**2022년 3월 QSMS Monthly Seminar****Date:**25 March Fri PM 2:00 ~ 5:00**Place:**129-101 (SNU)

**Speaker:**유화종 Hwajong Yoo (PM 2:00 ~ 3:00)**Title:**The dimension of the kernel of an Eisenstein ideal**Abstract:**We introduce a notion of multiplicity one for modular Jacobian varieties, which is about the dimension of the kernel of a certain maximal ideal of the Hecke algebra. When such a maximal ideal is non-Eisenstein (which will be explained), then multiplicity one holds (which means that the dimension is 2.) On the other hand, as first noticed by Calegari and Stein, multiplicity one often fails for Eisenstein maximal ideals. We propose a conjecture about the dimension of the kernel of an Eisenstein ideal. If time permits, we sketch the proof of my work with Ken Ribet.

**Speaker:**조창연 Chang-Yeon Chough (PM 4:00 ~ 5:00)**TiTle:**Twisted equivalences in spectral algebraic geometry**Abstract:**Derived equivalence has been an interesting subject in relation to Fourier-Mukai transform, Hochschild homology, and algebraic K-theory, just to name a few. On the other hand, the attempt to classify schemes by their derived categories twisted by elements of Brauer groups is very restrictive as we have a positive answer only for affines. I'll talk about how we can extend this result to a broader class of algebro-geometric objects in the setting of derived/spectral algebraic geometry at the expense of a stronger notion of twisted equivalences than that of ordinary twisted derived equivalences. I'll convince you that the new notion is not only reasonable but also indispensable from this point of view. The first half will be mainly devoted to giving brief expository accounts of some background materials needed to understand the notion of twisted derived equivalence in the setting of derived/spectral algebraic geometry. The remaining half will cover a derived/spectral analog of Rickard's theorem, which shows that derived equivalent associative rings have isomorphic centers. I'll try to avoid technicalities related to using the language of derived/spectral algebraic geometry.