Jungkai Chen (NCTS, Taiwan)

Meng Chen (Fudan, China)

Kiryong Chung (Kyungpook National Univ., Korea)

Baohua Fu (Chinese Academy of Science, China)

Yoshinori Gongyo (Tokyo, Japan)

Phung Ho Hai (VAST, Vietnam)

Yujiro Kawamata (Tokyo, Japan)

JongHae Keum (KIAS, Korea)

Conan Leung (CUHK, Hong Kong)

Hsueh-Yung Lin (NTU, Taiwan)

Yusuke Nakamura (Tokyo, Japan)

Xiaotao Sun (Tianjing Univ., China)

De-Qi Zhang (NUS, Singapore) **Sponsors: **

**2022/12/23 15:00-16:00 (Beijing Time)****Speaker: **Tasuki Kinjo (Kavli IPMU)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Cohomological Donaldson-Thomas theory for 2-Calabi--Yau categories**Abstract: **Cohomological Donaldson-Thomas (CoDT) invariants were introduced by Kontsevich-Soibelman and Brav-Bussi-Dupont-Joyce-Szendroi as categorifications of the Donaldson-Thomas invariants counting objects in 3-Calabi-Yau categories. In this talk, I will explain applications of the CoDT theory to the cohomological study of the moduli of objects in 2-Calabi-Yau categories. Among other things, I will construct a coproduct on the Borel-Moore homology of the moduli stack of objects in these categories and establish a PBW-type statement for the Kapranov-Vasserot cohomological Hall algebras. This talk is based on a joint work in progress with Ben Davison.

**2022/12/23**** 16:15-17:15 ****(Beijing Time)****Speaker: **Hyeonjun Park (Korea Institute For Advanced Study)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Counting surfaces on Calabi-Yau 4-folds**Abstract: **In this talk, we discuss counting surfaces on Calabi-Yau 4-folds. Besides the Hilbert scheme of 2-dimensional subschemes, we introduce two types of moduli spaces of stable pairs. We show that all three moduli spaces are related by GIT wall-crossing and parametrize polynomial Bridgeland stable objects in the bounded derived category. We construct reduced Oh-Thomas virtual cycles on the moduli spaces via Kiem-Li cosection localization and prove that they are deformation invariant along Hodge loci. We show that the cosections can be enhanced to (-1)-shifted closed 1-forms by generalizing the integration map of Pantev-Toen-Vaquie-Vezzosi which yields reduced (-2)-shifted symplectic derived enhancements on the moduli spaces. As an application, we show that the variational Hodge conjecture holds for any family of Calabi-Yau 4-folds supporting a non-zero reduced virtual cycle. **2022/12/09 15:00-16:00 (Beijing Time)****Speaker: **Jaehyun Kim (Ewha Womans University)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Some numbers on certain pencils of rational curves in del Pezzo surfaces**Abstract: **An open subset in a normal projective variety X is called a cylinder if it is isomorphic to A1 × Z for some affine variety Z. With an effective condition on the boundary of the cylinder, this A1-ruledness ensures that a nontrivial unipotent group action on the affine cone of the corresponding ample polarization (X, H). In particular, for del Pezzo surfaces, there are good properties to determine its ample polar cylindricity. Here we will remark on some results for del Pezzo surfaces known so far containing recent own research in progress. We consider smooth case only and work over an algebraically closed field of characteristic zero.

**2022/12/09**** 16:15-17:15 ****(Beijing Time)****Speaker: **Olivier Benoist (CNRS, École Normale Supérieure)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Smooth subvarieties of Jacobians**Abstract: **I will present new examples of algebraic cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of these examples have dimension 6, the lowest possible. More precisely, I will consider the case of minimal cohomology classes on Jacobians of very general curves. This is joint work with Olivier Debarre.

**2022/11/25 15:00-16:00 (Beijing Time)****Speaker: **Ziming Ma (Southern University of Science and Technology)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Smoothing, scattering diagram and a conjecture of Fukaya**Abstract: **In this talk, we will discuss the reconstruction problem of the mirror manifold, starting with Fukaya's conjectural constructionusing counting gradient flow trees/counting holomorphic disks. We will prove an important part of this conjecture and relate it to the famous Gross-Siebert construction, using a newly constructed dgBV algebra of polyvector field. This is a joint work with Kwokwai Chan and Naichung Conan Leung.

**2022/11/25**** 16:15-17:15 ****(Beijing Time)****Speaker: **Zheng Hua (University of Hong Kong)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Moduli space of complexes on Gorenstein Calabi-Yau curves**Abstract: **Together with Alexander Polishchuk, we prove that the (derived) moduli stack of complexes of vector bundles over a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Many interesting Poisson varieties are components of this moduli stack, including Hilbert scheme of points on Fano surfaces, semi-classical limits of Feigin-Odesskii elliptic algebras, Log canonical Poisson structures on projective spaces, etc. We are able to prove several new results in classical Poisson geometry using this modular interpretation. **2022/11/11 15:00-16:00 (Beijing Time)****Speaker: **Yao Yuan (Capital Normal University)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Rank zero Segre integrals on Hilbert schemes of points on surfaces**Abstract: **The generating function of the Segre integrals on Hilbert schemes of points on a surface X can be determined by five universal series A_0, A_1, A_2, A_3, A_4, due to the result of Ellingsrud-Göttsche-Lehn. These five series do not depend on the surface X, and depend on the element of K(X) to which the Segre integrals are associated, only through the rank. Marian-Oprea-Pandharipande have determined A_0, A_1, A_2 for all ranks. For rank 0, it is easy to see A_4=1. They also conjectured that A_3 = A_0A_1 for rank 0. We prove this conjecture by showing that when X is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Very recently Göttsche-Mellit have obtained the explicit expression for A_3 for all rank r >2 using localization, and hence by polynomiality their result also gives the expression for A_3 for all ranks. But our method is totally different from theirs and contains more geometric concept.

**2022/11/11**** 16:15-17:15 ****(Beijing Time)****Speaker: **Rong Du (East China Normal University)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Algebraic vector bundles on rational homogeneous spaces**Abstract: **I will introduce the background of algebraic vector bundles on rational homogeneous spaces and some open problems related to them in algebraic geometry. In particular, I will focus on two types of algebraic vector bundles, uniform bundles and homogeneous bundles, on special rational homogeneous spaces. This talk is from a joint work with Xinyi Fang and a joint work with Xinyi Fang and Peng Ren.

