In computer science and mathematics, isomorphism problems for algebraic structures form a fundamental family of algorithmic problems. They have been extensively studied by researchers across various fields, such as cryptography, computer algebra, quantum information, and pure mathematics. Understanding the computational complexity of isomorphism problems has been a fruitful research line and has given rise to many new areas in theoretical computer science. The most famous problem of this kind is the graph isomorphism problem, which asks to decide if two graphs are the same up to a permutation of vertices. Babai’s quasipolynomial-time algorithm for graph isomorphism has been regarded as one of the major breakthroughs in computer science of the past decade.

In light of Babai’s breakthrough, it is natural to consider “what’s next?” for isomorphism problems. Babai states that the Group Isomorphism Problem is one of the main bottlenecks towards more efficient algorithms for Graph Isomorphism. Recently, the theory of Tensor Isomorphism complexity class offers a unifying perspective on many isomorphism problems for algebraic structures, revealing previously unrecognized connections between these problems. This theory has enabled progress on long-standing open problems and has practical applications in information security. The meeting is envisaged to concentrate on recent developments in tensor isomorphism. By inviting leading researchers from computer science, mathematics and physics, we anticipate that the meeting will foster in-depth discussions on algorithm design, geometric structures and applications of tensor isomorphism problems. The meeting will hopefully strengthen the collaborations among various research communities, especially between Asian and non-Asian researchers.
The meeting will focus on the following three topics.

1. Algorithms for the Tensor Isomorphism Problem
The Tensor Isomorphism problem (TI) asks to decide if two given tensors are isomorphic up to local general linear group actions. Grochow and Qiao (SICOMP 23) established a fruitful list of algebraic isomorphism testing problems that are polynomial-time equivalent or reducible to TI. In particular, the reduction from the Group Isomorphism problem to TI has been utilized by Xiaorui Sun to obtain substantial time complexity improvements on the Group Isomorphism problem (STOC 23, FOCS 24). The objective of this seminar will be to both give an overview of this field and further investigate faster algorithms for TI and related problems. A first target will be investigating more connections between TI and isomorphism testing problems of different algebraic structures. A second goal will be finding more applications of Xiaorui Sun’s algorithm in other isomorphism testing problems, including testing general p-group isomorphisms. For this purpose, the seminar will invite participants with different backgrounds, such as experts from computational group theory and experts from complexity theory and algorithms, to exchange and generate new ideas from mathematics and computer science.

2. Algebraic and geometric structure of tensor isomorphism
Multi-way arrays lend themselves to various algebraic and geometric tools. For example, Brooksbank, Maglione, Wilson (Journal of Algebra 2017, 2020, & 2022) introduced many families of algebras associated to tensors and used them to develop efficient algorithms for both the Group and Tensor Isomorphism Problems. Central to this is the associated determinantal hypersurface (or Pfaffian hypersurface in some cases), which controls the geometry relevant to the tensor. These hypersurfaces were exploited to develop efficient algorithms in Brooksbank, Maglione, Wilson (Journal of Algebra2017) and Maglione, Stanojkovski (Algebra & Number Theory 2024+). The seminar will invite experts from communities studying algebraic and geometric properties of tensors to develop stronger tools and more efficient algorithms for the Tensor Isomorphism Problem.

3. Applications of the Tensor Isomorphism Problem
Due to its vast connections with different fields, it is promising to apply the algorithmic and structural results of TI to other research areas. The developments of TI have bridged many disciplines, including graph theory (the notorious and well-studied graph isomorphism problem), theoretical computer science (decidability and complexity questions), computer algebra (practical algorithms for computer algebra systems), quantum information theory (tensors), group and semigroup theory (undecidability, theoretical and practical algorithms), and representation theory (modules). For instance, in cryptography, several NIST PQC standardization proposals have utilized TI and related algorithmic problems to design post quantum cryptosystems. In quantum physics, TI has been utilized to classify multipartite quantum entanglements of many-body quantum systems. In mathematics, enumerating certain finite groups is equivalent to enumerating isomorphism classes of tensors of certain structures over finite fields. The goal of this topic is to investigate more applications of TI in different scientific fields. For this purpose, the seminar will invite experts to share recent advances and applications of TI in cryptography, quantum physics and mathematics.

On Monday, Tuesday and Thursday, we will host tutorial-style lectures in the morning, each dedicated to one of the three central themes outlined above. These lectures are designed to provide accessible entry points for researchers who may not be deeply familiar with all themes, while also offering a valuable orientation for early-career participants. After the tutorial lectures, we will continue with 4-5 40-minute talks given by senior and junior faculties for the corresponding topics. These talks will be focused on the recent progress on each topic, and indicate the connections between algorithms, geometry, and applications. After these talks, we will have free discussion session on each topic, which stimulates further research collaboration.
On Wednesday morning and Friday morning, we will organize lightning talks given by postdocs and research associates. These talks will be helpful for them to summarize their research outputs and receive advice from senior researchers. Our goal is to create an environment conducive to both learning and the formation of new research partnerships. We will also run an open problem session on Wednesday morning.