Fall 2005

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Math Matters: IMA Public Lectures

The final talk in the 2004–2005 public lecture series was Computers and the Future of Mathematical Proof, given by Thomas C. Hales (University of Pittsburgh) on March 30. Dr. Hales opened his talk by sharing with his audience some amazing examples of ‘obvious’ mathematical arguments that eventually proved to be false. His demonstration of the potential for unwarranted assumptions to slip into supposedly rigorous proofs set the stage for his introduction to computer-assisted proofs. He outlined some of the key challenges involved in gaining acceptance of this powerful new approach in the mathematical community—the astronomical number of steps taken by current state-of-the-art software in deriving even simple results from first principles leads to voluminous proofs inaccessible to conventional review processes by even the most persistent referees, while many mathematicians claim that the risk of software bugs is somehow fundamentally different from the possibility of oversight and error in conventional proofs and hence disqualifies computer-assisted proofs from publication in mainstream mathematical journals.

Mathematicians left Dr. Hales’ lecture with the inescapable realization that it's not just a pencil and paper world anymore—the computer promises to alter pure mathematics as fundamentally as it has changed applied mathematics. While predicting an increasing role for machines in mathematics, Dr. Hales helped humanize math for the non-scientists in the audience by showing that mathematicians argue, make mistakes, and have prejudices, ambitions and dreams, just like everyone else.

The 2005–2006 IMA Public Lecture Series opened on December 8 with the lecture Does Math Matter to Brain Matter? by Philip Holmes, Professor of Mechanics and Applied Mathematics at Princeton University. The human brain contains about 100 billion neurons, each making about 1000 synaptic connections with other neurons. This huge dynamical system communicates with itself and its environment via electrical impulses called spikes. Dr. Holmes showed how mathematics helps us model and analyze events ranging from single neural spikes to decisions that change our lives.

Dr. Holmes first introduced his audience to some of the basics of brain activity, establishing three general time scales for decision-making: immediate (‘Shall I cross the road now?’), short-term (‘Should I stop playing the slots?’), and long-term (‘Should I do a PhD in Math, or Applied Math, or ... Neuroscience?’). He described the electrical brain phenomena associated to each class of decision and proposed mathematical models for these phenomena. In addition to explaining some fascinating neurological quirks and features—some illustrated via audience participation in experiments—he demonstrated some key techniques in the mathematical modeling of complex phenomena, showing how qualitative understanding can be extracted from initially formidable-looking systems of equations.

Philip Holmes’ own brain is clearly a very versatile and powerful organ: during his two-day visit to the University of Minnesota, in addition to his public lecture, he gave a poetry reading (sponsored by the English Department) and a lecture on cockroach locomotion (sponsored by the Department of Aerospace Engineering and Mechanics).

The other three talks in the 2005–2006 Math Matters public lecture series will be