• Speaker: Lucas Hall, MSU
• Title: Approximately Finite Dimensional C*-algebras
• Date: 01/30/2023
• Time: 4:00 PM - 5:30 PM
• Place: C517 Wells Hall
• Contact: Brent Nelson (banelson@msu.edu)

I’ll tour through the study of finite dimensional C*-algebras and homomorphisms between them, and use this as a basis to define and study approximately finite dimensional (AF) algebras.

• Speaker: Theodore Voronov, University of Manchester
• Title:  From homotopy Lie brackets to thick morphisms of supermanifolds and non-linear functional-algebraic duality (NOTE UNUSUAL DAY)
• Date: 01/31/2023
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall
• Contact: Michael Shapiro (mshapiro@msu.edu)

I will give a motivation for homotopy Lie brackets and the corresponding morphisms preserving brackets "up to homotopy" (more precisely, for L-infinity morphisms and L-infinity algebras), and show how to describe them using supergeometry. So, instead of a single Poisson or Lie bracket, there is a whole sequence of operations with n arguments, n=1,2,3,..., satisfying a linked infinite sequence of identities replacing the familiar Jacobi identity for a Lie bracket; and, instead of a morphism as a linear map mapping a bracket to a bracket, there is a sequence of multi-linear mappings mixing brackets with different numbers of arguments, and, in particular, the binary bracket is preserved only up to an (algebraic) homotopy. Geometrically, such a sequence of multi-linear mappings assembles into one non-linear map of supermanifolds. For the case of homotopy brackets of functions ("higher Poisson" or "homotopy Poisson" structure), this leads us to the question about a natural construction of non-linear mappings between algebras of smooth functions generalizing the usual pull-backs. I discovered such a construction some years ago. These are "thick morphisms" of (super)manifolds generalizing ordinary smooth maps. From a more general perspective, we arrive in this way at a non-linear analog of the classical functional-algebraic duality between spaces and algebras.

• Speaker: Stephen Lacina, University of Oregon
• Title: Maximal Chain Descent Orders
• Date: 02/01/2023
• Time: 3:00 PM - 3:50 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Bruce E Sagan (bsagan@msu.edu)

We introduce a new partial order called the maximal chain descent order on the maximal chains of any finite, bounded poset with an EL-labeling. We prove that the maximal chain descent order encodes via its linear extensions all shellings of the order complex induced by the EL-labeling strictly including the well-known lexicographic shellings. We show that the standard EL-labeling of the Boolean lattice has maximal chain descent order isomorphic to the type A weak order. We also prove that natural EL-labelings of intervals in Young's lattice give maximal chain descent orders isomorphic to partial orders on the standard Young tableaux or standard skew tableaux of a fixed shape given by swapping certain entries. We additionally show that the cover relations of maximal chain descent orders are generally more subtle than one might first expect, but we characterize the EL-labelings with the expected cover relations including many well-known families of EL-labelings.

• Speaker: Igor Rapinchuk, MSU
• Title: Profinite groups and infinite Galois theory
• Date: 02/01/2023
• Time: 3:00 PM - 4:00 PM
• Place: C329 Wells Hall
• Contact: Igor Rapinchuk (rapinchu@msu.edu)

No abstract available.

• Speaker: Jie Yang, MSU
• Title: Potentially orthonormalizable modules
• Date: 02/02/2023
• Time: 2:00 PM - 3:00 PM
• Place: C204A Wells Hall
• Contact: Jie Yang (yangji79@msu.edu)

I will discuss basics of potentially orthonormalizable modules and some related concepts, which are preliminaries for the theory of Fredholm's determinant of compact operators in non-archimedean setting.

• Speaker: James Murphy, Tufts University
• Title: ZOOM TALK (password the smallest prime > 100) - Towards Intrinsically Low-Dimensional Models in Wasserstein Space: Geometry, Statistics, and Learning
• Date: 02/02/2023
• Time: 2:30 PM - 3:30 PM
• Place: C304 Wells Hall
• Contact: Mark A Iwen ()

We consider the problems of efficient modeling and representation learning for probability distributions in Wasserstein space. We consider a general barycentric coding model in which data are represented as Wasserstein-2 (W2) barycenters of a set of fixed reference measures. Leveraging the Riemannian structure of W2-space, we develop a tractable optimization program to learn the barycentric coordinates when given access to the densities of the underlying measures. We provide a consistent statistical procedure for learning these coordinates when the measures are accessed only by i.i.d. samples. Our consistency results and algorithms exploit entropic regularization of the optimal transport problem, thereby allowing our barycentric modeling approach to scale efficiently. We also consider the problem of learning reference measures given observed data. Our regularized approach to dictionary learning in Wasserstein space addresses core problems of ill-posedness and in practice learns interpretable dictionary elements and coefficients useful for downstream tasks. Applications to image and natural language processing will be shown throughout the talk.