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1. Conformal blocks for Galois covers of algebraic curves

   https://doi.org/10.1112/S0010437X23007418  

   Hong, Jiuzu; Kumar, Shrawan (October 2023, Compositio Mathematica)

   We study the spaces of twisted conformal blocks attached to a$$\Gamma$$-curve$$\Sigma$$with marked$$\Gamma$$-orbits and an action of$$\Gamma$$on a simple Lie algebra$$\mathfrak {g}$$, where$$\Gamma$$is a finite group. We prove that if$$\Gamma$$stabilizes a Borel subalgebra of$$\mathfrak {g}$$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$$\Gamma$$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$$\mathscr {G}$$be the parahoric Bruhat–Tits group scheme on the quotient curve$$\Sigma /\Gamma$$obtained via the$$\Gamma$$-invariance of Weil restriction associated to$$\Sigma$$and the simply connected simple algebraic group$$G$$with Lie algebra$$\mathfrak {g}$$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$$\mathscr {G}$$-torsors on$$\Sigma /\Gamma$$when the level$$c$$is divisible by$$|\Gamma |$$(establishing a conjecture due to Pappas and Rapoport). 

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2. 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. 

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3. Non-isomorphic smooth compactifications of the moduli space of cubic surfaces

   https://doi.org/10.1017/nmj.2023.27  

   Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus; Laza, Radu (June 2024, Nagoya Mathematical Journal)

   Abstract The well-studied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient$${\mathcal {M}}^{\operatorname {GIT}}$$, as a Baily–Borel compactification of a ball quotient$${(\mathcal {B}_4/\Gamma )^*}$$, and as a compactifiedK-moduli space. From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup$${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$$, whereas from the ball quotient point of view, it is natural to consider the toroidal compactification$${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$$. The spaces$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in factnotthe case. Indeed, we show the more refined statement that$${\mathcal {M}}^{\operatorname {K}}$$and$${\overline {\mathcal {B}_4/\Gamma }}$$are equivalent in the Grothendieck ring, but notK-equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients. 

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4. On the ergodic theory of the real Rel foliation

   https://doi.org/10.1017/fmp.2024.6  

   Chaika, Jon; Weiss, Barak (January 2024, Forum of Mathematics, Pi)

   Abstract Let$${{\mathcal {H}}}$$be a stratum of translation surfaces with at least two singularities, let$$m_{{{\mathcal {H}}}}$$denote the Masur-Veech measure on$${{\mathcal {H}}}$$, and let$$Z_0$$be a flow on$$({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$$obtained by integrating a Rel vector field. We prove that$$Z_0$$is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces$$({\mathcal L}, m_{{\mathcal L}})$$, where$${\mathcal L} \subset {{\mathcal {H}}}$$is an orbit-closure for the action of$$G = \operatorname {SL}_2({\mathbb {R}})$$(i.e., an affine invariant subvariety) and$$m_{{\mathcal L}}$$is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of$$Z_0$$with respect to any of the measures$$m_{{{\mathcal L}}}$$is zero. 

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5. $L^p$ regularity of the Bergman projection on the symmetrized polydisc

   https://doi.org/10.4153/S0008414X24000634  

   Huo, Zhenghui; Wick, Brett D (October 2024, Canadian Journal of Mathematics)

   Abstract We study the$$L^p$$regularity of the Bergman projectionPover the symmetrized polydisc in$$\mathbb C^n$$. We give a decomposition of the Bergman projection on the polydisc and obtain an operator equivalent to the Bergman projection over antisymmetric function spaces. Using it, we obtain the$$L^p$$irregularity ofPfor$$p=\frac {2n}{n-1}$$which also implies thatPis$$L^p$$bounded if and only if$$p\in (\frac {2n}{n+1},\frac {2n}{n-1})$$. 

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