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1. The Nitsche phenomenon for weighted Dirichlet energy

   https://doi.org/10.1515/acv-2017-0060  

   Iwaniec, Tadeusz; Onninen, Jani; Radice, Teresa (July 2020, Advances in Calculus of Variations)

   null (Ed.)

   Abstract The present paper arose from recent studies of energy-minimal deformations of planar domains.We are concerned with the Dirichlet energy. In general the minimal mappings need not be homeomorphisms. In fact, a part of the domain near its boundary may collapse into the boundary of the target domain. In mathematical models of nonlinear elasticity this is interpreted as interpenetration of matter . We call such occurrence the Nitsche phenomenon , after Nitsche’s remarkable conjecture (now a theorem) about existence of harmonic homeomorphisms between annuli. Indeed the round annuli proved to be perfect choices to grasp the nuances of the problem. Several papers are devoted to a study of deformations of annuli. Because of rotational symmetry it seems likely that the Dirichlet energy-minimal deformations are radial maps. That is why we confine ourselves to radial minimal mappings. The novelty lies in the presence of a weight in the Dirichlet integral. We observe the Nitsche phenomenon in this case as well, see our main results Theorem 1.4 and Theorem 1.7. However, the arguments require further considerations and new ingredients. One must overcome the inherent difficulties arising from discontinuity of the weight. The Lagrange–Euler equation is unavailable, because the outer variation violates the principle of none interpenetration of matter. Inner variation, on the other hand, leads to an equation that involves the derivative of the weight. But our weight function is only measurable which is the main challenge of the present paper. 

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2. On L-functions of modular elliptic curves and certain K3 surfaces

   https://doi.org/10.1007/s11139-021-00388-w  

   Amir, Malik; Hong, Letong (March 2022, The Ramanujan Journal)

   Abstract Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan $$\tau $$ τ -function, one may ask whether an odd integer $$\alpha $$ α can be equal to $$\tau (n)$$ τ ( n ) or any coefficient of a newform f ( z ). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $$k\ge 4$$ k ≥ 4 . We use these methods for weight 2 and 3 newforms and apply our results to L -functions of modular elliptic curves and certain K 3 surfaces with Picard number $$\ge 19$$ ≥ 19 . In particular, for the complete list of weight 3 newforms $$f_\lambda (z)=\sum a_\lambda (n)q^n$$ f λ ( z ) = ∑ a λ ( n ) q n that are $$\eta $$ η -products, and for $$N_\lambda $$ N λ the conductor of some elliptic curve $$E_\lambda $$ E λ , we show that if $$|a_\lambda (n)|<100$$ | a λ ( n ) | < 100 is odd with $$n>1$$ n > 1 and $$(n,2N_\lambda )=1$$ ( n , 2 N λ ) = 1 , then $$\begin{aligned} a_\lambda (n) \in&\{-5,9,\pm 11,25, \pm 41, \pm 43, -45,\pm 47,49, \pm 53,55, \pm 59, \pm 61,\\&\pm 67, -69,\pm 71,\pm 73,75, \pm 79,\pm 81, \pm 83, \pm 89,\pm 93 \pm 97, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , ± 41 , ± 43 , - 45 , ± 47 , 49 , ± 53 , 55 , ± 59 , ± 61 , ± 67 , - 69 , ± 71 , ± 73 , 75 , ± 79 , ± 81 , ± 83 , ± 89 , ± 93 ± 97 , 99 } . Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving $$\begin{aligned} a_\lambda (n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{aligned}$$ a λ ( n ) ∈ { - 5 , 9 , ± 11 , 25 , - 45 , 49 , 55 , - 69 , 75 , ± 81 , ± 93 , 99 } . 

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3. Automatic continuity of pure mapping class groups

   Dickmann, R (July 2024, The Womens business resource guide)

   We completely classify the orientable infinite-type surfaces S such that PMap(S), the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of surfaces with finitely many ends and no noncompact boundary components, we prove the mapping class group Map(S) does not have automatic continuity. We also completely classify the surfaces such that PMapc (S), the subgroup of the pure mapping class group composed of elements with representatives that can be approximated by compactly supported homeomorphisms, has automatic continuity. In some cases when PMapc (S) has automatic continuity, we show any homomorphism from PMapc (S) to a countable group is trivial. 

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4. Orientation-preserving homeomorphisms of Euclidean space are commutators

   https://doi.org/10.1090/bproc/258  

   Bhat, Megha; Vlamis, Nicholas (April 2025, Proceedings of the American Mathematical Society, Series B)

   We prove that every orientation-preserving homeomorphism of Euclidean space can be expressed as a commutator of two orientation-preserving homeomorphisms. We give an analogous result for annuli. In the annulus case, we also extend the result to the smooth category in the dimensions for which the associated sphere has a unique smooth structure. As a corollary, we establish that every orientation-preserving diffeomorphism of the real line is the commutator of two orientation-preserving diffeomorphisms. 

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5. Non-planar Ends are Continuously Unforgettable

   https://doi.org/10.1093/imrn/rnaf026  

   Aramayona, Javier; de_Pool, Rodrigo; Skipper, Rachel; Tao, Jing; Vlamis, Nicholas G; Wu, Xiaolei (March 2025, International Mathematics Research Notices)

   Abstract We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups. 

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