More Like this

1. DERIVATIVES OF THETA FUNCTIONS AS TRACES OF PARTITION EISENSTEIN SERIES

   https://doi.org/10.1017/nmj.2024.30  

   AMDEBERHAN, TEWODROS; ONO, KEN; SINGH, AJIT (June 2025, Nagoya Mathematical Journal)

   Abstract In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences ofq-series$$\{U_{2t}(q)\}$$and$$\{V_{2t}(q)\}$$that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series$$E_{2k}(q),$$defined by$$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$ For functions$$\phi : \mathcal {P}\mapsto {\mathbb C}$$on partitions, theweight$$2n$$partition Eisenstein traceis$$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$ For allt, we prove that$$U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$$and$$V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$$where$$\phi _U$$and$$\phi _V$$are natural partition weights, giving the first explicit quasimodular formulas for these series. 

   more » « less
2. Almost all orbits of the Collatz map attain almost bounded values

   https://doi.org/10.1017/fmp.2022.8  

   Tao, Terence (January 2022, Forum of Mathematics, Pi)

   Abstract Define theCollatz map$${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$$on the positive integers$$\mathbb {N}+1 = \{1,2,3,\dots \}$$by setting$${\operatorname {Col}}(N)$$equal to$$3N+1$$whenNis odd and$$N/2$$whenNis even, and let$${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$$denote the minimal element of the Collatz orbit$$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $$. The infamousCollatz conjectureasserts that$${\operatorname {Col}}_{\min }(N)=1$$for all$$N \in \mathbb {N}+1$$. Previously, it was shown by Korec that for any$$\theta> \frac {\log 3}{\log 4} \approx 0.7924$$, one has$${\operatorname {Col}}_{\min }(N) \leq N^\theta $$for almost all$$N \in \mathbb {N}+1$$(in the sense of natural density). In this paper, we show that foranyfunction$$f \colon \mathbb {N}+1 \to \mathbb {R}$$with$$\lim _{N \to \infty } f(N)=+\infty $$, one has$${\operatorname {Col}}_{\min }(N) \leq f(N)$$for almost all$$N \in \mathbb {N}+1$$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$$3$$-adic cyclic group$$\mathbb {Z}/3^n\mathbb {Z}$$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency. 

   more » « less
3. TORSION-FREE ABELIAN GROUPS OF FINITE RANK AND FIELDS OF FINITE TRANSCENDENCE DEGREE

   https://doi.org/10.1017/jsl.2025.14  

   HO, MENG-CHE TURBO; KNIGHT, JULIA FRANDSEN; MILLER, RUSSELL GEDDES (February 2025, The Journal of Symbolic Logic)

   Abstract Let$$\operatorname {TFAb}_r$$be the class of torsion-free abelian groups of rankr, and let$$\operatorname {FD}_r$$be the class of fields of characteristic$$0$$and transcendence degreer. We compare these classes using various notions. Considering the Scott complexity of the structures in the classes and the complexity of the isomorphism relations on the classes, the classes seem very similar. Hjorth and Thomas showed that the$$\operatorname {TFAb}_r$$are strictly increasing under Borel reducibility. This is not so for the classes$$\operatorname {FD}_r$$. Thomas and Velickovic showed that for sufficiently larger, the classes$$\operatorname {FD}_r$$are equivalent under Borel reducibility. We try to compare the groups with the fields, using Borel reducibility, and also using some effective variants. We give functorial Turing computable embeddings of$$\operatorname {TFAb}_r$$in$$\operatorname {FD}_r$$, and of$$\operatorname {FD}_r$$in$$\operatorname {FD}_{r+1}$$. We show that under computable countable reducibility,$$\operatorname {TFAb}_1$$lies on top among the classes we are considering. In fact, under computable countable reducibility, isomorphism on$$\operatorname {TFAb}_1$$lies on top among equivalence relations that are effective $$\Sigma _3$$. 

   more » « less
4. Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers

   https://doi.org/10.1017/fms.2025.5  

   Bezrukavnikov, Roman; Varshavsky, Yakov (January 2025, Forum of Mathematics, Sigma)

   Abstract LetGbe a connected reductive group over an algebraically closed fieldk, and let$$\operatorname {Fl}$$be the affine flag variety ofG. For every regular semisimple element$$\gamma $$of$$G(k((t)))$$, the affine Springer fiber$$\operatorname {Fl}_\gamma $$can be presented as a union of closed subvarieties$$\operatorname {Fl}^{\leq w}_{\gamma }$$, defined as the intersection of$$\operatorname {Fl}_{\gamma }$$with an affine Schubert variety$$\operatorname {Fl}^{\leq w}$$. The main result of this paper asserts that if elements$$w_1,\ldots ,w_n$$are sufficiently regular, then the natural map$$H_i(\bigcup _{j=1}^n \operatorname {Fl}^{\leq w_j}_{\gamma })\to H_i(\operatorname {Fl}_{\gamma })$$is injective for every$$i\in \mathbb Z$$. It plays an important role in our work [BV], where our result is used to construct good filtrations of$$H_i(\operatorname {Fl}_{\gamma })$$. Along the way, we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits. 

   more » « less
5. Asymptotic freeness in tracial ultraproducts

   https://doi.org/10.1017/fms.2024.93  

   Houdayer, Cyril; Ioana, Adrian (January 2024, Forum of Mathematics, Sigma)

   Abstract We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever$$M = M_1 \ast M_2$$is a tracial free product von Neumann algebra and$$u_1 \in \mathscr U(M_1)$$,$$u_2 \in \mathscr U(M_2)$$are Haar unitaries, the relative commutants$$\{u_1\}' \cap M^{\mathcal U}$$and$$\{u_2\}' \cap M^{\mathcal U}$$are freely independent in the ultraproduct$$M^{\mathcal U}$$. Our proof relies on Mei–Ricard’s results [MR16] regarding$$\operatorname {L}^p$$-boundedness (for all$$1 < p < +\infty $$) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan–Ioana–Kunnawalkam Elayavalli’s recent construction [CIKE22] to provide the first example of a$$\mathrm {II_1}$$factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras. 

   more » « less