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1. On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

   https://doi.org/10.4171/JEMS/1485  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (July 2024, Journal of the European Mathematical Society)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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2. Bohr sets in sumsets II: countable abelian groups

   https://doi.org/10.1017/fms.2023.49  

   Griesmer, John T.; Le, Anh N.; Lê, Thái Hoàng (July 2023, Forum of Mathematics, Sigma)

   Abstract We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, lettingGbe a countable discrete abelian group and$$\phi _1, \phi _2, \phi _3: G \to G$$be commuting endomorphisms whose images have finite indices, we show that(1)If$$A \subset G$$has positive upper Banach density and$$\phi _1 + \phi _2 + \phi _3 = 0$$, then$$\phi _1(A) + \phi _2(A) + \phi _3(A)$$contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$$\mathbb {Z}$$and a recent result of the first author.(2)For any partition$$G = \bigcup _{i=1}^r A_i$$, there exists an$$i \in \{1, \ldots , r\}$$such that$$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$$contains a Bohr set. This generalizes a result of the second and third authors from$$\mathbb {Z}$$to countable abelian groups.(3)If$$B, C \subset G$$have positive upper Banach density and$$G = \bigcup _{i=1}^r A_i$$is a partition,$$B + C + A_i$$contains a Bohr set for some$$i \in \{1, \ldots , r\}$$. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss. All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices$$[G:\phi _j(G)]$$, the upper Banach density ofA(in (1)), or the number of sets in the given partition (in (2) and (3)). 

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3. Scalar Curvature, Entropy, and Generalized Ricci Flow

   https://doi.org/10.1093/imrn/rnad002  

   Streets, Jeffrey (February 2023, International Mathematics Research Notices)

   Abstract We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction–diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles. 

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4. Eigenvalue lower bounds and splitting for modified Ricci flow

   https://doi.org/10.1515/ans-2022-0083  

   Colding, Tobias Holck; Minicozzi_II, William P (March 2024, Advanced Nonlinear Studies)

   We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality. 

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5. Exploring and Controlling Nonlinear Phi-Bit Modes in Elastic Systems for Quantum-Analogue Computing

   Faiaz, Abrar Nur_E; Ige, Akinsanmi S; Mahmood, Kazi Tahsin; Hasan, M Arif; Deymier, Pierre; Runge, Keith; Levine, Josh (March 2024, The American Institute of Physics, publisher, Bulletin of the American Physical Society)

   Phi-bits are classical mechanical analogues of qubits. Comprehending the nonlinear phenomena that underlie the control and relationships between phi-bits is of utmost importance for advancing phi-bit-based quantum-analogue computing systems. Phi-bits are acoustic waves in externally driven nonlinearly coupled arrays of waveguides, that can exist in a coherent superposition of two states. Tuning the frequency, amplitude, and phase of external drivers is a means of controlling the phi-bit states. We have developed a discrete element model to analyze and predict the nonlinear phi-bit response to external drivers that may result from different types, strengths, and orders of nonlinearity due to the presence of (i) intrinsic medium (epoxy) coupling the waveguides and (ii) external factors such as signal generators/transducers/ultrasonic couplant assembly. Key findings include the impact of nonlinearity type, strength, and order as well as damping on the modulus and phases of the complex amplitudes of the phi-bit coherent superposition of states. This research serves as an exploration for control of design parameters in the creation of phi-bits, which will enable the preparation and manipulation of superpositions of states essential for developing phi-bit-based quantum analogue information processing platforms. 

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