• Date: 1/17/25
  • Darren Creutz, Vanderbilt University
  • Title: Subshifts of non-superlinear complexity cannot be mixing
  • Abstract: I will present the companion result to my proof of the existence of mixing subshifts with word complexity arbitrarily close to linear, namely that any subshift which does not have superlinear word complexity must be partially rigid hence cannot be (strongly) mixing.
    More specifically, if p(q) is the number of words of length q appearing in a subshift and \liminf p(q)/q < infty then there exists a constant 0 < c \leq 1 such that every invariant probability measure \mu for the subshift, there is measure-theoretic partial rigidity of at least c: for all measurable sets B, \limsup \mu(T^{n} B \cap B) \geq c \mu(B). The tools involved range from graph theory and combinatorics to classical ergodic theory.

• Date: 1/24/25 (Zoom Meeting ID: 960 7137 8075)
  • Larissa Kroell, University of Waterloo
  • Title: The ideal intersection property for partial reduced crossed products
  • Abstract: Given a C*-dynamical system, a fruitful avenue to study its properties has been to study the dynamics on its injective envelope. This approach relies on the result of Kalantar and Kennedy (2017), who show that C*-simplicity can be characterized via the Furstenberg boundary using G-injective envelope techniques. In this talk, we will discuss consequences of this idea for partial C*-dynamical systems. In particular, we introduce partial G-injective envelopes and generalize techniques for ordinary C*-dynamical systems given in Kennedy and Schafhauser (2019) to partial C*-dynamical systems. As an application of this machinery, we give a characterization of the ideal intersection property for partial C*-dynamical systems. This is joint work with Matthew Kennedy and Camila Sehnem.

• Date: 1/31/25
  • Terry Adams, U.S. Government (retired)
  • Title: Is it reasonable to ask for universally convergent moving averages?
  • Abstract: This is joint work with Joe Rosenblatt. We consider pointwise convergence of moving averages of the form: \[ M(v_n, L_n)^T f = \frac{1}{L_n} \sum_{i=v_n+1}^{v_n+L_n} f \circ T^i \] where $L_n \geq n$, $T$ is an ergodic transformation preserving a measure $\mu$ and $f \in L^p(\mu)$ for some $p \geq 1$. For non-zero $f\in L^1(\mu)$, there is a generic class of ergodic transformations $T$ such that each transformation has an associated moving average which does not converge pointwise. By solving the coboundary equation, we show if $f \in L^2(\mu)$, there always exists an ergodic transformation $T$ with universally convergent moving averages for $L_n \geq n$. However, this does not generalize to $L^p(\mu)$ for $p<2$. We explicitly define functions $f\in \bigcap_{p<2} L^p(\mu)$ such that for each ergodic measure preserving $T$, there exists a moving average which does not converge pointwise. Several of the results are generalized to the case of moving averages where $L_n$ has polynomial growth. This is joint work with Joe Rosenblatt.

• Date: 2/7/25
  • Itamar Vigdorovich, UC San Diego
  • Title: The trace simplex of groups and C*-algebras
  • Abstract: The space of traces/characters of a group, and more generally a C*-algebra, is well known to be a simplex. For a group with Kazhdan property (T), the space of traces is a Bauer simplex - its extreme points are closed. For the free group, and more general free products, the space of traces is a Poulsen simplex - its extreme points are dense. I will discuss these results, proof ideas, and their applications. The talk is based on several works with co-authors including Ioana, Levit, Orovitz, Slutsky, Spaas.

• Date: 2/14/25
  • Caleb Eckhardt, Miami University
  • Title: Hilbert-Schmidt stability and residually finite amenable groups
  • Abstract: Loosely speaking, a group G is Hilbert-Schmidt (HS) stable if every approximate homomorphism from G to a finite-dimensional unitary group (equipped with the HS-norm) is close to an actual homomorphism. It is straightforward to see that any finitely generated amenable HS-stable group is residually finite. I'll describe some groups that show the converse does not hold and the connections of these examples to other forms of stability and topological dynamics. The examples are easy to describe, so depending on the patience of the audience, we could construct the almost homomorphisms that cannot be perturbed to true homomorphisms. Time permitting I'll also talk about related joint work with Tatiana Shulman by describing some amenable groups that are HS-stable but not permutation stable.

