Abstract: We reduce the termination of flips to the termination of terminal flips and the ACC conjecture for minimal log discrepancies (mlds) of enc pairs. As a consequence, the ACC conjecture for mlds of enc pairs implies the termination of flips in dimension 4. We show that, in any fixed dimension, the termination of flips follows from the lower-semicontinuity for mlds of terminal pairs, and the ACC for mlds of terminal and enc pairs. Moreover, in dimension 3, we give a rough classification of enc singularities, and prove the ACC for mlds of enc pairs. These two results provide a second proof of the termination of flips in dimension 3 which does not rely on any difficulty function. This is a joint work with Jihao Liu.

Abstract: A Calabi-Yau fibration is a fibration of projective varieties X->Z such that the canonical bundle K_X is numerically trivial over Z. This class of varieties plays a significant role in algebraic geometry, appearing naturally in contexts such as good minimal models and elliptic Calabi-Yau varieties. In this talk, I will present some results on the boundedness of Calabi-Yau fibrations under certain natural conditions, based on joint work with Minzhe Zhu and Xiaowei Jiang.

Abstract: This talk introduces the birational geometry of the automorphisms of projective fibered varieties.

Abstract: Let X be a complex smooth projective variety. Roughly speaking, the action of automorphisms group of X on its Chow groups are controlled by the action on cohomology groups predicted by Bloch-Beilinson conjecture.
    In this talk, we will introduce this kind of results for surfaces of general type with a genus two fibration. This is a joint work with Wenfei Liu.

Abstract: A folklore conjecture asserts that a Fano manifold of Picard number one admitting a non-isomorphic surjective endomorphism is the projective space. This conjecture is known up to dimension three by Amerik-Rovinsky-Van de Ven and Hwang-Mok independently. In this talk, I will report our recent progress on this conjecture especially when the tangent bundle is big. Certain specific dynamical restriction on the positivity of tangent bundles will also be discussed. This talk is partly based on the joint work with Feng shao.

Abstract: In this talk, we give an introduction to S.Pande's work on the positive characteristic analogy of Tian's alpha invariant. The main reference is arXiv:2311.00989.

Abstract: In this talk, I will report Andreasson-Berman's work on a conjectural bounds on the height of arithmetic Fano varieties under the assumption that its complexification is K-semistable. This is inspired by Fujita-Liu's algebro-geometric result that the complex projective space has maximal volume among all K-semistable Fano varieties. Reference: R. Andreasson and R. Berman, Sharp bounds on the height of K-semistable Fano varieties I, the toric case. arXiv:2205.00730.

Abstract: The abundance conjecture is one of the most important conjectures in algebraic geometry, particularly in birational geometry. In characteristic 0, it was proven in the 1990s that the conjecture holds for log canonical pairs of dimension 3. In this talk, we will discuss recent progress toward proving the abundance conjecture for threefolds in positive characteristic.

Abstract: In this talk, I will explain a joint work with Marion Jeannin, where we establish a general framework of Harder-Narasimhan theory, in which we prove the existence and uniqueness of coprimary filtration.

Abstract: The weight-cscK metric is initially defined and studied by Lahdili, Apostolov, etc. The definition of such metric could provide a universal framework for the study of different canonical metrics, e.g, extremal metrics, Kahler Ricci soliton metrics, \mu-cscK metrics. In a joint work with Yaxiong Liu, we prove that on a smooth Kahler manifold X, the G-coercivity of weighted Mabuchi functional implies the existence of weighted-cscK metrics. In particular, there exists a weighted-cscK metric if X is a projective manifold that is weighted K-stable for models.

Abstract: In this talk, I will report the work of Andreasson-Berman on canonical heights of arithmetric log pair with relatively ample (or anti-ample) log canonical line bundle. Canonical heights is defined as the (Arekelov-theoretic) height metrized by the Kahler-Einstein metic on the complexification of infinite place. The main result of Andreasson-Berman's work is the canonical height can be expressed as a limit of periods on the N-fold products, as N tends to infinity. As a corollary, the canonical heights of certain arithmetic log surfaces can be computed explicitly in terms of the Hurwitz zeta function. <\br> Reference: R. Andreasson and R. Berman, Canonical heights, periods and the Hurwitz zeta function. arXiv:2406.19785.

Abstract: The quintic Del Pezzo manifolds are hyperplane sections of Gr(2,5). It was proved by K. Fujita that the quintic Del Pezzo fourfolds and fivefolds are K-unstable, which are the first examples of K-unstable Fano manifolds of Picard number one. In this talk, I will compute the delta invariants of the quintic Del Pezzo fourfolds and fivefolds. This is a joint work with Yuchen Liu.

Abstract: Motivated by Shafarevich’s conjecture, Arakelov and Parshin established a significant finiteness result: for any curve C, the set of isomorphism classes of non-constant morphisms C → M_g is finite for g≥2. However, for moduli stacks parametrizing higher-dimensional varieties, the Arakelov-Parshin finiteness theorem fails due to the presence of non-rigid families. In this talk, I will review recent advances in rigidity problems for moduli spaces of polarized manifolds, focusing on two main topics: the 1-pointed rigidity property of moduli spaces of polarized manifolds, as well as the distribution of non-rigid families in moduli spaces.

Abstract: Past 50 years have witnessed the marriage of contemporary mathematics and modern physics, with String Theory serving as the central arena for this union. The need for constructing realistic models of our Universe in string theory has been relying heavily on advances of algebraic geometry. I shall in this talk motivate why this marriage is indispensable and introduce basic concepts bridging the two realms. My perspective is inevitably biased, but I hope to inspire the audience to explore the unknowns.

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