International Conference on Algebra, Geometry and Mathematical Physics

Bao, Yan-Hong (Anhui University)

Title: Operads and Cohomology of algebras with various operators

Abstract: Operad theory is a powerful tool for constructing deformation complexes of algebraic systems. In this talk, we will introduce the cohomology theory and deformation theory for several classes of algebras equipped with various operators. This work is joint with Jiang-Nan Xu and Yuan-Feng Zhang.

Dong, Chongying (Sichuan University)

Title: Rationality and modularity of vertex operator algebras

Abstract: Rational vertex operator algebras, whose module categories are semisimple, are analogues of finite-dimensional semisimple associative algebras and semisimple Lie algebras. In this talk, we will discuss how the modularity of VOA characters can be used to characterize rationality.

Fan, Jiuyu (Sichuan University)

Title: Schubert Calculus in Algebraic Combinatorics

Abstract: Schubert calculus occupies a vibrant crossroads of algebraic geometry, representation theory, and algebraic combinatorics—classical in its origins yet continually evolving. Although its core has long centered on intersection theory for flag varieties and the combinatorial structures of Schubert polynomials, contemporary developments have broadened the scope to encompass equivariant cohomology, K-theory, and more generalised cohomological frameworks, including Chern–Schwartz–MacPherson classes and motivic Chern classes. The first half of this talk will offer a self-contained overview of the foundational material, outlining key notions and currently open questions. In the second half, we will report on s everal recent advances achieved in our ongoing work. This presentation draws on collaborative efforts with Yiming Chen, Peter L. Guo, Changjian Su, Rui Xiong, Ming Yao, and Zelin Ye.

Hua, Zheng (Hong Kong University)

Title: Positroids and Vector Bundles on Curves

Abstract: Positroids are matroids that admit a totally nonnegative realization. They naturally index the cells of the totally nonnegative Grassmannian, projected open Richardson strata in the complex Grassmannian, and affine Schubert cells in the central fiber of local models of Shimura varieties of type A. We provide a new perspective on positroids and these geometric constructions via vector bundles on Neron polygons, which naturally gives rise to a Poisson structure whose T-orbits of symplectic leaves are positroid varieties. As we replace Neron polygons by other curves of genus 1, we get new Poisson geometry of grassmannian and its degeneration. This is a joint work with Sasha Polishchuk.

Huang, Hualin (Huaqiao University)

Title: On Compositions of Forms

Abstract: The composition problem of forms focuses on algebraic phenomena, intrinsic structures and inherent regularities derived from the multiplication of homogeneous polynomials. This talk will introduce a brief history, core achievements and open problems of this subject, and elaborate on several advances we have obtained in the research of compositions of diagonal forms and the Waring problem of polynomials.

Wang, Lizhong (Peking University)

Title: A criterion for a bimodule embedding

Abstract: In this talk, we will present a criterion for a bimodule embedding. This bimodule comes from the representation theory of symmetric groups and this criterion is helpful in the problems related to plethysms.

Wang, Yanhua (Shanghai University of Finance and Economics)

Title: Zariski cancellation problem

Abstract: Zariski cancellation problem is related to the Automorphism Problem, Characterization Problem, Linearization Problem, Embedding Problem, Jacobian Conjecture. In this talk, I will introduce some recent developments on the Zariski cancellation problem of noncommutative algebras. Especially, I will introduce concepts and methods to handle the Zariski cancellation problem for noncommutative algebras.

Ren, Li (Sichuan University)

Title: Fusion actions and categorical orbifold theory

Abstract: I will discuss our recent investigation into the generalized symmetries of condensable algebras in modular tensor categories. The mian results inculde Schur-Weyl duality, Galois correspondence and categorical orbifold theory. This is a joint work with Chongying Dong, Siu-Hung Ng and Feng Xu.

Wu, Quanshui (Fudan University)

Title: Chevalley property of Hopf algebras with large centers

Abstract: Abstract：Any affine Cayley-Hamilton Hopf algebra, in the sense of De Concini-Procesi-Reshetikhin-Rosso, is finitely generated as a module over its center. We prove that any affine Hopf algebra admitting a large central Hopf subalgebra can be endowed with a trace map and a Cayley-Hamilton Hopf algebra structure. Cayley-Hamilton Hopf algebras serve as a general framework for the study of the irreducible representations of certain quantum groups at roots of unity. For any affine Cayley-Hamilton Hopf algebra, the identity fiber algebra naturally admits a Hopf algebra structure. The category of finite-dimensional modules over any fiber algebra of H forms an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra. The irreducible representations, for instance, the Chevalley property, of H are closely related to the discriminant ideals of H. In fact, we prove that H has the Chevalley property if and only if its identity fiber algebra has the Chevalley property and all discriminant ideals are trivial. The level of the lowest discriminant ideal of H is equal to the Frobenius-Perron dimension of the identity fiber algebra plus one. This talk is based on recent works with Huang, Mi, Qi, and Zhu, following an earlier work with Mi and Yakimov.

