spectra are among the most fundamental objects in algebraic topology and appear naturally in the study of generalized cohomology theories, algebraic k-groups and cobordism invariants.  i will first explain that spectra define a homotopical enlargement of algebra known as “higher algebra,” where one has topological analogues of algebraic structures like rings, modules, and tensor products.

a striking feature of higher algebra is that there are additional “chromatic characteristics” interpolating between characteristic 0 and characteristic p.  these intermediate characteristics have 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 within algebraic topology: i will discuss joint work with robert burklund and tomer schlank which proves a higher analogue of hilbert’s nullstellensatz, thus identifying the ‘’algebraically closed fields’’ of intermediate characteristic.  in addition to initiating the study of “chromatic algebraic geometry,” this work resolves a form of rognes’ chromatic redshift conjecture in algebraic k-theory.