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1. “Lagrangian disks” in M-theory

   https://doi.org/10.1007/JHEP11(2020)033  

   Franco, Sebastían; Gukov, Sergei; Lee, Sangmin; Seong, Rak-Kyeong; Sparks, James (November 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G 2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T 2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T [ M 4 ] on the other. 

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2. Maximal Brill–Noether Loci via K3 Surfaces

   https://doi.org/10.1093/imrn/rnaf313  

   Auel, Asher; Haburcak, Richard (October 2025, International Mathematics Research Notices)

   Abstract The Brill–Noether loci $$\mathcal{M}^{r}_{g,d}$$ parameterize curves of genus $$g$$ admitting a linear system of rank $$r$$ and degree $$d$$. When the Brill–Noether number is negative, they are proper subvarieties of the moduli space of genus $$g$$ curves. We explain a strategy for distinguishing Brill–Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill–Noether loci. Via an analysis of the stability of Lazarsfeld–Mukai bundles, we obtain new lifting results for line bundles of type $$g^{3}_{d}$$ that suffice to prove the maximal Brill–Noether loci conjecture in genus $$3$$–$19$, $22$, $23$, and infinitely many cases. 

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3. Asymptotic plateau problem in $$\mathbb{H}^2x\mathbb{R}$$: Tall curves

   https://doi.org/10.1512/iumj.2023.72.9051  

   Coskunuzer, Baris (November 2023, Indiana University Mathematics Journal)

   Demeter, Ciprian (Ed.)

   We study the asymptotic Plateau problem in H2 × R for area-minimizing surfaces, and give a fairly complete solution for finite curves. 

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4. Quantum representations and monodromies of fibered links.

   Renaud Detcherry, Efstratia Kalfagianni (July 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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5. Cosets of monodromies and quantum representations

   Renaud Detcherry, Efstratia Kalfagianni (April 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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