Xu, Fei (Shantou University)

Title: On lax functors in representations and their applications

Abstract: In representations of finite groups and finite-dimensional algebras, there are many finite (index) categories, used to glue modules and module categories. Let C be such a finite category. Besides strict functors, recent years lax functors have been considered in several interesting situations, by Asashiba, Balmer, Gantor and Kapranov et al. Motivated by these works, we introduce the concept of the modulated category algebra over a lax functor on C, as a common generalizations of the (skew) group algebra and the modulated path algebra, and then explain some of its applications.

Zhang, Yinhuo (University of Hasselt)

Title: Affineness of Noetherian Hopf algebras

Abstract: In 2003, Q.-S. Wu and J. J. Zhang posed a well-known open question: Is every Noetherian Hopf algebra over a field k affine? By employing mirror-reduction orderings, reduction￾factorization, prime-word rewriting, and the generalized lifting technique, we establish a definitive necessary and sufficient criterion for the affineness of Noetherian Hopf algebras, representing a substantial step toward resolving this long-standing question. Specifically, we prove that a left or right Noetherian Hopf algebra over k is affine if and only if its Hopf coradical is affine, thereby reducing verification to the first filtration step. To establish necessity, we prove that the Hopf coradical of an affine Hopf algebra inherits affineness. As applications of this equivalence, we show that a left or right Noetherian Hopf algebra H is affine provided that its coradical H₍₀₎ forms a subalgebra (i.e., satisfies the dual Chevalley property), its coradical H₍₀₎ is cocommutative, or its Hopf coradical H_[0] is commutative. This talk is based on joint work with Huan Jia.

Zhou, Guisong (Ningbo University)

Title: The structure of iterated Hopf Ore extensions

Abstract: Iterated Hopf Ore extensions (IHOEs) over a field k of characteristic zero form an important class of connected Hopf algebras with finite Gelfand–Kirillov dimension. In this talk, we investigate the structural properties of IHOEs. We prove that the class of IHOEs is closed under Hopf subalgebras and quotient Hopf algebras, and that every one-sided coideal subalgebra of an IHOE is itself an iterated Ore extension of the base field. The key ingredient is a PBW inheritance theorem developed through the thin replacement technique originating from Kharchenko’s work. This theorem provides a systematic method for constructing PBW generating systems of subalgebras from those of ambient algebras under mild conditions. Building on these results, we obtain an explicit combinatorial algorithm for classifying one-sided coideal subalgebras of arbitrary IHOEs. As an application, we classify all right coideal subalgebras of noncocommutative connected Hopf algebras of GK-dimension three. This talk is based on a joint work with Bofan Yu (arXiv: 2607.28234).

Zhou, Guodong (East China Normal University)

Title: Yoneda Ext-groups of twisted tensor products

Abstract: We consider Yoneda Ext algebras over twisted tensor products which generalise skew group algebras, Ore extensions, smash products, Tacheuchi smash products etc. Under a rigidity condition and some mild finiteness conditions, we show that the Ext algebra of a generalised twisted tensor product of two algebras is isomorphic to the twisted tensor product of the Yoneda Ext algebras of these two algebras. This talk is based on a joint work with Hongguang Nie, Chaoyuan Zhou and Ruipeng Zhu.

Zhou, Yuanyang (Central China Normal University)

Title: Blocks with only one irreducible Brauer character orbit

Abstract: In this talk, we confirm the Kessar-Linckelmann conjecture for blocks with abelian defect groups. More generally, we prove that if a covered block with abelian defect groups has its irreducible Brauer characters forming a single orbit under the block stabilizer, then the covering block is inertial; we term these blocks with a single irreducible Brauer character orbit. Consequently, we establish Broué's abelian defect conjecture for such blocks. As a byproduct, we show that blocks of finite quasisimple groups with a single irreducible Brauer character orbit necessarily have abelian defect groups. As applications, we verify the blockwise Alperin weight conjecture for blocks with a unique irreducible Brauer character and prove Puig's long-standing conjecture in full generality. All results rely essentially on the Classification of finite simple groups.