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1. On inscribed trapezoids and affinely 3-regular maps

   https://doi.org/10.1017/prm.2023.44  

   Frick, Florian; Harrison, Michael (May 2023, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   We show that any embedding $$\mathbb {R}^d \to \mathbb {R}^{2d+2^{\gamma (d)}-1}$$ inscribes a trapezoid or maps three points to a line, where $$2^{\gamma (d)}$$ is the smallest power of $$2$$ satisfying $$2^{\gamma (d)} \geq \rho (d)$$ , and $$\rho (d)$$ denotes the Hurwitz–Radon function. The proof is elementary and includes a novel application of nonsingular bilinear maps. As an application, we recover recent results on the nonexistence of affinely $$3$$ -regular maps, for infinitely many dimensions $$d$$ , without resorting to sophisticated algebraic techniques. 

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2. Linear growth of quantum circuit complexity

   https://doi.org/10.1038/s41567-022-01539-6  

   Haferkamp, Jonas; Faist, Philippe; Kothakonda, Naga B.; Eisert, Jens; Yunger Halpern, Nicole (May 2022, Nature Physics)

   Abstract The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits. 

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3. Birational maps with transcendental dynamical degree

   https://doi.org/10.1112/plms.12573  

   Bell, Jason P.; Diller, Jeffrey; Jonsson, Mattias; Krieger, Holly (December 2023, Proceedings of the London Mathematical Society)

   Abstract We give examples of birational selfmaps of , whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation. 

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4. Topological Bounds on Hyperkähler Manifolds

   https://doi.org/10.1080/10586458.2023.2172630  

   Sawon, Justin (April 2023, Experimental Mathematics)

   We conjecture that certain curvature invariants of compact hyperkähler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an “experimental proof” in higher dimensions, and verify it for all known hyperkähler manifolds up to dimension eight. As an application, we show that our conjecture leads to a bound on the second Betti number in all dimensions. 

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5. A Schmidt–Nochka theorem for closed subschemes in subgeneral position

   https://doi.org/10.1515/crelle-2024-0085  

   Heier, Gordon; Levin, Aaron (November 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   In previous work, the authors established a generalized version of Schmidt’s subspace theorem for closed subschemes in general position in terms of Seshadri constants.We extend our theorem to weighted sums involving closed subschemes in subgeneral position, providing a joint generalization of Schmidt’s theorem with seminal inequalities of Nochka.A key aspect of the proof is the use of a lower bound for Seshadri constants of intersections from algebraic geometry, as well as a generalized Chebyshev inequality.As an application, we extend inequalities of Nochka and Ru–Wong from hyperplanes in 𝑚-subgeneral position to hypersurfaces in 𝑚-subgeneral position in projective space, proving a sharp result in dimensions 2 and 3, and coming within a factor of 3/2 of a sharp inequality in all dimensions.We state analogous results in Nevanlinna theory generalizing the second main theorem and Nochka’s theorem (Cartan’s conjecture). 

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