• Date: 2/21/25
  • Behrang Forghani, College of Charleston
  • Title: Boundaries and Entropy of Random Walks on Groups
  • Abstract: One of the central questions in the theory of random walks on groups is how the long-term behavior of these walks relates to the algebraic or geometric structure of the group. In the 1960s, Furstenberg introduced the Poisson boundary to describe the stochastically significant behavior of a random walk at infinity. In this talk, we will begin with an introduction to random walks on groups, harmonic functions, the construction of the Poisson boundary, and characterizations of groups based on their Poisson boundaries. We will also discuss the role of entropy in identifying the Poisson boundary of random walks on groups, particularly those endowed with additional geometric, algebraic, or combinatorial structures (e.g., free groups or, more generally, groups acting on hyperbolic spaces).

• Date: 3/14/25
  • No Meeting, Spring Break

• Date: 3/21/25
  • Scott Schmieding, Penn State University
  • Title: Flow equivalence and PSL_2(Q)-equivalence
  • Abstract: A real number gives rise to a Sturmian system encoding a rotation of the circle, and there are several beautiful connections between these systems and arithmetic properties of the associated parameters. One is a result of Fokkink, which shows that two Sturmian subshifts with parameters \alpha and \beta are flow equivalent if and only if \alpha and \beta lie in the same orbit of the action of PSL_2(Z) on the set of reals via Mobius transformations, a condition which is itself characterized by the tails of their continued fraction expansions. I'll describe some recent work, joint with Christopher-Lloyd Simon, describing the action of PSL_2(Q) in terms of a certain relation on minimal Cantor systems called isogeny.

• Date: 3/28/25
  • Ronnie Pavlov, University of Denver
  • Title: Word complexity, automorphism groups, and some unexpected connections
  • Abstract: A general heuristic for a subshift is that if its word complexity function is small, then its group Aut(X) of automorphisms, which are self-homeomorphisms of X which commute with the shift action, should be "small." There are many interesting results in recent years of this sort. I will summarize some of them, particularly one of myself and Schmieding: if word complexity grows more slowly than n(log log n)^C for every C > 0, then Aut(X), after modding out by the subgroup generated by the shift, is locally finite. I'll give a brief outline of the proof of this result, which involved an interesting detour into geometric group theory and an auspicious Google search when all seemed lost.

• Date: 4/4/25
  • Gennadi Kasparov, Vanderbilt University
  • Title: Index theory on manifolds with a tangent Lie structure
  • Abstract: In recent years there was a significant progress in the theory of pseudo-differential operators on filtered manifolds. In my talk I will introduce a wider class of manifolds which I call manifolds with a tangent Lie structure. I will explain a coarse approach to pseudo-differential theory which gives a simplified pseudo-differential calculus containing only operators of order 0 and negative order. This calculus easily leads to the Atiyah-Singer type index theorem for operators of order 0 on manifolds with a tangent Lie structure. For filtered manifolds this calculus agrees with the known Hormander and van Erp - Yuncken calculi, which allows to extend the index theorem to operators of any order.

• Date: 4/11/25
  • Zhengwei Liu, Yau Mathematical Sciences Center, Tsinghua Universtiy
  • Title: How rare subfactors are?
  • Abstract: In various classifications, subfactors are quite rare comparing to the computational complexity. Based on a striking correspondence between exchange relations and forests, we propose a new classification scheme in which the rareness are quantified. We introduce a series of efficient sieving methods to classify and to discover subfactors. It is based on joint work with F. Lu, arXiv:2412.17790.

• Date: 4/18/25
  • Changying Ding, UCLA
  • Title: Relative solidity in measure equivalence and its applications
  • Abstract: In his seminal paper, Ozawa demonstrated the solidity property for ${\rm II}_1$ factor arising from Biexact groups. In this talk, I will discuss a relative version of the solidity property for biexact groups in the setting of measure equivalence and its applications to measure equivalence rigidity. This is a joint work with Daniel Drimbe.

• End of Spring Semester.

Past NCGOA and Subfactor seminars