# Mathematics

## DINOSAUR WARS: Extinction by Asteroid or Volcanism? Are we the Dinosaurs of the 6th Mass Extinction?

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.

## Geometricity and Galois actions on fundamental groups

Which local systems on a Riemann surface X arise from geometry, i.e. as (subquotients of) monodromy representations on the cohomology of a family of varieties over X? For example, what are the possible level structures on Abelian schemes over X? We describe several new results on this topic which arise from an analysis of the outer Galois action on etale fundamental groups of varieties over finitely generated fields.

## The stable cohomology of the moduli space of curves with level structures

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.

## The Grothendieck ring of varieties, and stabilization in the algebro-geometric setting - 2 of 2

A central theme of this workshop is the fact that arithmetic and topological structures become best behaved “in the limit”. The Grothendieck ring of varieties (or stacks) gives an algebro-geometric means of discovering, proving, or suggesting such phenomena.

In the first lecture of this minicourse, Ravi Vakil will introduce the ring, and describe how it can be used to prove or suggest such stabilization in several settings.

In the second lecture of the minicourse, Aaron Landesman will use these ideas to describe a stability of the space of low degree covers (up to degree 5) of the projective line (joint work with Vakil and Wood). The results are cognate to Bhargava’s number field counts, the philosophy of Ellenberg-Venkatesh-Westerland, and Anand Patel’s fever dream.

This is the second lecture in a two part series: part 1.

## Conjectures, heuristics, and theorems in arithmetic statistics - 2 of 2

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.

## Stable cohomology of complements of discriminants

The discriminant of a space of functions is the closed subset consisting of the functions which are singular in some sense. For instance, for complex polynomials in one variable the discriminant is the locus of polynomials with multiple roots. In this special case, it is known by work of Arnol'd that the cohomology of the complement of the discriminant stabilizes when the degree of the polynomials grows, in the sense that the k-th cohomology group of the space of polynomials without multiple roots is independent of the degree of the polynomials considered. A more general set-up is to consider the space of non-singular sections of a very ample line bundle on a fixed non-singular variety. In this case, Vakil and Wood proved a stabilization behaviour for the class of complements of discriminants in the Grothendieck group of varieties. In this talk, I will discuss a topological approach for obtaining the cohomological counterpart of Vakil and Wood's result and describe stable cohomology explicitly for the space of complex homogeneous polynomials in a fixed number of variables and for spaces of smooth divisors on an algebraic curve.

## Stable cohomology of complements of discriminants

The discriminant of a space of functions is the closed subset consisting of the functions which are singular in some sense. For instance, for complex polynomials in one variable the discriminant is the locus of polynomials with multiple roots. In this special case, it is known by work of Arnol'd that the cohomology of the complement of the discriminant stabilizes when the degree of the polynomials grows, in the sense that the k-th cohomology group of the space of polynomials without multiple roots is independent of the degree of the polynomials considered. A more general set-up is to consider the space of non-singular sections of a very ample line bundle on a fixed non-singular variety. In this case, Vakil and Wood proved a stabilization behaviour for the class of complements of discriminants in the Grothendieck group of varieties. In this talk, I will discuss a topological approach for obtaining the cohomological counterpart of Vakil and Wood's result and describe stable cohomology explicitly for the space of complex homogeneous polynomials in a fixed number of variables and for spaces of smooth divisors on an algebraic curve.

## The circle method and the cohomology of moduli spaces of rational curves

The cohomology of the space of degree d holomorphic maps from the complex projective line to a sufficiently nice algebraic variety is expected to stabilize as d goes to infinity. The limit is expected to be the cohomology of the double loop space, i.e. the space of degree d continuous maps from the sphere to that variety. This was shown for projective space by Segal, and there has been further subsequent work. In joint work with Tim Browning, we give a new approach to the problem for smooth affine hypersurfaces of low degree (which should also work for projective hypersurfaces, complete intersections, and/or higher genus curves), based on methods from analytic number theory. We take an argument of Birch that solves the number-theoretic analogue of this problem and translate it, step by step, into the language of ell-adic sheaf theory using the sheaf-function dictionary. This produces a spectral sequence that computes the cohomology, whose degeneration would imply that the rational compactly-supported cohomology matches that of the double loop space.

## $E_2$ algebras and homology - 2 of 2

Block sum of matrices define a group homomorphism $GL_n(R) \times GL_m(R) \to GL_{n+m}(R)$, which can be used to make the direct sum of $H_s(BGL_t(R);k)$ over all $s, t$ into a bigraded-commutative ring. A similar product may be defined on homology of mapping class groups of surfaces with one boundary component, as well as in many other examples of interest. These products have manifestations on various levels, for example there is a product on the level of spaces making the disjoint union of $BGL_n(R)$ into a homotopy commutative topological monoid. I will discuss how it, and other concrete examples, may be built by iterated cell attachments in the category of topological monoids, or better yet $E_2$ algebras, and what may be learned by this viewpoint. This is all joint work with Alexander Kupers and Oscar Randal-Williams.

This is the second lecture in a two part series: part 1

## Coincidences between homological densities, predicted by arithmetic - 2 of 2

In this talk I'll describe some remarkable coincidences in topology that were found only by applying Weil's (number field)/(f unction field) analogy to some classical density theorems in analytic number theory, and then computing directly. Unlike the finite field case, here we have only analogy; the mechanism behind the coincidences remains a mystery. As a teaser: it seems that under this analogy the (inverse of the) Riemann zeta function at $(n+1)$ corresponds to the 2-fold loop space of $P^n$. This is joint work with Jesse Wolfson and Melanie Wood.

This is the second lecture in a two part series: part 1