Fall 2025

• 08/20/2025 - Naihuan Jing (NC State)
  Title: q-immanants and Capelli identities
  Abstract: Capelli identity is a classical identity about the central elements of the enveloping algebra gl(n). It is one of the main tools in Weyl's famous book on classical groups. Immanant of a square matrix A is a generalization of det(A) and per(A) introduced by Littlewood. In 1996, Okounkov found new basis elements in Z(U(gl(n)) using the quantum immanant of the generator matrix of U(gl(n)). In this talk, I will discuss my recent joint work with M. Liu and A. Molev on quantization of Okounkov's quantum immanants and quantum Capelli identities for the quantum enveloping algebra of gl(n). 
  
  

• 08/27/2025 - David Green (NC State)
  Title: Braidings for non-split Tambara-Yamagami Categories over the reals
  Abstract: Non-split Real Tambara-Yamagami categories are a family of fusion categories over the real numbers that were recently introduced and classified by Plavnik, Sanford, and Sconce. Owing to the fact that the reals are not algebraically closed, these categories involve nontrivial division algebras along with interesting complex conjugation actions. Extending the work of Siehler, we consider which of these categories admit braidings, and classify the resulting braided equivalence classes in terms of quadratic forms. This is recent joint work with Sean Sanford and Yoyo Jiang. 
  
  

• 09/03/2025 - Seth Sullivant (NC State)
  Title: Lattice supported distributions and graphical models
  Abstract:  For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. While these results are about statistical models, the proof techniques come from combinatorics and algebraic geometry.  This is joint work with Thomas Kahle.
  
  

• 09/10/2025 - Dietmar Bisch (Vanderbilt University)
  Title: Subfactors yearn for Analysis
  Abstract:  The representation theory of an inclusion of II_1 factors with finite Jones index, also called a subfactor, is a unitary tensor category of bimodules. If the subfactor is hyperfinite, and the category is fusion, it classifies the subfactor by a result of Popa. However, this is a rather special situation, and analysis is required in general to understand the subfactor.

I will explain all these notions and show explicit examples of hyperfinite subfactors arising from actions of property (T) groups that have different analytical properties, but the representation categories are the same.

• 09/17/2025 - Alex Betz (NC State)
  Title: Actions of Fusion Category on Path Algebras
  Abstract: Fusion categories are rich mathematical objects that generalize finite groups and their representation categories. What does it mean for a fusion category to act on an algebra? Do these actions generalize group actions on algebras? This question has been study by many in the context of operator algebras but poses an interesting question for an associative algebra A. In this talk, we will define an action of a fusion category on an algebra and show how it relates to group actions on algebras. Then, as an application of this theory, we will apply our results to study actions of fusion categories on path algebras.
  
  

• 09/24/2025 - Francisco Ponce-Carrion (NC State)
  Title: Using Newton homotopies to find real solutions of square polynomial systems
  Abstract: A common tool in numerical algebraic geometry to find complex solutions of polynomial systems is homotopy continuation. If we wish to find only real solutions, then we could consider using Newton homotopy. Much like Newton’s method, Newton homotopy depends on a start point, or an initial guess. One of the shortcomings of Newton homotopy is that the sets of start points for which Newton homotopy fails to find a real solution can be full dimensional, and complicated to describe algebraically. In this talk, we will discuss some results on classical Newton homotopy and present a generalization of Newton homotopy that addresses some of the issues from the classical case.
  
  

• 10/01/2025 - Camryn Thompson (NC State)
  Title: Domino Tableaux and a New Cyclic Sieving Phenomenon
  Abstract: The cyclic sieving phenomenon (CSP) provides valuable data about symmetry classes of cyclic actions and has applications to representation theory. In this talk, we motivate the study of the CSP and introduce a new instance of the phenomenon on domino tableaux of shape 2xn. To do so, we first enumerate the collection of domino tableaux of this shape, then use this result to find candidates for a generating function and a cyclic action necessary for cyclic sieving. With this information, we will state our new CSP and discuss some ideas behind its proof. 
  
  

• 10/08/2025 - Anna Shapiro (NC State)
  Title: Settling two conjectures on polymatroids using a new valuative function  
  Abstract: We introduce a new valuative function called the cave polynomial of a polymatroid. Using the cave polynomial, we reduce a 2012 conjecture of Bandari, Bayati, and Herzog regarding homological shift ideals to a more combinatorial conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montaño from 2022. In this talk we will discuss the relevant background on polymatroids, the history of the two conjectures, and sketch the proof of the 2022 conjecture. Parts of this work are joint with Yairon Cid-Ruiz and Jacob P. Matherne.  
  
  

• 10/15/2025 - Daniel Profili (NC State)
  Title: Conditions for eigenvalue configurations of two real symmetric matrices  
  Abstract: Given two real symmetric matrices, their eigenvalue configuration is the relative arrangement of their eigenvalues on the real line. We consider the following problem: given two real symmetric matrices and an eigenvalue configuration, find a simple condition on the entries of the matrices so that their eigenvalues are arranged according to the given configuration. We give an algorithm which solves the problem by producing a set of related matrices whose signatures (i.e. the difference between the positive and negative eigenvalues) determine the eigenvalue configuration. The eigenvalue configuration problem generalizes Descartes' rule of signs and has many potential applications.
  
  

• 10/22/2025 - Avery St. Dizier (Michigan State University)
  Title: Presenting the Cohomology of Schubert Varieties
  Abstract: The integral cohomology ring of the complete flag variety classically admits a short presentation due to Borel. Later work of Akyildiz, Lascoux, and Pragacz extended this presentation to all Schubert varieties, though with redundancy in the relations. By introducing a type-independent generalization of Fulton’s essential set, Reiner, Woo, and Yong gave a refined and combinatorially explicit presentation for the cohomology of any Schubert variety. They conjectured that their generating sets were minimal in certain cases, which would imply exponential lower bounds on the number of generators needed in general. I will give an overview of the results of Reiner, Woo, and Yong, and discuss the ideas behind the proof of their conjecture. Based on joint work with Alexander Yong.
  
  

• 10/29/2025 - Michael Brown (Auburn University)
  Title: A Horrocks splitting criterion for noncommutative vector bundles
  Abstract: The Horrocks splitting criterion states that a vector bundle on projective space splits as a sum of line bundles if and only if its intermediate cohomology vanishes. The goal of this project is to generalize this result to noncommutative projective schemes, in the sense of Artin-Zhang. Along the way, I will explain a method for computing noncommutative sheaf cohomology, generalizing an algorithm due to Eisenbud-Fløystad-Schreyer. This is joint work with Greg Smith.
  
  

• 11/05/2025 - Ian Klein
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• 11/12/2025 - Tom Braden
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• 11/19/2025 - Karlee Westrem
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