Mar 2025, Volume 3 Issue 2

    

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• Braid Group Representations from Twisted Tensor Products of Algebras

  Paul Gustafson, Andrew Kimball, Eric C. Rowell, Qing Zhang

  2020, 3(2): 103-130. https://doi.org/10.1007/s42543-020-00023-5

  We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. We provide some general characterizations and classification of these representations, focusing on the size of their images, which are typically finite groups. The well-studied Gaussian representations associated with metaplectic modular categories can be understood in this framework, and we give some new examples to illustrate their ubiquity. Our results suggest a relationship between the braiding on the G-gaugings of a pointed modular category ${\mathcal {C}}(A,Q)$ and that of ${\mathcal {C}}(A,Q)$ itself.

• An $L^2$ to $L^\infty $ Framework for the Landau Equation

  Jinoh Kim, Yan Guo, Hyung Ju Hwang

  2020, 3(2): 131-202. https://doi.org/10.1007/s42543-019-00018-x

  Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}\ \text{to}\ L^{\infty }$ framework to construct global unique solutions near Maxwellian with small $L^{\infty }$ norm. The first step is to establish global $L^{2}$ estimates with strong velocity weight and time decay, under the assumption of $L^{\infty }$ bound, which is further controlled by such $L^{2}$ estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv:1505.04608 (2015)). The second step is to employ estimates in $S_{p}$ spaces to control velocity derivatives to ensure uniqueness, which is based on Hölder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv:1506.01908 (2015), Imbert and Mouhot in arXiv:1505.04608 (2015)).

• Quantitative Estimates on the Singular Sets of Alexandrov Spaces

  Nan Li, Aaron Naber

  2020, 3(2): 203-234. https://doi.org/10.1007/s42543-020-00026-2

  Let $X\in \text {Alex}\,^n(-1)$ be an n-dimensional Alexandrov space with curvature $\ge -1$. Let the r-scale $(k,\epsilon )$-singular set ${\mathcal {S}}^k_{\epsilon ,\,r}(X)$ be the collection of $x\in X$ so that $B_r(x)$ is not $\epsilon r$-close to a ball in any splitting space $\mathbb {R}^{k+1}\times Z$. We show that there exists $C(n,\epsilon )>0$ and $\beta (n,\epsilon )>0$, independent of the volume, so that for any disjoint collection $\big \{B_{r_i}(x_i):x_i\in {\mathcal {S}}_{\epsilon ,\,\beta r_i}^k(X)\cap B_1, \,r_i\le 1\big \}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates ${\mathcal {H}}^k({\mathcal {S}}^k_\epsilon (X)\cap B_1)\le C$ and ${\mathcal {H}}^n\big (B_r ({\mathcal {S}}^k_{\epsilon ,\,r}(X))\cap B_1(p)\big )\le C\,r^{n-k}$. This answers an open question in Kapovitch et al. (Metric-measure boundary and geodesic flow on Alexandrov spaces. arXiv:1705.04767 (2017)). We also show that the k-singular set $\textstyle{\mathcal {S}}^k(X)=\bigcup_{\epsilon>0}\big(\bigcap_{r>0}{\mathcal {S}}^k_{\epsilon ,\,r}\big)$ is k-rectifiable and construct examples to show that such a structure is sharp. For instance, in the $k=1$ case we can build for any closed set $T\subseteq \mathbb {S}^1$ and $\epsilon >0$ a space $Y\in \text {Alex}^3(0)$ with ${\mathcal {S}}^{1}_\epsilon (Y)=\phi (T)$, where $\phi :\mathbb {S}^1\rightarrow Y$ is a bi-Lipschitz embedding. Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable, 1-Cantor set with positive 1-Hausdorff measure.