Mini-courses on Representation Theory (by Dr. Baptiste Rognerud & Dr. Shijie Zhu)

These two introductory mini-courses are offered by experts in Representation Theory, as part of the thematic program TDA PARTI: Topological Data Analysis, Persistence And Representation Theory Intertwined (TP25TD).

• Posets and representation theory​ (by Dr. Baptiste Rognerud)
• Monomorphism categories and applications (by Dr. Shijie Zhu)

Each mini-course consists of four lectures. Each one of the two lecturers will give a 1-hour talk on Monday, Tuesday, Thursday and Friday (there will be no talk on Wednesday). Lectures will be between 9:30-12:00, with a 30-minute break between the two lectures.

***Further information on the mini-courses and the schedule will be posted on the main website of the thematic program TDA PARTI (TP25TD): https://www.oist.jp/visiting-program/tp25td

※ Please note that these lecture series will be streamed via Zoom and may be recorded. Upon the consent of the lecturer, the mini-course may be uploaded on the OIST website※

• Miniourse by Dr. Baptiste Rognerud​.
  Title: Posets and representation theory
  Abstract: Representation theory and combinatorics of partially ordered sets (posets) interacts in two main ways: posets can be used to organize an (most of the time) infinite amount of data coming from the study of the representation theory of a given object: algebra, quiver, finite group etc. On the other hand, we can study the representation theory of posets, for example by viewing them as categories and looking at functors with values in a category of vector spaces.
  
  In this mini-course we propose to have a look at the two aspects. We will start by an introduction to the representation theory of posets which goes back to the work of Mitchell in the sixties and to the classification of the finite posets of finite representation type by Chaptal and Loupias soon after. In contrast with Gabriel’s theorem for quivers, the classification is rather complicated and difficult to use, this is probably why this theory more or less disappeared for 30 years before reappearing quite unexpectedly in algebraic combinatorics through derived categories and in topological data analysis through persistence modules. In the second part of the course, we will see an example where modern representation theory of algebra, through the so-called τ-tilting theory led to a representation theorem in lattice theory. In the last part of the minicourse we will look at recent aspects of the representation theory of posets involving Coxeter matrices and derived categories.
  
  There is another point of view on poset representations which goes back to Nazarova and Kleiner that we will not touch in these course. However, it is deeply related to Shijie Zhu’s mini-course, which will be presented parallel to this mini-course!
   
• Miniourse by Dr. Shijie Zhu.
  Title: Monomorphism categories and applications
  Abstract: Classifying subobjects is a fundamental problem in mathematics, such as the classification of 2-manifolds in 3-dimensional spaces and the classification of d-dimensional subspaces of an n-dimensional linear space. Usually such classifications are up to a certain equivalence relation. All these questions can be formalized using the language of category theory, where we try to classify all the monomorphisms in a category, and this motivates the study of monomorphism categories. We will see several such questions, the subspace problem and the Birkhoff problem, which are seemingly easy but quite difficult to solve.
  
  The modern development of monomorphism categories initiated from Ringel and Schmidmeier’s study of invariant subspaces of nilpotent linear operators, where they gave a complete description of the indecomposable objects in the submodule category of k[x]/⟨x⁶⟩-mod, using Auslander-Reiten theory. Their work was soon developed and applications in various mathematics fields were found. As a generalization of submodule category, X. Luo and P. Zhang. introduced the notion of separated monic representations over an acyclic quiver, in order to describe so-called Gorenstein-projective modules. In fact, their work suggests a strong connection between cotorsion pairs via the separated monic correspondence. In the second and third talk, I will give a brief introduction to the theory of monomorphism categories and separated monic correspondence.
  
  Representations for grid quivers have significant applications in topological data analysis. In particular, representations with all the morphisms monic or epic along one or both directions in the grid arises naturally from hierarchical clustering methods. A recent work by U.Bauer, M.B. Botnan, S. Oppermann and J. Steen reveals certain reductive formula for these categories. We will recover their result from the perspective of separated monic correspondences.
  
  There has been some work by J.Asadollahi, P. Bahiraei, R. Hafezi and R. Vahed about monic representations over rooted posets, which is related to Baptiste Rognerud’s mini-course, presented parallel to this mini-course.