**Speaker: **Bao Viet Le Hung (Northwestern University)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Geometry of moduli spaces of local Galois representations**Abstract: **Moduli spaces of representations of Galois groups of p-adic fields with p-adic Hodge theoretic conditions traditionally play a pivotal role in establishing automorphy, and have recently gained even more prominence with the advent of the categorical p-adic Langlands program. Unfortunately, the geometry of these moduli spaces is poorly understood, and conjecturally must be sufficiently complicated to encode the modular representation theory of finite groups of Lie type. In this talk, I will describe some special but important cases where it is possible to probe the geometry of these moduli spaces, by relating them to more amenable (but still complicated) geometric objects. **2022/10/28**** 10:00-11:00 ****(Beijing Time)****Speaker: **Hiromu Tanaka (University of Tokyo)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **On quasi-F-splitting**Abstract: **F-splitting is a notion for algebraic varieties of positive characteristic, which is defined by splitting of the Frobenius endomorphism. Recently, Yobuko has introduced a generalisation of F-splitting, called quasi-F-splitting, via Witt rings. In this talk, we will discuss which algebraic varieties are (should be) quasi-F-split. This is joint work with T. Kawakami, T. Takamatsu, J. Witaszek, F. Yobuko, and S. Yoshikawa. **2022/10/14 15:00-16:00 (Beijing Time)****Speaker: **Yen-An Chen (National Center for Theoretical Science)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Foliated MLD and LCT**Abstract: **There are more and more studies on foliations in the viewpoint of the Minimal Model Program, in which singularities play a vital role. In order to measure the foliation singularities, we introduce the foliated version of minimal log discrepancies (MLD) and log canonical thresholds (LCT). In this talk, we will show the sets of MLD and of LCT satisfy the ascending chain condition when the dimension is small. **2022/10/14**** 16:15-17:00 ****(Beijing Time)****Speaker: **Dang Quoc Huy (Vietnam Institute for Advanced Study in Mathematics)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Deforming cyclic covers in towers**Abstract: **There are many interesting phenomena in the case of positive or mixed characteristics that defy the geometric intuition obtained from classical complex geometry. For instance, there exist covers of curves whose number of branch points are different but lie in the same flat family. In this talk, we briefly discuss the process of showing that a smooth equal-characteristic p deformation of a cyclic sub-covering extends to that of the whole tower. The result indicates that the p-fibers of the canonical maps between the moduli space of cyclic coverings and one of the sub-coverings are surjective at any closure. The crucial technique is a study of local covering’s degeneration using Kato-Saito-Abbes’ refined Swan conductor, which generalizes the classical perfect residue case. **2022/09/30 8:00-9:00 (Beijing Time)****Speaker: **Yuchen Liu (Northwestern University)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Wall crossing for K-moduli spaces**Abstract: **Recent developments in K-stability provide a nice moduli space, called a K-moduli space, for log Fano pairs. When the coefficient of the divisor varies, these K-moduli spaces demonstrate wall crossing phenomena. In this talk, I will discuss the general principle of K-moduli wall crossings, and show in examples that it provides a bridge connecting various moduli spaces of different origins, such as GIT, KSBA, and Hodge theory. Based on joint works with Kenny Ascher and Kristin DeVleming. **2022/09/30**** 9:15-10:15****(Beijing Time)****Speaker: **Ming Hao Quek (Brown University)**Place: **Online** **(Zoom ID: 851 1452 8712 Password: 608225)**Title: **Around the motivic monodromy conjecture for non-degenerate hypersurfaces**Abstract: **I will discuss my ongoing effort to comprehend, from a geometric viewpoint, the motivic monodromy conjecture for a "generic" complex multivariate polynomial f, namely any polynomial f that is non-degenerate with respect to its Newton polyhedron. This conjecture, due to Igusa and Denef--Loeser, states that for every pole s of the motivic zeta function associated to f, exp(2πis) is a "monodromy eigenvalue" associated to f. On the other hand, the non-degeneracy condition on f ensures that the singularity theory of f is governed, up to a certain extent, by faces of the Newton polyhedron of f. The extent to which the former is governed by the latter is one key aspect of the conjecture, and will be the main focus of my talk. **2022/07/01 GMT 7:00-8:00****Speaker: **Do Viet Cuong (University of Science, Vietnam National University, Hanoi)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **On the moduli spaces of parabolic Higgs bundles on a curve**Abstract: **Let $C$ be a projective curve. The moduli space of Higgs bundles on $C$, introduced by Hitchin, is an interesting object of study in geometry. If $C$ is defined over the complex numbers, the moduli space of Higgs bundles is diffeomorphic to the space of representations of the fundamental group of the curve. If $C$ is defined over finite fields, the adelic description of the stack of Higgs bundles on $C$ is closely related to spaces occurring in the study of the trace formula. It is a start point to Ngo's proof for the fundamental lemma for Lie algebras. **2022/07/01 GMT 8:15-9:15****Speaker:** Nguyen Tat Thang (Hanoi Institute of Mathematics)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Contact loci and Motivic nearby cycles of nondegenerate singularities**Abstract: **In this talk, we study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. We introduce an explicit description of these quantities in terms of the face functions. As a consequence, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci. This is a joint work with Le Quy Thuong.

