학과 세미나/콜로퀴엄 - 위상수학 세미나: Extremal problems and probabilistic methods in hyperbolic geometry
by Bram Petri(Institut de Mathématiques de Jussieu-Paris Rive Ga)
Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few:
- How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V?
- How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V?
- Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space?
Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces.
In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.
2020-08-19 / 10:30 ~ 11:30
학과 세미나/콜로퀴엄 - 콜로퀴엄: Holomorphy of adjoint L-function of GL(3)
by 장칭()
Riemann zeta functions and Dirichlet L-functions are first several examples of L-functions. Automorphic L-functions are vast generalizations of these L-functions. In this talk, I will give a quick survey of these L-functions and some related topics, including our recent work on holomorphy of adjoint L-function for GL(3) joint with Joseph Hundley.
In this talk, I will show that secant varieties of a smooth projective curve embedded by a sufficiently large degree has normal Cohen-Macaulay Du Bois singularities. I will also prove that the curve is rational if and only if any secant variety has log terminal singularities, and the curve is elliptic if and only if any secant variety has log canonical singularities that are not log terminal (not even rational). This talk is based on joint work with Lawrence Ein and Wenbo Niu.
In this talk, I will give a brief survey on singularities and log pairs in birational geometry. Log terminal singularities, log canonical singularities, rational singularities, and Du Bois singularities naturally appear in many areas in algebraic geometry such as birational geometry, moduli theory, and Hodge theory.
Grothendieck posed a question of whether the natural map from the Brauer group of a scheme to its cohomological one is an isomorphism of abelian groups. It’s not true in general, but we have some positive results from Grothendieck and Gabber (and de Jong), among many others. After a brief review of Brauer groups in algebraic geometry, I’ll talk about some recent progress in the setting of derived and spectral algebraic geometry, where we can provide an affirmative answer for quasi-compact and quasi-separated (derived/spectral) schemes, and my work which extends the previous results to spectral algebraic stacks.
