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1. Enriched string-net models and their excitations

   https://doi.org/10.22331/q-2024-03-28-1301  

   Green, David; Huston, Peter; Kawagoe, Kyle; Penneys, David; Poudel, Anup; Sanford, Sean (March 2024, Quantum)

   Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $\mathcal{A}$representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an $\mathcal{A}$-enriched unitary fusion category $\mathcal{X}$on a 2D boundary of the 3D Walker-Wang model associated to $\mathcal{A}$. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer $Z^{\mathcal{A}}\left(\mathcal{X}\right)$of $\mathcal{A}$in $Z\left(\mathcal{X}\right)$.In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center $Z_2\left(\mathcal{A}\right)$, and we construct bulk-to-boundary hopping operators $Z_2\left(\mathcal{A}\right)\to Z^{\mathcal{A}}\left(\mathcal{X}\right)$reflecting how the UMTC of boundary excitations $Z^{\mathcal{A}}\left(\mathcal{X}\right)$is symmetric-braided enriched in $Z_2\left(\mathcal{A}\right)$.This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal $6j$symbol viewpoint. 

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2. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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3. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

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4. Mapping class groups of exotic tori and actions by 𝑆𝐿_{𝑑}(𝐙)

   https://doi.org/10.1090/btran/207  

   Bustamante, Mauricio; Krannich, Manuel; Kupers, Alexander; Tshishiku, Bena (November 2024, Transactions of the American Mathematical Society, Series B)

   We determine for which exotic tori $\mathcal{T}$of dimension $d\ne <\#comment/>4$the homomorphism from the group of isotopy classes of orientation-preserving diffeomorphisms of $\mathcal{T}$to ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$given by the action on the first homology group is split surjective. As part of the proof we compute the mapping class group of all exotic tori $\mathcal{T}$that are obtained from the standard torus by a connected sum with an exotic sphere. Moreover, we show that any nontrivial ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$-action on $\mathcal{T}$agrees on homology with the standard action, up to an automorphism of ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$. When combined, these results in particular show that many exotic tori do not admit any nontrivial differentiable action by ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$. 

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5. Intersections of dual 𝑆𝐿₃-webs

   https://doi.org/10.1090/tran/9349  

   Shen, Linhui; Sun, Zhe; Weng, Daping (August 2025, Transactions of the American Mathematical Society)

   We introduce a topological intersection number for an ordered pair of ${\mathrm{SL}}_3$-webs on a decorated surface. Using this intersection pairing between reduced $({\mathrm{SL}}_3,\mathcal{A})$-webs and a collection of $({\mathrm{SL}}_3,\mathcal{X})$-webs associated with the Fock–Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced $({\mathrm{SL}}_3,\mathcal{A})$-webs to the tropical set ${\mathcal{A}}_{{\mathrm{PGL}}_3,\stackrel{^<\#comment/>}{S}}^{+}\left({\mathbb{Z}}^t\right)$, as established by Douglas and Sun in [Forum Math. Sigma 12 (2024), p. e5, 55]. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock–Goncharov duality conjecture of higher Teichmüller spaces for ${\mathrm{SL}}_3$. 

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