Mathematics

A random nxm matrix gives a random linear transformation from $\mathbb{Z}^m$ to $\mathbb{Z}^n$ (between vectors with integral coordinates). Asking for the probability that such a map is injective is a question of the non-vanishing of determinants. In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question. We show that when $m=n+u$, for $u$ at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function. This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields. This talk is on joint work with Hoi Nguyen.

When subjects are exposed to full field flicker in certain frequencies, they perceive a variety of complex geometric patterns that are often called flicker hallucinations. On the other had, when looking at high contrast geometric patterns like op art, shimmering and flickering is observed. In some people, flicker or such op art can induce seizures. In this talk, I describe a simple network model of excitatory and inhibitory neurons that comprise the visual area of the brain. I show that these phenomena are reproduced and then give an explanation based on symmetry breaking bifurcations and Floquet theory. Symmetric bifurcation theory also shows why one expects a different class of patterns at high frequencies from those at low frequencies.

On the other hand, the visual system is also very sensitive to specific spatial frequencies and this sensitivity can be pathological in the case of so-called pattern-senstive epilepsy. It has been shown that certain types of "op art"can cause visual discomfort. We show that the network that we used in flicker is also sensitive to spatially periodic inputs and suggest that a Hopf bifurcation instability is responsible for the discomfort and seizures.

Speaker Biography

Bard Ermentrout received his PhD in Theoretical Biology at the University of Chicago and was a postdoctoral fellow at the NIH from 1979-1982. He has been at the University of Pittsburgh since then. He is the author of over 200 papers and two books as well as the simulation package, XPPAUT. He is a Sloan Fellow and a SIAM Fellow and received the Mathematical Neuroscience Prize in 2015.

For the past 40 years the demise of the dinosaurs has been attributed to an asteroid impact on Yucatan, a theory that is imaginative, popular and even sexy. From the very beginning, scientists who doubted this theory were threatened into silence or their careers destroyed by the main theory proponents. Thus began the Dinosaur wars in 1980 – and still continuing. As in any war, there are two sides to the Dinosaur wars. The majority believes an asteroid hit Yucatan and instantaneously wiped out 75% of all life including the dinosaurs in a global firestorm and nuclear type winter. A small minority tested this theory and found contrary evidence that supported Deccan volcanism in India that caused rapid climate warming due to greenhouse gases (CO2), environmental stress, acid rain and ocean acidification culminating in the mass extinctions. This lecture highlights the four decades of the dinosaur wars, the increasing acceptance of volcanism’s catastrophic effects and likely cause of the mass extinction and the resulting ad hoc revisions to accommodate the impact theory. The talk ends with the ongoing 6th mass extinction initiated by rapid fuel burning that is causing the most rapid climate warming in Earth’s history and ocean acidification, which is predicted to reach the 6th mass extinction in as little as 50-75 years and maximum of 250 years. We could be the Dinosaurs of the 6th mass extinction.

Speaker Biography

Gerta Keller is Professor of Paleontology and Geology in the Geosciences Department of Princeton University since 1984. She was born on March 7, 1945 in Liechtenstein. She grew up on a small farm in Switzerland as the sixth of a dozen children with no prospect for education. At age 14 she entered apprenticeship as dressmaker, at 17 she worked for the DIOR Fashion House in Zurich. With no prospect for advancement she began her adventure travels through North Africa and the Middle East, supporting herself by waitressing. She immigrated to Australia at 21, was shot by a bank robber and nearly died at 22. After recovery, she resumed her adventure travels through Southeast Asia and arrived in San Francisco in 1968. There she found the first opportunity for education and entered City College, continued her undergraduate studies at San Francisco State College majoring in Anthropology and Geology, concentrating on climate and environmental changes and their effects on mass extinctions. She was awarded a Danforth Fellowship for her graduate studies at Stanford University in 1974 and received her PhD in 1978. She continued her work at Stanford University and the U.S. Geological Survey in Menlo Park and steadily researched climatic and environmental effects on life all the way back to the dinosaur mass extinction 66 million years ago. In 1984 she was hired as tenured faculty at Princeton University.

Prof. Gerta Keller’s major research and discoveries ranged from climate change and its effects on ocean circulation, ocean anoxic events, polar warming, Deccan volcanism, comet showers, extraterrestrial impacts, the dinosaur mass extinction, the age of the Chicxulub impact and the 6th mass extinction. Her research frequently challenged accepted scientific dogma and placed her at the center of acrimonious debates fighting for survival of truth-based evidence. All but the cause of the Chicxulub impact were soon accepted by scientists and integrated into new research. After four decades, impact proponents still fiercely defend the impact theory, deny contrary evidence and at best incorporate volcanism as ad hoc revisions, proclaiming the impact triggered volcanism that caused the mass extinction.

Gerta Keller has over 260 scientific publications in international journals and is a leading authority on catastrophes and mass extinctions, and the biotic and environmental effects of impacts and volcanism. She has co-authored and edited several books and she has been featured in many films and documentaries by very popular TV channels and Film Corporations, including BBC, The History Channel, and Hollywood.

I will prove that in a stable range, the rational cohomology of the moduli space of curve with level structures is the same as the ordinary moduli space of curves: a polynomial ring in the Miller-Morita-Mumford classes.

We will begin by surveying some conjectures and heuristics in arithmetic statistics, most relating to asymptotic questions for number fields and elliptic curves. We will then focus on one method that has been successful, especially in recent years, in studying some of these problems: a combination of explicit constructions of moduli spaces, geometry-of-numbers techniques, and analytic number theory.

This is the second lecture of two: first lecture.