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Many Cheerful Facts — Summer 2020 Mathematics Seminar

Date: Monday, June 8 at 3:00pm
Speaker: Eric Rowland
Title: Cheerful facts about Pascal's triangle
Abstract: Pascal's triangle comes up everywhere in mathematics. Pascal studied it in the 1600s in connection with probability, but it had been described in other parts of the world at least a thousand years earlier. In the 1870s, Édouard Lucas studied Pascal's triangle from a number theory perspective. He obtained a beautiful formula for the remainder left after dividing a given number in Pascal's triangle by a prime p. Variants and generalizations of this formula have been actively investigated ever since, including a new result this year that uses some hidden rotational symmetry in Pascal’s triangle.

Date: Monday, June 29 at 3:00pm
Speaker: Johanna Franklin
Title: Mathematics Wrapped in a Mystery Inside an Enigma
Abstract: The Enigma machine was used by the German military to encipher their communications during World War II. I'll talk about the mathematics that made the Germans believe the Enigma was secure, the practical and mathematical reasons they should not have, and how the Allies were able to crack it by exploiting its weaknesses.

Date: Monday, July 6 at 3:00pm
Speaker: Stefan Waner
Title: The boundaries of mathematics
Abstract: The mathematicians Alan Turing and Kurt Gödel each precipitated a kind of existential crisis in mathematics; specifically in the theory of computation and in the foundations of mathematics respectively. Each exposed a startling limitation in the power of these disciplines that reverberated way beyond mathematics and indeed throughout the philosophical and intellectual world: the unsolvability of the so-called Halting Problem in computing theory and Gödel’s First Incompleteness Theorem, on the existence of true statements in mathematics for which there is no possible proof. What I will try to convey is roughly based on Chaitin’s argument on the equivalence of versions of these two results. It is that same argument I will present here. It is neither precise nor rigorous—the proof of the pudding is in the details I will simply gloss over—but the conceptual relationship is startling and understandable at a level assuming no knowledge of either formal mathematical systems or recursive function theory.

Date: Monday, July 20 at 3:00pm
Speaker: Steve Warner
Title: Strips, rectangles, and limits
Abstract: One of the first topics that math students often struggle with is the εδ definition of the limit of a real-valued function. In this talk, I will present an approach to such limits that is more visual in nature than the traditional approach, but still rigorous. This approach has the benefit of avoiding mysterious Greek letters and absolute value inequalities, while providing a single framework for both finite and infinite limits.

Date: Monday, August 3 at 3:00pm
Speaker: Abe Mantell
Title: Linear regression via recursion (i.e., without calculus!)
Abstract: The least-squares-best-fit linear function is initially developed in the usual way, but rather than use multivariable calculus to minimize the sum of the squares of the deviations, precalculus mathematics is used to obtain recurrence relations for the slope and y-intercept. These recurrence relations can then easily be entered into a spreadsheet to show convergence to the true solution. Moreover, the recurrence relations can be solved to yield the familiar regression formulas for the slope and y-intercept.

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