**2022/06/17 GMT 7:00-8:00****Speaker: **Qifeng Li (Shandong University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Deformation rigidity of wonderful group compactifications**Abstract: **For a complex connected semisimple linear algebraic group G of adjoint type, De Concini and Procesi constructed its wonderful compactification, which is a smooth Fano equivariant embedding of G enjoying many interesting properties. In this talk, we will discuss on the properties of wonderful group compactifications, especially the deformation rigidity of them. This is a joint work with Baohua Fu. **2022/06/17 GMT 8:15-9:15****Speaker: **Xin Lu (East China Normal University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Sharp bound on the abelian automorphism groups of surfaces of general type**Abstract: **We prove that the order of any abelian (resp. cyclic) automorphism group of a smooth complex projective of general type is bounded from above by $12.5c_1^2+100$ (resp. $12.5c_1^2+90$) provided that its geometric genus $p_g>6$. The upper bounds can be both reached by infinitely many examples whose geometric genera can be arbitrarily large. This is a joint work with Sheng-Li Tan. **2022/05/20 GMT 7:00-8:00****Speaker: **Tong Zhang (East China Normal University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Noether-Severi inequality and equality for irregular threefolds of general type**Abstract: **In this talk, I will introduce the optimal Noether-Severi inequality for all smooth and irregular threefolds of general type. It answers in dimension three an open question of Z. Jiang. I will also present a complete description of canonical models of smooth and irregular threefolds of general type attaining the Noether-Severi equality. This is a joint work with Yong Hu. **2022/05/20 GMT 8:15-9:15****Speaker: **Hélène Esnault (Freie Universität Berlin)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: ** Recent developments on rigid local systems**Abstract: **We shall review some of the general problems which are unsolved on rigid local systems and arithmetic $\ell$-adic local systems. We'll report briefly on a proof (2018 with Michael Groechenig) of Simpson's integrality conjecture for {\it cohomologically rigid local systems}. While all rigid local systems in dimension $1$ are cohomologically rigid (1996, Nick Katz), we did not know until last week of a single example in higher dimension which is rigid but not cohomologically rigid. We'll present one series of examples (2022, joint with Johan de Jong and Michael Groechenig) **2022/05/06 GMT 7:00-8:00****Speaker: **Adeel Khan (Academia Sinica)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Microlocalization and Donaldson-Thomas theory**Abstract: **I will discuss a certain categorification of Kontsevich's virtual fundamental class, which I call derived microlocalization. Based on joint work with Tasuki Kinjo, I will explain how this formalism can be used to prove a conjecture of D. Joyce about categorified Donaldson-Thomas theory of Calabi-Yau threefolds. This has several consequences, including the existence of cohomological Hall algebras à la Kontsevich-Soibelman for Calabi-Yau threefolds. **2022/05/06 GMT 8:15-9:15****Speaker: **Hiroki Matsui (Tokushima University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Spectra of derived categories of Noetherian schemes**Abstract: **The spectrum of a tensor triangulated category (i.e., a triangulated category with a tensor structure) has been introduced and studied by Balmer in 2005. Balmer applied it to the perfect derived category with the derived tensor products for a Noetherian scheme and proved that the tensor triangulated category structure of the perfect derived category completely determines the original scheme.

In this talk, I will introduce the notion of the spectrum of a triangulated category without tensor structure and develop a "tensor-free" analog of Balmer's theory. Also, I will apply this to derived categories of Noetherian schemes. **2022/04/22 GMT 7:00-8:00****Speaker: **Jun-Muk Hwang (IBS Center for Complex Geometry)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Partial compactification of metabelian Lie groups with prescribed varieties of minimal rational tangents**Abstract: **We study minimal rational curves on a complex manifold that are tangent to a distribution. In this setting, the variety of minimal rational tangents (VMRT) has to be isotropic with respect to the Levi tensor of the distribution. Our main result is a converse of this: any smooth projective variety isotropic with respect to a vector-valued anti-symmetric form can be realized as VMRT of minimal rational curves tangent to a distribution on a complex manifold. The complex manifold is constructed as a partial equivariant compactification of a metabelian group, which is a result of independent interest. **2022/04/22 GMT 8:15-9:15****Speaker: **Qizheng Yin (Peking University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Perverse-Hodge symmetry for Lagrangian fibrations**Abstract: **For a Lagrangian fibration from a projective irreducible symplectic variety, the perverse numbers of the fibration are equal to the Hodge numbers of the source variety. In my talk I will first explain how this fact is related to hyper-Kähler geometry. Then I will focus on the symplectic side of the story, especially on how to enhance/categorify the perverse-Hodge symmetry. Joint work with Junliang Shen.

**2022/04/08 GMT 1:00-2:00****Speaker: **Christopher Hacon (The University of Utah)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Boundedness of polarized Calabi-Yau fibrations and generalized pairs**Abstract: **In this talk we will discuss recent results and work in progress related to the boundedness of polarized Calabi-Yau fibrations and to the failure of the boundedness of moduli spaces of glc generalized pairs. **2022/04/08 GMT 2:15-3:15****Speaker: **Ngô Bao Châu (Vietnam Institute for Advanced Study in Mathematics)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **On the functional equation of automorphic L-functions**Abstract: **Automorphic L-functions introduced by Langlands in the late 60' are expected to satisfy a functional equation similar to the functional equation of Riemann's zeta function. The functional equation would follow from the Langlands' functoriality conjecture, which is one of the far-reaching goals of the Langlands program, and in a sense is equivalent to it. Around 2000, Braverman and Kazhdan formulated a new approach to the functional equation not following the route of functoriality but attempting to generalize the Fourier analysis on adeles used by Tate to prove the functional equation of the Riemann zeta function. I will report some recent progress in this approach.

**2022/03/25 GMT 7:00-8:00****Speaker:** Yang Zhou (Fudan University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Wall-crossing for K-theoretic quasimap invariants**Abstract: **For a large class of GIT quotients, the moduli of epsilon-stable quasimaps is a proper Delinge-Mumford stack with a perfect obstruction theory. Thus K-theoretic epsilon-stable quasimap invariants are defined.

As epsilon tends to infinity, it recovers the K-theoretic invariants; and as epsilon decreases, fewer and fewer rational tails are allowed in the domain curves. There is a wall and chamber structure on the space of stability conditions.

In this talk, we will decribe a master space construction involoving the moduli spaces on the two sides of a wall, leading to the proof of a wall-crossing formula.

