Trivial notion II

Spring 2024

Schedule

Abstracts

• • Jan 242024
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  4-Manifolds and Quadratic Reciprocity

  Ollie

  In this talk, I will discuss the amazing G-Signature Theorem of Atiyah and Singer, which computes a certain homological invariant (the g-signature) of an isometry of a manifold in terms of characteristic classes integrated over its fixed-point set. For the sake of concreteness, I will only cover the case of 4-dimensional manifolds, where the theorem simplifies to a beautiful combinatorial formula. After describing this formula and extracting simple corollaries, I will illustrate its power through a very surprising application due to Hirzebruch and Zagier to prove the quadratic reciprocity law.
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• • Feb 12024
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  Schemes as functor of points

  Tauek

  Growing up, most of us are taught algebraic geometry in the following standard way: The spectrum of a ring Spec(R) is the set of prime ideals of R equipped with the Zariski topology. It has a natural sheaf of rings called the structure sheaf given by localizations of R. We call these affine schemes. A scheme X is a locally ringed space that is locally an affine scheme. We continue to call its topology the Zariski topology and its sheaf of rings the structure sheaf O_X. Given a ring R and an R-module M, we can define a sheaf of modules over the structure sheaf of Spec(R) called the associated sheaf of M, given by localizations of M. A quasicoherent sheaf on a scheme is a sheaf of O_X-modules on the Zariski topology of X that is locally given by associated sheaves of modules. We will discuss an alternative approach to algebraic geometry that does not involve locally ringed spaces. The speaker would like to express their gratitude to Elden Elmanto for rescuing them from the above indoctrination.
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• • Feb 82024
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  15, 290, and All That

  Frank

  One fundamental line of questions in the theory of quadratic forms revolves around what integer values a quadratic form can take. In this talk, I will discuss a few related surprising results in this area, centered on the 15 Theorem. This theorem states that a positive-definite integer-matrix quadratic form represents all positive integers if and only if they represent every positive integer at most 15. I will discuss the main ideas in the proof of this theorem, as well as the ideas in the similar 290 Theorem, which applies for a slightly wider set of quadratic forms. If time permits, I will also discuss a more general statement which can be made along these lines.
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• • Feb 152024
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  Convex projective geometry

  Charlie

  Convex projective geometry studies discrete subgroups of PGL(n,R) which act properly on a convex subset of RP^{n-1}. I will focus on the case when the quotient of the convex set by the subgroup is compact. Many things are known, and many things are not known about the classification of these subgroups.
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• • Feb 222024
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  11/8 conjecture

  Runze

  The 11/8-conjecture proposed by Y. Matsumoto gives an inequality that obstructs the existence of smooth structures on closed oriented spin 4-manifolds, and is closely related to the classical problem of the classification of 4-manifolds. In this talk, I will define a Pin(2)-equivariant version of the Seiberg-Witten equations and explain how M. Furuta uses it and equivariant K-theory to prove a (10/8+2)-theorem.
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• • Feb 292024
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  McKay correspondence

  Merrick

  There’s a very surprising and deep way to enumerate finite subgroups of SL(2,C). It turns out to be intricately linked to the geometry of the group as well.
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• • Mar 72024
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  The beginning of theta correspondence

  Daniel

  I will try to give a sampler of the local theta correspondence, which allows one to relate irreducible representations for certain pairs of reductive Lie groups. The story starts with the birth of modern quantum mechanics.
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