m.
Dan Kalman, American University

This talk surveys interesting problems, contexts, and results I have encountered over the years. They are united by two common themes. Each is closely connected with ideas in the undergrauate math curriculum, and in some way, each focusses on properties of ellipses or ellipsoids. The topics considered will include the "ladder round a corner" problem, the minimum flight-path angle of a satellite, the structural safety of coal mines, and the optimal speed for a cow running in the rain.

Geometry of the Earth and Universe

3:00 p.m. - 3:20 p.m.
Sarah Greenwald, Appalachian State University

The quest to understand the shape of our earth and universe began thousands of years ago, when mathematicians and astronomers used mathematical models to try and explain their observations. We'll explore historical and current theories as we focus on various ways that the classification of objects, the sum of the angles in a triangle, and other geometric ideas relate. In the process we will see connections to algebra, abstract algebra, physics, philosophy and art.

The Graphic Nature of Gaus sian Periods

3:30 p.m. - 3:50 p.m.
Sephan Garcia, Pomona College

At the age of eighteen, Gauss established the constructibility of the 17-gon, a result that had eluded mathematicians for two millennia. At the heart of his argument was a keen study of certain sums of complex exponentials, known now as Gaussian periods. It turns out that these classical objects, when viewed appropriately, exhibit dazzling array of visual patterns of great complexity and remarkable subtlety.

Gaining Perspective on Homographies

4:00 p.m. - 4:20 p.m.
Annalisa Crannell, Franklin & Marshall College

This is a talk about how to look at the world---and at pictures of the world---with geometry. If we have a perspective picture of objects in a plane, we want to answer questions about where in the world the artist was, where in the world the picture was, and just how distorted the image was. To do this, we'll use a function called a homography, a map from a plane to a plane that preserves lines. The most familiar kinds of homographies to mathematicians are affine maps, which preserve parallel lines, but perspective maps (for example, photographs of parallel railroad tracks that appear to intersect on the horizon) are another form of homographies. In this talk, we'll show that in some precise sense, affine maps and perspective maps are the only kinds of homographies. Specifically, if a homography is non-affine, it must be a perspective mapping composed with an isometry. (In essence, if we put the image in the right location, and stand in the correct spot in the world, then the image will appear to line up exactly with the world).