A key ingredient is keeping track of the S_n-equivariant structure on the K-theoretic invariants. **2022/03/25 GMT 8:15-9:15****Speaker: **Yong Hu (Shanghai Jiao Tong University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Algebraic threefolds of general type with small volume **Abstract: **It is known that the optimal Noether inequality $\vol(X) \ge \frac{4}{3}p_g(X) - \frac{10}{3}$ holds for every $3$-fold $X$ of general type with $p_g(X) \ge 11$. In this talk, we give a complete classification of $3$-folds $X$ of general type with $p_g(X) \ge 11$ satisfying the above equality by giving the explicit structure of a relative canonical model of $X$. This model coincides with the canonical model of $X$ when $p_g(X) \ge 23$. I would also introduce the second and third optimal Noether inequalities for $3$-folds $X$ of general type with $p_g(X) \ge 11$. This is a joint work with Tong Zhang. **2022/02/25 GMT 7:00-8:00****Speaker:** Quoc Ho (Hong Kong University of Science and Technology)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Revisiting mixed geometry**Abstract: **I will present joint work with Penghui Li on our theory of graded sheaves on Artin stacks. Our sheaf theory comes with a six-functor formalism, a perverse t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the six-functor formalism, perverse t-structures, and Frobenius weights on ell-adic sheaves. The theory of graded sheaves has a natural interpretation in terms of mixed geometry à la Beilinson--Ginzburg--Soergel and provides a uniform construction thereof. In particular, it provides a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Historically, constructions of graded lifts were done on a case-by-case basis and were technically subtle, due to Frobenius' non-semisimplicity. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. As an application, I will conclude the talk by showing that the category of constructible B-equivariant graded sheaves on the flag variety G/B is a geometrization of the DG-category of bounded chain complexes of Soergel bimodules. **2022/02/25 GMT 8:15-9:15****Speaker: **Jinhyun Park (KAIST)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **On motivic cohomology of singular algebraic schemes **Abstract: **Motivic cohomology is a hypothetical cohomology theory for algebraic schemes, including algebraic varieties, over a given field, that can be seen as the counterpart in algebraic geometry to the singular cohomology theory in topology. It‘s construction was completed for smooth varieties, but for singular ones the situation was not clear. In this talk, I will sketch some recent attempts of mine to provide an algebraic-cycle-based functorial model for the motivic cohomology of singular algebraic schemes, via formal schemes and some ideas from derived algebraic geometry. As this is very complicated, as an illustration I will give an example on the concrete case of the fat points, where the situation is simpler, but not still trivial. **2022/02/11 GMT 7:00-8:00****Speaker: **Hsin-Ku Chen (NTU)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Classification of three-dimensional terminal divisorial contractions to curves **Abstract: **We classify all divisorial contractions to curves between terminal threefolds by describing them as weighted blow-ups. This is a joint work with Jungkai Alfred Chen and Jheng-Jie Chen. **2022/02/11 GMT 8:15-9:15****Speaker: **Iacopo Brivio (NCTS)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Invariance of plurigenera in positive and mixed characteristic**Abstract: **A famous theorem of Siu states that the m-plurigenus P_m(X) of a complex projective manifold is invariant under deformations for all m\geq 0. It is well-known that in positive or mixed characteristic this can fail for m=1. In this talk I will construct families of smooth surfaces over a DVR X/R such that P_m(X_k)>>P_m(X_K) for all m>0 divisible enough. If time permits, I will also explain how the same ideas can be used to prove (asymptotic) deformation invariance of plurigenera for certain families of threefold pairs in positive and mixed characteristic. **2022/01/14 GMT 7:00-8:00****Speaker: **Kien Huu Nguyen (KU Leuven, Belgium)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Exponential sums modulo $p^m$ for Deligne polynomials**Abstract: **TBA **2022/01/14 GMT 8:15-9:15****Speaker: **Xuan Viet Nhan Nguyen (BCAM, Spain)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Smooth approximation in polynomially bounded o-minimal structures**Abstract: **Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^p$ semialgebraic function , $p \in \mathbb{N}$. Shiota (1986) proved that given a positive continuous semi-algebraic function $\varepsilon: \mathbb{R}^n \to \mathbb{R}$, there is a $C^\infty$ semialgebraic function $g: \mathbb{R}^n \to \mathbb{R}$ such that $|D^\alpha (f - g)| < \varepsilon$ for every $|\alpha|\leq p$. In this talk, we show that the theorem still holds if we replace semialgebraic functions by definable functions in polynomially bounded o-minimal structures that allow smooth cell decompositions. Some applications are also given. This is a joint work with Anna Valette. **2021/12/31 GMT 7:00-8:00****Speaker: **Lei Wu (KU Leuven, Belgium)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title:** D-modles, motivic integral and hypersurface singularities **Abstract: **This talk is an invitation to the study of monodromy conjecture for hypersurfaces in complex affine spaces. I will recall two different ways to understand singularities of hypersurfaces in complex affine spaces. The first one is to use D-modules to define the b-function (also known as the Bernstein-Sato polynomial) of a polynomial (defining the hypersurface). The other one uses motivic integrals and resolution of singularities to obtain the motivic/topological zeta function of the hypersurface. The monodromy conjecture predicts that these two ways of understanding hypersurface singularities are related. Then I will discuss some known cases of the conjecture for hyperplane arrangements. **2021/12/31 GMT 8:15-9:15****Speaker: **Wenhao Ou (AMSS, CAS)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title:** Projective varieties with strictly nef anticanonical divisor**Abstract: **A conjecture of Campana-Peternell presumes that, if the anticanonical divisor of a projective variety X has strictly positive intersection with all curves, then the manifold is Fano. We show that if X is klt, then it is rationally connected. This provides an evidence to the conjecture. Furthermore, if the dimension is at most three, then we prove that X is Fano. This is joint with Jie Liu, Juanyong Wang, Xiaokui Yang and Guolei Zhong. **2021/12/17 GMT 7:00-8:00****Speaker: **Shuai Guo (Peking University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title:** Structure of higher genus Gromov-Witten invariants of the quintic threefolds**Abstract: **The computation of the Gromov-Witten (GW) invariants of the compact Calabi Yau 3-folds is a central and yet difficult problem in geometry and physics. In a seminal work in 1993, Bershadsky, Cecotti, Ooguri and Vafa (BCOV) introduced the higher genus B-model in physics. During the subsequent years, a series of conjectural formulae was proposed by physicists based on the BCOV B-model, which effectively calculates the higher genus GW potential from lower genus GW potentials and a finite ambiguity. In this talk, we will introduce some recent mathematical progresses in this direction. This talk is based on the joint works with Chang-Li-Li and the joint works with Janda-Ruan. **2021/12/17 GMT 8:15-9:15****Speaker: **Yang Zhou (Fudan University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title:** Wall-crossing for K-theoretic quasimap invariants**Abstract: **For a large class of GIT quotients, the moduli of epsilon-stable quasimaps is a proper Delinge-Mumford stack with a perfect obstruction theory. Thus K-theoretic epsilon-stable quasimap invariants are defined. As epsilon tends to infinity, it recovers the K-theoretic invariants; and as epsilon decreases, fewer and fewer rational tails are allowed in the domain curves. There is a wall and chamber structure on the space of stability conditions. In this talk, we will decribe a master space construction involoving the moduli spaces on the two sides of a wall, leading to the proof of a wall-crossing formula. A key ingredient is keeping track of the S_n-equivariant structure on the K-theoretic invariants. **2021/12/03 GMT 7:00-8:00****Speaker: **Jinhyun Park (KAIST)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title:** On motivic cohomology of singular algebraic schemes**Abstract: **Motivic cohomology is a hypothetical cohomology theory for algebraic schemes, including algebraic varieties, over a given field, that can be seen as the counterpart in algebraic geometry to the singular cohomology theory in topology. Its construction was completed for smooth varieties, but for singular ones the situation was not clear.

