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1. Almost isotropic Kähler manifolds

   https://doi.org/10.1515/crelle-2019-0030  

   Schmidt, Benjamin ; Shankar, Krishnan ; Spatzier, Ralf ( October 2020 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let M be a complete Riemannian manifold and suppose {p\in M} . For each unit vector {v\in T_{p}M} , the Jacobi operator , {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds , i.e. manifolds having the property that for each {p\in M} , there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}} . 

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2. Rigidity of nonpositively curved manifolds with convex boundary

   https://doi.org/10.1090/proc/16475  

   Ghomi, Mohammad ; Spruck, Joel ( November 2023 , Proceedings of the American Mathematical Society)

   We show that a compact Riemannian$3$-manifold$M$with strictly convex simply connected boundary and sectional curvature$K\leq a\leq 0$is isometric to a convex domain in a complete simply connected space of constant curvature$a$, provided that$K\equiv a$on planes tangent to the boundary of$M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard$3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for$K\geq a\geq 0$.

    

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3. Borel and Volume Classes for Dense Representations of Discrete Groups

   https://doi.org/10.1093/imrn/rnab078  

   Farre, James ( April 2021 , International Mathematics Research Notices)

   Abstract We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties. 

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4. Individual differences and long-term consequences of tDCS augmented cognitive training

   Katz, B. ; Au, J. ; Buschkuehl, M. ; Abagis, T. ; Zabel, C. ; Jaeggi, S. ; Jonides, J. ( October 2017 , Journal of cognitive neuroscience)

   A gr e at d e al of i nt er e st s urr o u n d s t h e u s e of tr a n s cr a ni al dir e ct c urr e nt sti m ul ati o n (t D C S) t o a u g m e nt c o g niti v e tr ai ni n g. H o w e v er, eff e ct s ar e i n c o n si st e nt a cr o s s st u di e s, a n d m et aa n al yti c e vi d e n c e i s mi x e d, e s p e ci all y f o r h e alt h y, y o u n g a d ult s. O n e m aj or s o ur c e of t hi s i n c o n si st e n c y i s i n di vi d u al diff er e n c e s a m o n g t h e p arti ci p a nt s, b ut t h e s e diff er e n c e s ar e r ar el y e x a mi n e d i n t h e c o nt e xt of c o m bi n e d tr ai ni n g/ sti m ul ati o n st u di e s. I n a d diti o n, it i s u n cl e ar h o w l o n g t h e eff e ct s of sti m ul ati o n l a st, e v e n i n s u c c e s sf ul i nt er v e nti o n s. S o m e st u di e s m a k e u s e of f oll o w- u p a s s e s s m e nt s, b ut v er y f e w h a v e m e a s ur e d p erf or m a n c e m or e t h a n a f e w m o nt hs aft er a n i nt er v e nti o n. H er e, w e utili z e d d at a fr o m a pr e vi o u s st u d y of t D C S a n d c o g niti v e tr ai ni n g [ A u, J., K at z, B., B u s c h k u e hl, M., B u n arj o, K., S e n g er, T., Z a b el, C., et al. E n h a n ci n g w or ki n g m e m or y tr ai ni n g wit h tr a n scr a ni al dir e ct c urr e nt sti m ul ati o n. J o u r n al of C o g niti v e N e u r os ci e n c e, 2 8, 1 4 1 9 – 1 4 3 2, 2 0 1 6] i n w hi c h p arti ci p a nts tr ai n e d o n a w or ki n g m e m or y t as k o v er 7 d a y s w hil e r e c ei vi n g a cti v e or s h a m t D C S. A n e w, l o n g er-t er m f oll o w- u p t o a ss es s l at er p erf or m a n c e w a s c o n d u ct e d, a n d a d diti o n al p arti ci p a nt s w er e a d d e d s o t h at t h e s h a m c o n diti o n w a s b ett er p o w er e d. W e a s s e s s e d b a s eli n e c o g niti v e a bilit y, g e n d er, tr ai ni n g sit e, a n d m oti v ati o n l e v el a n d f o u n d si g nifi c a nt i nt er a cti o ns b et w e e n b ot h b as eli n e a bilit y a n d m oti v ati o n wit h c o n diti o n ( a cti v e or s h a m) i n m o d els pr e di cti n g tr ai ni n g g ai n. I n a d diti o n, t h e i m pr o v e m e nt s i n t h e a cti v e c o nditi o n v er s u s s h a m c o n diti o n a p p e ar t o b e st a bl e e v e n a s l o n g a s a y e ar aft er t h e ori gi n al i nt er v e nti o n. ■ 

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5. Gromov norm and Turaev-Viro invariants of 3-manifolds

   Detcherry, Renaud ; Kalfagianni, Efstratia ( December 2020 , Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $TV_r $and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $log|TVr(M)|$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $log|TVr(M)|\geq B r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$- Turaev-Viro invariants grow exponentially. These constructions are essential for the results of article [3] where we make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants TVr and hyperbolic volume are preserved under Dehn filling. Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm. 

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