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Allen Yuan, Columbia University

Abstract:  Spectra are among the most fundamental objects in algebraic topology and appear naturally in the study of generalized cohomology theories, higher K-groups and cobordism invariants.  My goal is to explain the modern perspective that spectra are natural homotopical analogues of abelian groups in a theory of “higher algebra,” where one has topological analogues of algebraic structures like rings, modules, and tensor products.

This perspective promotes new interactions with other areas of mathematics.  On the one hand, the existence of additional “chromatic primes” in higher algebra (interpolating between characteristic 0 and characteristic p) has shed light on mod p phenomena in geometry, number theory, and representation theory.  On the other hand, the extension of algebraic ideas to higher algebra has been fruitful for topology: among other things, I will discuss work, joint with Robert Burklund and Tomer Schlank, which proves a higher algebra analogue of Hilbert’s Nullstellensatz.  In addition to initiating the study of “chromatic algebraic geometry,” this work resolves a form of Rognes’ “chromatic redshift” conjecture in algebraic K-theory.

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