In this talk, I will sketch some recent attempts of mine to provide an algebraic-cycle-based functorial model for the motivic cohomology of singular algebraic schemes, via formal schemes and some ideas from derived algebraic geometry. As this is very complicated, as an illustration I will give an example on the concrete case of the fat points, where the situation is simpler, but not still trivial. **2021/11/19 GMT 7:00-8:00****Speaker: **Keiji Oguiso (University of Tokyo) **Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Smooth complex projective rational varieties with infinitely many real forms**Abstract: **This is a joint work with Professors Tien-Cuong Dinh and Xun Yu. The real form problem asks how many different ways one can describe a given complex variety by polynomial equations with real coefficients up to isomorphisms over the real number field. For instance, the complex projective line has exactly two real forms up to isomorphisms. This problem is in the limelight again after a breakthrough work due to Lesieutre in 2018. In this talk, among other relevant things, we would like to show that in each dimension greater than or equal to two, there is a smooth complex projective rational variety with infinitely many real forms. This answers a question of Kharlamov in 1999.

**2021/11/19 GMT**** 8:15-9:15****Speaker: **Yuki Hirano (Kyoto University)**Place: **Online** **(Zoom ID: 896 0321 0669 Password: 340252)**Title: **Equivariant tilting modules, Pfaffian varieties and noncommutative matrix factorizations**Abstract: **It is known that a tilting bundle T on a smooth variety X induces a derived equivalence of coherent sheaves on X and finitely generated modules over the endomorphism algebra End(T). We prove that, in a suitable setting, a tilting bundle also induces an equivalence of derived matrix factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived matrix factorization category of a noncommutative gauged Landau-Ginzburg model. **2021/10/08 GMT 7:00-8:00****Speaker: **Christian Schnell (Stony Brook University)**Place: **Online** **(Zoom ID: 827 7229 2046 Password: 830392) **Title: **Finiteness for self-dual classes in variations of Hodge structure**Abstract: **I will talk about a new finiteness theorem for variations of Hodge structure. It is a generalization of the Cattani-Deligne-Kaplan theorem from Hodge classes to so-called self-dual (and anti-self-dual) classes. For example, among integral cohomology classes of degree 4, those of type (4,0) + (2,2) + (0,4) are self-dual, and those of type (3,1) + (1,3) are anti-self-dual. The result is suggested by considerations in theoretical physics, and the proof uses o-minimality and the definability of period mappings. This is joint work with Benjamin Bakker, Thomas Grimm, and Jacob Tsimerman.

**2021/10/08 GMT**** 8:15-9:15****Speaker: **Nguyen-Bac Dang (Université Paris-Saclay)**Place: **Online** **(Zoom ID: 827 7229 2046 Password: 830392) **Title: **Spectral interpretations of dynamical degrees**Abstract: **This talk is based on a joint work with Charles Favre. I will explain how one can control the degree of the iterates of rational maps in arbitrary dimension by applying method from functional analysis. Namely, we endow some particular norms on the space of b-divisors and on the spaces of b-classes and study the eigenvalues of the pullback operator induced by a rational map.

**2021/09/24 GMT 7:00-8:00****Speaker:** Kenta Hashizume (University of Tokyo)**Place: **Online** **(Zoom ID: 827 7229 2046 Password: 830392) **Title: **Adjunction and inversion of adjunction**Abstract: **Finding a relation between singularities of a variety and singularities of subvarietes is a natural problem. An answer to the problem, called adjunction and inversion of adjunction for log canonical pairs, plays a critical role in the recent developments of the birational geometry. In this talk, I will introduce a generalization of the result, that is, adjunction and inversion of adjunction for normal pairs. This is a joint work with Osamu Fujino.

**2021/09/24 GMT 8:15-9:15****Speaker:** Takuzo Okada (Saga University)**Place: **Online** **(Zoom ID: 827 7229 2046 Password: 830392) **Title: **Birational geometry of sextic double solids with cA points**Abstract: **A sextic double solid is a Fano 3-fold which is a double cover of the projective 3-space branched along a sextic surface. Iskovskikh proved that a smooth sextic double solid is birationally superrigid, that is, it does not admit a non-biregular birational map to a Mori fiber space. Later on Cheltsov and Park showed that the same conclusion holds for sextic double solids with ordinary double points. In this talk I will explain birational (non-)superrigidity of sextic double solids with cA points. This talk is based on a joint work with Krylov and Paemurru.

**2021/9/10 GMT 1:00-2:00****Speaker:** Kyoung-Seog Lee (Miami University) **Title: **Derived categories and motives of moduli spaces of vector bundles on curves**Abstract: **Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon and M. S. Narasimhan.

**2021/9/10 GMT 2:15-3:15****Speaker:** Insong Choe (Konkuk University) **Title:** Symplectic and orthogonal Hecke curves**Abstract: **A Hecke curve is a rational curve on the moduli space $SU_C(r,d)$ of vector bundles over an algebraic curve, constructed by using the Hecke transformation. The Hecke curves played an important role in Jun-Muk Hwang's works on the geometry of $SU_C(r,d)$. Later, Xiaotao Sun proved that they have the minimal degree among the rational curves passing through a general point. We construct rational curves on the moduli spaces of symplectic and orthogonal bundles by using symplecitic/orthogonal version of Hecke transformation. It turns out that the symplectic Hecke curves are special kind of Hecke curves, while the orthogonal Hecke curves have degree $2d$, where $d$ is the degree of Hecke curves. Also we show that those curves have the minimal degree among the rational curves passing through a general point. This is a joint work with Kiryong Chung and Sanghyeon Lee.

**2021/9/3 GMT 1:00-2:00****Speaker:** Jingjun Han (Johns Hopkins University) **Title: **Shokurov's conjecture on conic bundles with canonical singularities**Abstract:** A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that the anit-canonical divisor is relatively ample. In this talk, I will prove a conjecture of Shokurov which predicts that, if $X\to Z$ is a conic bundle such that $X$ has canonical singularities, then base variety $Z$ is always $\frac{1}{2}$-lc, and the multiplicities of the fibers over codimension $1$ points are bounded from above by $2$. Both values $\frac{1}{2}$ and $2$ are sharp. This is a joint work with Chen Jiang and Yujie Luo.

