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1. Constructions of Lagrangian cobordisms

   https://doi.org/10.1007/978-3-030-80979-9  

   Blackwell, Sarah; Legout, Noémie; Leverson, Caitlin; Limouzineau, Maÿlis; Myer, Ziva; Pan, Yu; Pezzimenti, Samantha; Simone Suárez, Lara; Traynor, Lisa (January 2021, Association for Women in Mathematics series)

   Acu, Bahar; Cannizzo, Catherine; McDuff, Dusa; Myer, Ziva; Pan, Yu; Traynor, Lisa (Ed.)

   Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known ``elementary'' building blocks for Lagrangian cobordisms that are smoothly the attachment of 0- and 1-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is ``decomposable'' into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms. 

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2. Confined monopoles and failure of the Lattice Weak Gravity Conjecture

   https://doi.org/10.1007/JHEP10(2025)186  

   Etheredge, Muldrow; Heidenreich, Ben; Pittman, Nicholas; Rauch, Sebastian; Reece, Matthew; Rudelius, Tom (October 2025, Journal of High Energy Physics)

   Almost all known theories of quantum gravity satisfy the Lattice Weak Gravity Conjecture (LWGC), which posits that a consistent theory of quantum gravity must have a superextremal particle at every site in the charge lattice. However, a number of theories have been observed to violate the LWGC; such theories exhibit only a (finite index) sublattice of superextremal particles. This paper aims to identify universal features and patterns associated with LWGC violation across numerous examples in effective field theory, string theory, and M-theory. Some of these examples have appeared previously in the literature, while others are novel. In all such examples, we observe that LWGC failure is accompanied by the existence of fractionally charged monopoles confined by flux tubes, where superextremal particles exist everywhere in the sublattice dual to the superlattice of fractional confined monopole charges. The confining flux tubes become light when the failure of the LWGC becomes more extreme, so monopoles deconfine in the limit where LWGC-violating particles become infinitely massive. We also identify similarities between these confined monopoles, non-invertible symmetries, and the Hanany-Witten effect. 

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3. Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

   https://doi.org/10.1515/Crelle-2024-0003  

   Lim, Geunho; Weinberger, Shmuel (February 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   .We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated. 

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4. Non‐Flat Universes and Black Holes in Asymptotically Free Mimetic Gravity

   https://doi.org/10.1002/prop.201900103  

   Chamseddine, Ali H.; Mukhanov, Viatcheslav; Russ, Tobias B. (December 2019, Fortschritte der Physik)

   Abstract The recently proposed theory of “Asymptotically Free Mimetic Gravity” is extended to the general non‐homogeneous, spatially non‐flat case. We present a modified theory of gravity which is free of higher derivatives of the metric. In this theory asymptotic freedom of gravity implies the existence of a minimal black hole with vanishing Hawking temperature. Introducing a spatial curvature dependent potential, we moreover obtain non‐singular, bouncing modifications of spatially non‐flat Friedmann and Bianchi universes. 

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5. Topological operators and completeness of spectrum in discrete gauge theories

   https://doi.org/10.1007/JHEP12(2020)172  

   Rudelius, Tom; Shao, Shu-Heng (December 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions, we show that completeness of the spectrum is equivalent to the absence of certain Gukov-Witten topological operators. We further extend our analysis to four and higher spacetime dimensions. Since topological operators are natural generalizations of global symmetries, we discuss evidence for their absence in a consistent theory of quantum gravity. 

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