Recently, Beijing Institute of Mathematical Sciences and Applications (BIMSA) has made a series of significant research achievements in the core field of pure mathematics. A research paper entitled "Super Gromov–Witten Invariants via Torus Localization", co-authored by Professor Shing-Tung Yau, Director of BIMSA, Professor Artan Sheshmani, Dr. Enno Keßler of the Max Planck Institute for Mathematics, and Professor Dimitri Zvonkine of Université Paris-Saclay, has been published in the Journal of the European Mathematical Society (JEMS), one of the flagship journals of the European Mathematical Society.

Another paper titled "Rigid Schubert Classes in Partial Flag Varieties", co-authored by Assistant Researcher Yuxiang Liu, Professor Artan Sheshmani, and Professor Shing-Tung Yau, has been accepted for publication in the Journal of Differential Geometry (JDG), a leading international journal in geometry. These two achievements have established new mathematical frameworks in the fields of super enumerative geometry and deformation theory of homogeneous spaces respectively.

Paper 1: Bypassing Traditional Technical Barriers to Establish a Rigorous Definition of Super Gromov-Witten Invariants

For decades, Gromov–Witten (GW) invariants have been a cornerstone of enumerative geometry. They encode counts of curves in geometric spaces and play a central role in algebraic geometry, symplectic geometry, mirror symmetry, and string theory. Since their development in the 1990s, GW invariants have served as an important bridge between mathematics and theoretical physics.

Motivated by supersymmetric theories, physicists have long predicted the existence of a richer "super" analogue of GW invariants. Unlike classical GW invariants, which describe only bosonic degrees of freedom, super Gromov–Witten invariants incorporate both bosonic and fermionic contributions. They are expected to provide fundamental mathematical structures for superstring theory and supersymmetric quantum field theories.

Establishing a rigorous mathematical theory of super Gromov–Witten invariants, however, has remained a difficult problem. While substantial progress has been made in the study of moduli spaces of super stable curves and super stable maps, a complete supergeometric intersection theory and the corresponding cohomological framework are still not fully understood. This has posed a major obstacle to the mathematical development of super enumerative geometry.

Rather than pursuing the traditional route of first constructing a full supergeometric intersection theory, the authors adopt a different perspective. The central idea of the paper is to postulate a torus localization property for the odd (fermionic) directions of the moduli spaces of super stable maps. This allows the construction of super Gromov–Witten invariants using classical intersection theory, bypassing the need for a fully developed supergeometric framework.

More specifically, the invariants are defined by integrating pullbacks of homology classes along evaluation maps and dividing by the equivariant Euler class of the normal bundle associated with the embedding of the moduli space of stable spin maps into the moduli space of super stable maps. The entire construction is based on the framework of classical intersection theory, and the authors prove that the newly defined invariants satisfy a generalization of the Kontsevich–Manin axioms.

"The key point is that the relevant SUSY normal bundles possess the compatibility properties needed for the theory," said Artan Sheshmani. "These bundles behave well with respect to forgetful maps and gluing maps, which in turn ensures that the resulting invariants satisfy a generalized form of the Kontsevich–Manin axioms."

Building on these properties, the authors show that the resulting invariants form a consistent theory and can be used to construct a super small quantum cohomology ring, one of the fundamental algebraic structures associated with super Gromov–Witten theory.

Beyond the theoretical construction, the paper demonstrates that the new framework is computationally effective. Using torus localization techniques, the authors perform explicit calculations of super Gromov–Witten invariants for projective spaces ℙⁿ in genus zero and degree one. Concrete examples are worked out for small dimensions and small numbers of marked points.

The authors view this work as an important step toward establishing a rigorous mathematical foundation for super enumerative geometry. By circumventing the need for a fully developed supergeometric intersection theory, the proposed framework makes it possible to define and compute super Gromov–Witten invariants within existing mathematical machinery. At the same time, it provides new foundations for future investigations in superstring theory, mirror symmetry, and related questions in supersymmetric geometry.

Paper 2: Solving a Long-Standing Open Problem to Establish a Unified Criterion Framework for Schubert Class Rigidity in Classical Partial Flag Varieties

Yuxiang Liu, Artan Sheshmani, Shing-Tung Yau, Rigid Schubert classes in partial flag varieties, Accepted by Journal of Differential Geometry (2026)

In algebraic geometry, flag varieties are fundamental homogeneous spaces with highly symmetric structures, while Schubert varieties, defined by incidence conditions, form an important class of algebraic subvarieties in flag varieties. Their classes generate the Chow ring and cohomology ring of flag varieties, and play fundamental roles in enumerative geometry, representation theory, and algebraic combinatorics.

The rigidity problem for Schubert classes asks whether every subvariety representing a fixed Schubert class must itself be a Schubert variety, or equivalently, whether there exist non-Schubert subvarieties that are rationally equivalent to a given Schubert variety. This property is central to understanding the global geometric structure of flag varieties and has close connections with deformation problems in enumerative geometry. While substantial progress has been made for Grassmannians and several classical cases, the problem exhibits more intricate structural features in general classical partial flag varieties, where a unified characterization has long been unavailable.

This work conducts a systematic investigation of the long-standing open problem of Schubert class rigidity in classical partial flag varieties. It establishes a unified working framework for type A partial flag varieties, type B and D orthogonal partial flag varieties, as well as type C symplectic Grassmannians and symplectic partial flag varieties. The framework characterizes rigidity in terms of numerical conditions on Schubert indices, together with compatibility relations among essential incidence conditions. In particular, it provides an effective criterion for determining the rigidity of a given Schubert class and achieves a systematic description of rigidity phenomena in classical cases.

Methodologically, the core innovation of the work is a reduction technique that utilizes the natural projection maps $\pi_t: F(d_1,\dots,d_k;n) \to G(d_t,n) $ to reduce the more complicated geometry of partial flag varieties to known rigidity results for Grassmannians, while carefully tracking the behavior and compatibility of incidence conditions at different levels. A new partial order on essential sub-indices is introduced to characterize the compatibility and combinatorial patterns of different incidence conditions, enabling the unified organization of scattered local numerical conditions within a coherent global framework.

For orthogonal partial flag varieties, where additional complications arise from the existence of non-isotropic incidence conditions, a new relational structure is introduced to systematically describe the mutual constraints between non-isotropic parts and successfully integrate them into the unified analytical framework. In the symplectic case, the paper further presents several explicit construction results and clarifies the projection relations between symplectic partial flag varieties and symplectic Grassmannians.

In addition, the work systematically studies the phenomenon referred to as multi-rigidity: certain Schubert classes are not only rigid themselves, but every positive integral multiple of such a class can only be represented by unions of Schubert varieties. This property reveals a strong intrinsic stability of Schubert classes within the structure of the Chow ring. The study also constructs and analyzes a class of non-smoothable Schubert classes, providing new canonical examples for the singularity theory of algebraic varieties and smoothing problems.

Overall, this work develops a systematic approach to rigidity problems in classical partial flag varieties, extending from Grassmannians to more general flag varieties. By unifying numerical criteria, reduction techniques, and the combinatorics of sub-indices within a single theoretical framework, it provides new tools and perspectives for understanding the deformation behavior of Schubert classes in homogeneous spaces.