**2021/9/3 GMT 2:15-3:15****Speaker:** Jia Jia (National University of Singapore) **Title: **Surjective Endomorphisms of Affine and Projective Surfaces.**Abstract:** In this talk, we will give structure theorems of finite surjective endomorphisms of smooth affine surfaces and normal projective surfaces. Combining with some local dynamics and known results, we will talk about their applications to Zariski Dense Orbit and Kawaguchi-Silverman Conjectures. These are joint work with Takahiro Shibata, Junyi Xie and De-Qi Zhang.

**2021/8/13 GMT 7:00-8:00****Speaker:** Jihao Liu (University of Utah) **Title:** Minimal model program for generalized lc pairs**Abstract:** The theory of generalized pairs was introduced by C. Birkar and D.-Q. Zhang in order to tackle the effective Iitaka fibration conjecture, and has proven to be a powerful tool in birational geometry. It has recently become apparent that the minimal model program for generalized pairs is closely related to the minimal model program for usual pairs and varieties. A folklore conjecture proposed by J. Han and Z. Li and recently re-emphasized by Birkar asks whether we can always run the minimal model program for generalized pairs with at worst generalized lc singularities. In this talk, we will confirm this conjecture by proving the cone theorem, contraction theorem, and the existence of flips for generalized lc pairs. As an immediate consequence, we will complete the minimal model program for generalized lc pairs in dimension <=3 and the pseudo-effective case in dimension 4. This is joint work with C. D. Hacon.

**2021/8/13 GMT 8:15-9:15****Speaker:** Thomas Krämer (Humboldt-Universität zu Berlin) **Title: **Big Tannaka groups on abelian varieties**Abstract:** Lawrence and Sawin have shown that up to translation, any abelian variety over a number field contains only finitely many smooth ample hypersurfaces with given fundamental class and good reduction outside a given finite set of primes. A key ingredient in their proof is that certain Tannaka groups attached to smooth hypersurfaces are big. In the talk I will give a general introduction to Tannaka groups of perverse sheaves on abelian varieties and explain how to determine them for subvarieties of higher codimension (this is work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan).

**2021/7/30 GMT 1:00-2:00****Speaker:** Hsian-Hua Tseng (Ohio State University) **Title:** Relative Gromov-Witten theory without log geometry**Abstract: **We describe a new Gromov-Witten theory of a space relative to a simple normal-crossing divisor constructed using multi-root stacks.

**2021/7/30 GMT 2:15-3:15****Speaker:** Shusuke Otabe (Tokyo Denki University) **Title:** Universal triviality of the Chow group of zero-cycles and unramified logarithmic Hodge-Witt cohomology**Abstract:** Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety over a field has universally trivial Chow group of zero-cycles, then its cohomological Brauer group is trivial as well. Binda-Rülling-Saito recently prove that the same conclusion is true for all reciprocity sheaves. For example, unramified logarithmic Hodge-Witt cohomology has the structure of reciprocity sheaf. In this talk, I will discuss another proof of the triviality of the unramified cohomology, where the key ingredient is a certain kind of moving lemma. This is a joint work with Wataru Kai and Takao Yamazaki.

**2021/7/16 GMT 7:00-8:00****Speaker:** Qingyuan Jiang (University of Edinburgh) **Title:** On the derived categories of Quot schemes of locally free quotients**Abstract: **Quot schemes of locally free quotients of a given coherent sheaf, introduced by Grothendieck, are generalizations of projectivizations and Grassmannian bundles, and are closely related to degeneracy loci of maps between vector bundles. In this talk, we will discuss the structure of the derived categories of these Quot schemes in the case when the coherent sheaf has homological dimension $\le 1$. This framework not only allows us to relax the regularity conditions on various known formulae -- such as the ones for blowups, Cayley's trick, standard flips, projectivizations, and Grassmannain flips, but it also leads us to many new phenomena such as virtual flips, and blowup formulae for blowups along determinantal subschemes of codimension $\le 4$. We will illustrate the idea of proof in concrete cases, and if time allowed, we will also discuss the applications to the case of moduli of linear series on curves, and Brill-Noether theory for moduli of stable objects in K3 categories.

**2021/7/16 GMT 8:15-9:15****Speaker:** Le Quy Thuong (Vietnam National University) **Title:** The ACVF theory and motivic Milnor fibers**Abstract:** In this talk, I review recent studies on the theory of algebraically closed value fields of equal characteristic zero (ACVF theory) developed by Hrushovski-Kazhdan and Hrushovski-Loeser. More precisely, I consider a concrete Grothendieck ring of definable subsets in the VF-sort and prove the structure theorem of this ring which can be presented via materials from extended residue field sort and value group sort. One can construct a ring homomorphism HL from this ring to the Grothendieck ring of algebraic varieties, from which the motivic Milnor fiber can be described in terms of a certain definable subset in VF-sort. As applications, I sketch proofs of the integral identity conjecture and the motivic Thom-Sebastiani theorem using HL, as well as mention the recent work of Fichou-Yin in the same topic.

**2021/7/2 GMT 7:00-8:00****Speaker:** Han-Bom Moon (Fordham University, New York) **Title:** Point configurations, phylogenetic trees, and dissimilarity vectors**Abstract:** In 2004 Pachter and Speyer introduced the dissimilarity maps for phylogenetic trees and asked two important questions about their relationship with tropical Grassmannian. Multiple authors answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. In this talk, we present a weighted variant of the dissimilarity map and show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter-Speyer envisioned. This tropical variety has a geometric interpretation in terms of point configurations on rational normal curves. This is joint work with Alessio Caminata, Noah Giansiracusa, and Luca Schaffler.

**2021/7/2 GMT 8:15-9:15****Speaker:** Yifei Chen (Chinese Academy of Sciences)**Title: **Jordan property of automorphism groups of surfaces of positive characteristic**Abstract: **A classical theorem of C. Jordan asserts the general linear group G over a field of characteristic zero is Jordan. That is, any finite subgroup of G contains a normal abelian subgroup of index at most J, where J is an integer only depends on the group G. J.-P. Serre proved that the same property holds for the Cremona group of rank 2. In this talk, we will discuss Jordan property for automorphism groups of surfaces of positive characteristic. This is a joint work with C. Shramov.

