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SUMMARY:Noether-Lefschetz theory and class groups (John Brevik\, Cal State Long Beach)
DESCRIPTION:The classical Noether-Lefschetz Theorem states that a suitably general algebraic surface S of degree d ≥ 4 in complex projective 3-space P3 contains no curves besides complete intersections\, that is\, curves of the form S ∩ T where T is another surface. After discussing briefly Noether’s non-proof of this theorem and hinting at the idea behind Lefschetz’s proof\, I will sketch some of our recent progress in generalizing this theorem and its implications for global and local divisor class groups. We explore the question of what class groups are possible for local rings on surfaces in a particular analytic isomorphism class and show the ubiquitousness of unique factorization domains among such rings. Joint work with Scott Nollet (Texas Christian University).
URL:https://colleges.claremont.edu/ccms/event/antc-talk-john-brevik-cal-state-long-beach/
LOCATION:Estella 2113
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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LAST-MODIFIED:20241028T192316Z
UID:3341-1731413700-1731417000@colleges.claremont.edu
SUMMARY:Traces of Partition Eisenstein series (Ken Ono\, University of Virginia)
DESCRIPTION:Integer partitions are ubiquitous in mathematics\, arising in subjects as disparate as algebraic combinatorics\, algebraic geometry\, number theory\, representation theory\, to mathematics physics. Many of the deepest results on partitions have their origin in the work of Ramanujan. In this lecture\, we will describe a completely new and unexpected role for partitions that also arises from the mysterious “lost notebook” of Ramanujan. We discover and explain the role of new q-series called “partition Eisenstein series”. These functions magically pop up as the key device for solving a conjecture of Andrews and Berndt\, for studying symmetric functions of 2-dimensional lattice sums\, for determining the properties of Andrews-Garvan “crank statistic”\, and for representing the Taylor coefficients of virtually every interesting Jacobi automorphic form. This talk will tell the story of the recent discovery of these functions\, and will offer a brief tour of these applications.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-ken-ono-university-of-virginia/
LOCATION:Estella 2113
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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