Algebraic K-theory is a powerful invariant of rings (and more generally of ring spectra) which contains information about topology, geometry and number theory. In this talk, we will outline the basics of the theory and survey some applications to number theory, in particular to the Kummer–Vandiver conjecture. We aim to motivate why number theorists should care about an a priori homotopy theoretic notion. (Note: if you ask me questions about number theory I will forcefully eject you from J11).
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Fermat's Last Theorem was one of the most famous and oldest unsovled problems in mathematics until its resolution by Andrew Wiles in 1994. The final step of the proof was a highly technical proof of a special case of the modularity theorem that I could not hope to describe in an hour. Instead, I will introduce the relevant objects to explore how the modularity theorem allows you to deduce Fermat's last theorem, and discuss how they show up in much of modern number theory research.
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Bott periodicity was originally proved in the context of homotopy theory however in the realm of topological K-Theory it gains tremendous significance by making the K cohomology functor periodic. This is a very surprising fact as topological K-theory is defined geometrically through isomorphism classes of vector bundles. In the 1960s, Bott and Atiyah gave a proof of this flavour (for the real and complex cases) inspired by their work on elliptic boundary valued problems. The proof is a rich blend of analysis, topology and geometry, utilising the deep connection between vector bundles and functional analysis. In this seminar I aim to introduce vector bundles, Topological K-Theory and give a (very rough) sketch of the proof (with pictures).
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tba
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In 1945 Jacobson proved that if each element x of a ring satisfies x^n=x for some n>1, then the ring is commutative. This was the first of an ongoing series of celebrated commutativity theorems, and we will discuss some examples and a new approach introduced a few years ago. Beyond identities like Jacobson's, are there different kinds of properties that force a ring to be commutative? Ring theory often studies noncommutative areas which resemble the better understood world of commutative algebra. We will see that sometimes if you get close you end up inside.
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Research in mathematics is exploring a dark house; spending months tripping over furniture and slamming into walls until finally you find a light switch. Imagine if in your exploration, you found a second floor. (Alternative title: A Gentle introduction to Homological Algebra)
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For non-mathematicians, mathematics is a tool of abstraction - we take a sentence like "If I have 1 bean, and then add 2 more beans, what does that make?" and abstract it to make it both easier to solve and more generally applicable. In this case, we know the answer is "a few beans" and not "a very small casserole". For mathematicians however, the equivalent tool is logic! In this talk, I'm going to introduce a particularly useful branch of logic: _Universal Algebra_. I will explain some of the fundamental notions of the subject, and highlight some elementary theorems that are never the less incredibly powerful tools for understanding algebra, and discuss how we can use related notions to understand even more complex mathematics, such as algebraic geometry, analysis, and more! There is no assumption of familiarity with first-order logic in this talk, nor with anything above undergraduate algebra* , although more advanced knowledge will certainly be useful here. Additionally, no knowledge of category theory** or any other related subject knowledge is required. *I myself haven't got anything above an undergraduate understanding in damn near anything, so it would be remiss of me to assume anyone else does... **On a personal note, it is my own opinion that knowledge of category theory is reserved only for the truly mad, so having an understanding of it is a sign of a failing of character. That being said, category theorists are also welcome to attend.
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