**2021/6/18 GMT 7:00-8:00****Speaker:** Mingshuo Zhou (Tianjin University)**Title: **Moduli space of parabolic bundles over a curve**Abstract:** In this talk, we will review a program (by Narasinhan-Ramadas and Sun) on the proof of Verlinde formula by using degeneration of moduli space of parabolic bundles over a curve. We will also show how the degeneration argument can be used to prove F-splitting of moduli space of parabolic bundles (for generic choice of parabolic points) over a generic curve in positive charactersitic. This is a joint work with Professor Xiaotao Sun. **2021/6/18 GMT 8:15-9:15****Speaker:** Zhi Jiang (Shanghai Center for Mathematical Sciences)**Title: **On syzygies of homogeneous varieties**Abstract: **We discuss some recent progress on syzygies of ample line bundles on homogeneous varieties, including abelian varieties and rational homogeneous varieties. **2021/5/28 GMT 7:00-8:00****Speaker: **Zhiyu Tian (BICMR-Beijing University)**Title:** Some conjectures about Kato homology of rationally connected varieties and KLT singularities**Abstract:** A natural question about zero cycles on a variety defied over an arithmetically interesting field is the injectivity/surjectivity of the cycle class map. This leads to the study of a Gersten type complex defined by Bloch-Ogus and Kato. I will present some conjectures about this complex for rationally connected varieties and Kawamata log terminal (KLT) singularities. I will also present some evidence for the conjectures, and explain how they fit into a variety of conjectures about the stability phenomenon observed in topology and number theory. **2021/5/28 GMT 8:15-9:15****Speaker: **Joao Pedro dos Santos (Universite de Paris)**Title: **Group schemes from ODEs defined over a discrete valuation ring.**Abstract:** Differential Galois theory has the objective to study linear ODEs (or connections) with the help of algebraic groups. Roughly and explicitly, to a matrix $A\in \mathrm{Mat}_n( \mathbb C(x) )$ and a differential system $y'=Ay$, we associate a subgroup of $GL_n(\mathbb C)$, the differential Galois group, whose function is to measure the complexity of the solutions. There are three paths to this theory: Picard-Vessiot extensions, monodromy representations and Tannakian categories.

**2021/5/14 GMT 6:00-7:00****Speaker: **Yuji Odaka (Kyoto University) **Title:** On (various) geometric compactifications of moduli of K3 surfaces**Abstract:** What we mean by "geometric compactifications" in the title is it still parametrizes "geometric objects" at the boundary. In algebraic geometry, it is natural to expect degenerate varieties as such objects. For the moduli of polarized K3 surfaces (or K-trivial varieties in general) case, it is natural to expect slc and K-trivial degenerations, but there are many such compactifications for a fixed moduli component, showing flexibility / ambiguity / difficulty of the problem. This talk is planned to mainly focus the following. In K3 surfaces (and hyperKahler varieties), there is a canonical geometric compactification whose boundary and parametrized objects are Not varieties but tropical geometric or with more PL flavor. This is ongoing joint work with Y.Oshima (cf., arXiv:1810.07685, 2010.00416). **2021/5/14 GMT 7:15-8:15****Speaker: **YongJoo Shin (Chungnam National University)**Title:** Complex minimal surfaces of general type with pg= 0 and K2 = 7 via bidouble covers**Abstract:** Let S be a minimal surface of general type with pg(S) = 0 and K2S = 7 over the field of complex numbers. Inoue firstly constructed such surfaces S described as Galois Z2×Z2-covers over the four-noda cubic surface. Chen later found different surfaces S constructed as Galois Z2×Z2-covers over six nodal del Pezzo surfaces of degree one. In this talk we construct a two-dimensional family of surfaces S different from ones by Inoue and Chen. The construction uses Galois Z2×Z2-covers over rational surfaces with Picard number three, with eight nodes and with two elliptic fibrations. This is a joint work with Yifan Chen.** ****2021/4/30 GMT 7:00-8:00****Speaker: **Yi Gu (Suzhou University)**Title:** On the equivariant automorphism group of surface fibrations**Abstract: **Let f:X→C be a relatively minimal surface fibration with smooth generic fibre. We will discuss the finiteness of its equivariant automorphism group, which is the group of pairs {ÎAut(X)´Aut(C)|} with natural group law. We will give a complete classification of those surface fibrations with infinite equivariant automorphism group in any characteristic. As an application, we will show how this classification can be used to study the bounded subgroup property and the Jordan property for automorphism group of algebraic surfaces. **2021/4/30 GMT 8:15-9:15****Speaker: **Takehiko Yasuda (Osaka University)**Title:** On the isomorphism problem of projective schemes**Abstract:** I will talk about the isomorphism problem of projective schemes; is it algorithmically decidable whether or not two given projective (or, more generally, quasi-projective) schemes, say over an algebraic closure of Q, are isomorphic? I will explain that it is indeed decidable for the following classes of schemes: (1) one-dimensional projective schemes, (2) one-dimensional reduced quasi-projective schemes, (3) smooth projective varieties with either the canonical divisor or the anti-canonical divisor being big, and (4) K3 surfaces with finite automorphism group. Our main strategy is to compute Iso schemes for finitely many Hilbert polynomials. I will also discuss related decidability problems concerning positivity properties (such as ample, nef and big) of line bundles. **2021/4/16 GMT 1:00-2:00****Speaker: **Kuan-Wen Lai (University of Massachusetts Amherst)**Title: **On the irrationality of moduli spaces of K3 surfaces**Abstract: **As for moduli spaces of curves, the moduli space of polarized K3 surfaces of genus g is of general type and thus is irrational for g sufficiently large. In this work, we estimate how the irrationality grows with g in terms of the measure introduced by Moh and Heinzer. We proved that the growth is bounded by a polynomial in g of degree 15 and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree 10. These results are built upon the modularity of the generating series of these moduli spaces in certain ambient spaces, and also built upon the existence of Hodge theoretically associated cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds. This is a collaboration with Daniele Agostini and Ignacio Barros (arXiv:2011.11025). **2021/4/16 GMT 2:15-3:15****Speaker: **Yu-Shen Lin (Boston University)**Title: **Special Lagrangian Fibrations in Log Calabi-Yau Surfaces and Mirror Symmetry**Abstract: **Strominger-Yau-Zaslow conjecture predicts that the Calabi-Yau manifolds admit special Lagrangian fibrations and the mirror can be constructed via the dual torus fibration. The conjecture has been the guiding principle for mirror symmetry while the original conjecture has little progress. In this talk, I will prove that the SYZ fibration exists in certain log Calabi-Yau surfaces and their mirrors indeed admit the dual torus fibration under suitable mirror maps. The result is an interplay between geometric analysis and complex algebraic geometry. The talk is based on joint works with T. Collins and A. Jacob. **2021/4/2 GMT 7:00-8:00****Speaker: **Weizhe Zheng (Morningside Center of Mathematics) **Title:** Ultraproduct cohomology and the decomposition theorem**Abstract: **Ultraproducts of étale cohomology provide a large family of Weil cohomology theories for algebraic varieties. Their properties are closely related to questions of l-independence and torsion-freeness of l-adic cohomology. I will present recent progress in ultraproduct cohomology with coefficients, such as the decomposition theorem. This talk is based on joint work with Anna Cadoret. **2021/4/2 GMT 8:15-9:15****Speaker: **Kestutis Cesnavicius (U. Paris Sud) **Title:** Grothendieck--Serre in the quasi--split unramified case**Abstract: **The Grothendieck-Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. To overcome obstacles that have so far kept the mixed characteristic case out of reach, we adapt Artin's construction of "good neighborhoods" to the setting where the base is a discrete valuation ring, build equivariant compactifications of tori over higher dimensional bases, and study the geometry of the affine Grassmannian in bad characteristics. **2021/3/19 GMT 1:00-2:00****Speaker: **Zhiyuan Li (Shanghai Center for Mathematical Sciences) **Title:** Twisted derived equivalence for abelian surfaces**Abstract:** Over complex numbers, the famous global Torelli theorem for K3 surfaces says that two integral Hodge isometric K3 surfaces are isomorphic. Recently, Huybrechts has shown that two rational Hodge isometric K3 surfaces are twisted derived equivalent. This is called the twisted derived Torelli theorem for K3. Natural questions arise for abelian varieties. In this talk, I will talk about the twisted derived equivalence for abelian surfaces, including the twisted derived Torelli theorem for abelian surfaces (over all fields) and its applications. This is a joint work with Haitao Zou. **2021/3/19 GMT 2:15-3:15****Speaker: **Michael Kemeny (University of Wisconsin-Madison) **Title:** Universal Secant Bundles and Syzygies **Abstract: **We describe a universal approach to the secant bundle construction of syzygies provided by Ein and Lazarsfeld. As an application, we obtain a quick proof of Green's Conjecture on the shape of the equations of general canonical curves. Furthermore, we will explain how the same technique resolves a conjecture of von Bothmer and Schreyer on Geometric Syzygies of canonical curves.

