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1. Solution Existence and Uniqueness for Degenerate SDEs with Application to Schrödinger-Equation Representations

   Dower, P.M.; Kaise, H.; McEneaney, W.M.; Wang, T; Zhao, R. (April 2021, Comms. communications in Information and Systems.)

   null (Ed.)

   Existence and uniqueness results for solutions of stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coeffcient is nondegenerate. Here, existence and uniqueness of strong solutions is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solutions of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coeffcient is exactly half that of the state. In addition to this degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold. The second consists of singularities that may occur on a smoothly parameterized curve. 

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2. An upper Minkowski dimension estimate for the interior singular set of area minimizing currents

   https://doi.org/10.1002/cpa.22165  

   Skorobogatova, Anna (September 2023, Communications on Pure and Applied Mathematics)

   Abstract We show that for an area minimizingm‐dimensional integral currentTof codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most . This provides a strengthening of the existing ‐dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by‐product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximateTalong blow‐up scales. 

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3. Uniqueness and stability of Ricci flow through singularities

   https://doi.org/DOI: https://dx.doi.org/10.4310/ACTA.2022.v228.n1.a1  

   Bamler, Richard; Kleiner, Bruce (July 2022, Acta mathematica)

   We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper. 

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4. G2 manifolds with nodal singularities along circles

   https://doi.org/10.1007/s12220-019-002833-3  

   Chen, Gao (September 2019, The Journal of geometric analysis)

   This paper finds matching building blocks for the construction of a compact manifold with G2 holonomy and nodal singularities along circles using twisted connected sum method by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with G2 holonomy and isolated conical singularities with link S3×S3. 

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5. 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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