• Date: 1/16/26
  • Georgios Tsikalas, Vanderbilt University
  • Title: Holomorphic Interpolation: Old and New Results
  • Abstract: Suppose you are given points $z_1, \dots, z_n$ and  $w_1, \dots, w_n$ in the unit disk of the complex plane. Does there exist a holomorphic self-map $f$ of the disk such that $f(z_i)=w_i$ for all $i$? Pick's solution of this problem in 1915 impacted the development of function theory throughout the twentieth century. In 1967, Sarason gave an operator-theoretic reformulation of Pick's result and proved his seminal commutant lifting theorem, which was subsequently generalized into a powerful tool that encodes, unifies and extends a variety of classical interpolation theorems on the disk.
    
    In this talk, I will discuss Sarason's approach to the Pick problem and how it connects to a certain reproducing kernel Hilbert space. Further, I will explain how Sarason's theorem can be extended to more general spaces. 
    
    This is joint work with Scott McCullough (University of Florida).

• Date: 1/23/26 **Zoom Meeting @6:10pm**
  • Zishuo Zhao, Tsinghua University
  • Title: Relative Entropy for Quantum Channels
  • Abstract: We propose a notion of relative entropy between quantum channels with respect to an input state over von Neumann algebras. This notion generalizes the Pimsner-Popa entropy of finite index inclusions. Using the quantum Fourier transform, we make a connection between this relative entropy and the relative entropy between states. When an inclusion admits a downward Jones basic construction, we derive a formula for the relative entropy of quantum channels in terms of their Fourier multipliers.

• Date: 2/6/26
  • No meeting due to winter storm and its aftermath

• Date: 2/10/26 **Special day, 4:10pm-5:30pm, SC 1432**
  • David Gao, UCSD
  • Title: Handle constructions on von Neumann algebras
  • Abstract: The problem of classifying non-Gamma II_1 factors up to elementary equivalence is a challenging one. Recently, Chifan-Ioana-Kunnawalkam Elayavalli discovered an interesting new invariant and thereby found two non-Gamma factors with non-isomorphic ultrapowers. There are several technical challenges that underpin this work, including the usage of Property (T), deformation/rigidity methods, 1-bounded entropy, and lifting/perturbation to independent unitaries. I will shed light on some of these and also discuss several new developments that are able to bypass these and generalize the construction to arrive at various applications. This is based on joint work with Kunnawalkam Elayavalli and Patchell, and a new work joint with Jekel, Kunnawalkam Elayavalli and Patchell.

• Date: 2/20/26
  • Darren Creutz, Vanderbilt University
  • Title: Actions of Lattice Subgroups of Higher-Rank Semisimple Lie Groups
  • Abstract: Lattice subgroups—discrete subgroups with finite covolume—play a key role in number theory, dynamics, and a variety of other fields of math.  One striking feature is that every (continuous, minimal) action of such a lattice subgroup on a compact metric space is free (C ’23, also independently by Peterson et al).  I will give a historical overview of this and related results and discuss the key ideas in the proofs, most notably the Furstenberg-Poisson Boundary construction which exists for all (locally compact) groups and has been a key ingredient in a wide array of proofs.

• Date: 2/27/26
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• Date: 3/5/26 **Special day, Dissertation Defense Junwhi Lim, 10:00am-12:00pm, Garland 234**
  • Junhwi Lim, Vanderbilt Universty
  • Title: Planar algebras associated to cocommuting squares
  • Abstract: Dissertation Defense.

    The ‘generalized symmetries’ of subfactors are encoded by their planar algebras. A natural foundational question is ``What minimal algebraic structure must always be present in these symmetries?'' For arbitrary subfactors, Jones showed that the associated planar algebra contains the Temperley-Lieb-Jones algebra. In the presence of a single intermediate subfactor, the associated planar algebra was identified by Bisch and Jones. However, the case of two intermediate subfactors remains open. As a natural special case, we discuss cocommuting squares of factors with ‘group-like’ properties. We exhibit the skein relations of their associated planar algebras and show that these algebras extend the partition algebras. This talk is based on joint work with Dietmar Bisch.

• Date: 3/6/26
  • No meeting(tentative)
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• Date: 3/13/26
  • No meeting, Spring Break

• Date: 3/20/26
  • Madeline Brandt, Vanderbilt University
  • Title:
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• Date: 3/26/26 **Mathematics Colloquium, 4:10pm-5:15pm, SC 5206**
  • Feng Xu, UC Riverside
  • Title:
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• Date: 3/27/26
  • (talk reserved for a participant of the Shanks workshop)
  • Title:
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• Date: 3/28/26 & 3/29/26
  • Shanks workshop ``Von Neumann Entropy and Quantum Symmetries''

• Date: 4/3/26
  • Junhwi Lim, Vanderbilt University
  • Title: Subfactors, tensor categories, and the Minkowski integral inequality
  • Abstract: The construction of subfactors and unitary tensor categories typically requires solving large systems of equations, namely the biunitary and pentagon equations. These equations arise from inclusion matrices and based (or fusion) rings. There is no known systematic method for solving them, and in many cases no solution exists. We introduce an a priori criterion for excluding such cases without having to solve the equations. The criterion is based on a noncommutative analogue of the Minkowski integral inequality.

• Date: 4/10/26
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• Date: 4/17/26
  • Scott Schmieding, Penn State University (tentative)
  • Title:
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• End of Spring Semester.

Past NCGOA and Subfactor seminars