**2021/3/5 GMT 7:00-8:00****Speaker: **Junyi Xie (CNRS Rennes)**Title:** Some boundedness problems in Cremona group**Abstract: **This talk is based on a work with Cantat and Deserti. According to the degree sequence, there are 4 types (elliptic, Jonquieres, Halphen and Loxodromic) of elements f in Bir(P^2). For a fixed degree d>=1, we study the set of these 4 types of elements of degree d. We show that for Halphen twists and Loxodromic transformations, such sets are constructible. This statement is not true for elliptic and Jonquieres elements.We also show that for a Jonquieres or Halphen twist f of degree d, the degree of the unique f-invariant pencil is bounded by a constant depending on d. This result may be considered as a positive answer to the Poincare problem of bounding the degree of first integrals,but for birational twists instead of algebraic foliations. As a consequence of this, we show that for two Halphen twists f and g, if they are conjugate in Bir(f), then they are conjugate by some element of degree bounded by a constant depending on deg(f)+deg(g). This statement is not true for Jonquieres twists. **2021/3/5 GMT 8:15-9:15****Speaker: **Guolei Zhong (National University of Singapore)**Title:** Fano threefolds and fourfolds admitting non-isomorphic endomorphisms.**Abstract:** In this talk, we first show that a smooth Fano threefold X admits a non-isomorphic surjective endomorphism if and only if X is either toric or a product of a smooth rational curve and a del Pezzo surface. Second, we show that a smooth Fano fourfold Y with a conic bundle structure is toric if and only if Y admits an amplified endomorphism. The first part is a joint work with Sheng Meng and De-Qi Zhang, and the second part is a joint work with Jia jia.

National Center for Theoretical Sciences

A natural generalization of the Higgs bundles is the parabolic Higgs bundles (that we shall equip each bundle of a parabolic structure, i.e the choice of flags in the fibers over certain marked points, and some compatible conditions). Simpson proved that there is analogous relation between the space of representations of the fundamental group of a punctured curve (the marked points are the points that are took out from the curve) with the moduli space of parabolic Higgs bundles.

Despite their good applications, the cohomology of the moduli space of (parabolic) Higgs bundles has not yet been determined. In this talk, I shall explain an algorithm to calculate the (virtual) motive (i.e in a suitable Grothendieck group) of the moduli spaces of (parabolic) Higgs bundles. In the case when the moduli space is quasi-projective, the virtual motive allows us to read off the dimensions of its cohomology spaces.

In this talk, I will introduce the notion of the spectrum of a triangulated category without tensor structure and develop a "tensor-free" analog of Balmer's theory. Also, I will apply this to derived categories of Noetherian schemes.

As epsilon tends to infinity, it recovers the K-theoretic invariants; and as epsilon decreases, fewer and fewer rational tails are allowed in the domain curves. There is a wall and chamber structure on the space of stability conditions.

In this talk, we will decribe a master space construction involoving the moduli spaces on the two sides of a wall, leading to the proof of a wall-crossing formula.

A key ingredient is keeping track of the S_n-equivariant structure on the K-theoretic invariants.

In this talk, I will sketch some recent attempts of mine to provide an algebraic-cycle-based functorial model for the motivic cohomology of singular algebraic schemes, via formal schemes and some ideas from derived algebraic geometry. As this is very complicated, as an illustration I will give an example on the concrete case of the fat points, where the situation is simpler, but not still trivial.

If instead of working with complex coefficients we deal with a discrete valuation ring $R$, the construction of the differential Galois groups are less obvious and the theory of groups gives place to that of group schemes. This puts forward the Tannakian approach and relevant concepts from algebraic geometry like formal group schemes and blowups. In this talk, I shall explain how to associate to these differential equations certain flat $R$-group schemes, what properties these may have--what to expect from a group having a generically faithful representation which becomes trivial under specialisation?--and how to compute with the help of the analytic method of monodromy. The talk is a horizontal report on several works done in collaboration with P.H.Hai and his students N.D.Duong and P.T.Tam over the past years.

In general, there is a canonical PARTIAL compactification (quasi-projective variety) of moduli of polarized K-trivial varieties (essentially due to Birkar and Zhang), as a completion with respect to the Weil-Petersson metric. This is characterized by K-stability.

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