Heisenberg's Uncertainty Principle is one of the most celebrated features of quantum mechanics, which states that one cannot simultaneously obtain the precise measurements of two conjugated physical quantities such as the pair of position and momentum or the pair of electric potential and charge density. Among the different formulations of this fundamental quantum property, the uncertainty between energy and time has a special place. This is because the time is rather a variable parametrizing the system evolution than a physical quantity waiting for determination. Physicists working in quantum information theory have understood this energy-time relation by a universal bound of how fast any quantum system with given energy can evolve from one state to another in a distinguishable (orthogonal) way. Recently, there have been many arguing that this bound is not a pure quantum phenomenon but a general dynamical property of Hilbert space. In this talk, in contrast to the usual Hilbert space formalism, we will provide a viewpoint of this evolutional speed limit based on a persistence-like distance of the derived category of sheaves : during a fixed time period what is the minimal energy needed for a system to evolve from one sheaf to a status that is distinguishable from a given subcategory? As an application, we will discuss its relation to the dynamics of classical mechanics, namely the notion of symplectic displacement. We will show that such categorical energy gives rise to a nontrivial lower bounded of Hofer displacement energy.

It is known that an automorphism group of a K3 surface with Picard number two is either infinite cyclic group or infinite dihedral group if it is infinite. In this paper, we study the generators of an automorphism group. We use the eigenvector corresponding to the spectral radius of an automorphism of infinite order to determine the generators. We also determine whether an automorphism group is the infinite cyclic group or the infinite dihedral group with several examples. 

Abouzaid-Ganatra-Iritani-Sheridan computed asymptotics of integrations of holomorphic volume forms on toric Calabi--Yau hypersurfaces over Lagrangian sections of SYZ fibrations by using tropical geometry. They gave a new proof of the gamma conjecture for ambient line bundles on Batyrev pairs of mirror Calabi--Yau hypersurfaces. Their work also gave us a new perspective from SYZ conjecture and tropical geometry to the gamma conjecture. In particular, they observed that the Riemann zeta values (the effect of gamma classes) appearing in the subleading terms of periods are contributions from the discriminants of SYZ fibrations. In the talk,we review their work and discuss its generalization to the case of toric hypersurfaces which are not necessarily Calabi--Yau hypersurfaces.

A Higgs bundle is a pair (E, ϕ) consisting of a vector bundle E and a Higgs field ϕ. Such pairs and their moduli space were introduced by Nigel Hitchin in 1987. The moduli space of Higgs bundles over a smooth curve (or a Riemann surface) have been intensively studied in various directions. In this talk, I will introduce some topics in two directions, the nonabelian Hodge correspondence and spectral data for Higgs bundles. 

주제: 연세대학교 BK 수요 기하학 세미나 (유상범교수님 5월11일)

시간: 2022년 5월 11일  01:30 오후 서울

Zoom 회의 참가: https://yonsei.zoom.us/j/92529493525

회의 ID: 925 2949 3525

G2 manifold is one of the key players in the study of duality. In this talk, we discuss the geometry of G2 manifolds as manifolds with special holonomy groups. After the comparison with the geometry of Calaibi-Yau 3 fold, the early version of duality on G2 manifolds is presented.

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on $\mathbb{S}^5$ admits at most $3$ non-free orbits. Koll\'ar as well as Fintushel and Stern formulated its algebraic version by considering its orbit space. The algebraic version is verified in many cases by the joint work with JongHae Keum. In the first part of the talk, motivated by an observation that every known example of the remaining case has a special birational behavior called a cascade, we establish algebraic Montgomery-Yang problem assuming the cascade conjecture, which claims that every rational Q-homology projective planes with quotient singularities having ample canonical divisor admits a cascade. In the second part of the talk, we study log del Pezzo surfaces by using variants of cascades. In particular, we will discuss, for log del Pezzo surfaces of Picard rank one and toric log del Pezzo surfaces, the reconstruction problem: How to reconstruct spaces from given singularity information?