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1. A unipotent circle action on 𝑝-adic modular forms

   https://doi.org/10.1090/btran/52  

   Howe, Sean (November 2020, Transactions of the American Mathematical Society, Series B)

   null (Ed.)

   Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log ⁡ q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions. 

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2. Universal minimal flows of extensions of and by compact groups

   https://doi.org/10.1017/etds.2022.52  

   BARTOŠOVÁ, DANA (August 2023, Ergodic Theory and Dynamical Systems)

   Abstract Every topological group G has, up to isomorphism, a unique minimal G -flow that maps onto every minimal G -flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K . Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $$G/K\ltimes K$$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $$2^{\mathbb {N}}$$ , $$M(\mathbb {Z})$$ , or $$M(\mathbb {Z})\times 2^{\mathbb {N}}.$$ 

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3. Machine Learning and Invariant Theory

   https://doi.org/10.1090/noti2760  

   Blum-Smith, Ben; Villar, Soledad (September 2023, Notices of the American Mathematical Society)

   Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group G, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that G is a compact Lie group. 

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4. Quasinormalizers in crossed products of von Neumann algebras

   https://doi.org/10.1016/j.aim.2024.109535  

   Bannon, Jon P; Cameron, Jan; Chifan, Ionuţ; Mukherjee, Kunal; Smith, Roger; Wiggins, Alan (May 2024, Advances in Mathematics)

   We study the relationship between the dynamics of the action $$\alpha$$ of a discrete group $$G$$ on a von Neumann algebra $$M$$, and structural properties of the associated crossed product inclusion $$L(G) \subseteq M \rtimes_\alpha G$$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $$L(G) \subseteq M \rtimes_\alpha G$$ such as the Haagerup Approximation Property, and is essential to capturing ``almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras. 

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5. COUNTING CONJUGACY CLASSES IN

   https://doi.org/10.1017/S0004972718000047  

   HULL, MICHAEL; KAPOVICH, ILYA (June 2018, Bulletin of the Australian Mathematical Society)

   We show that if a finitely generated group $$G$$ has a nonelementary WPD action on a hyperbolic metric space $$X$$ , then the number of $$G$$ -conjugacy classes of $$X$$ -loxodromic elements of $$G$$ coming from a ball of radius $$R$$ in the Cayley graph of $$G$$ grows exponentially in $$R$$ . As an application we prove that for $$N\geq 3$$ the number of distinct $$\text{Out}(F_{N})$$ -conjugacy classes of fully irreducible elements $$\unicode[STIX]{x1D719}$$ from an $$R$$ -ball in the Cayley graph of $$\text{Out}(F_{N})$$ with $$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$$ of the order of $$R$$ grows exponentially in $$R$$ . 

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