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1. The effect of adding randomly weighted edges

   https://doi.org/10.1137/20M1335418  

   Frieze, A. ( January 2021 , SIAM journal on discrete mathematics)

   We consider the following question. We have a dense regular graph $G$ with degree $\a n$, where $\a>0$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $X(e),e\in E(G(m))$. We estimate the minimum weight of some specified combinatorial structures. We show that in certain cases, we can obtain the same estimate as is known for the complete graph, but scaled by a factor $\a^{-1}$. We consider spanning trees, shortest paths and perfect matchings in (pseudo-random) bipartite graphs. 

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2. Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

   https://doi.org/10.1093/imrn/rnac002  

   Iarrobino, Anthony ; Macias Marques, Pedro ; McDaniel, Chris ; Seceleanu, Alexandra ; Watanabe, Junzo ( February 2022 , International Mathematics Research Notices)

   Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript. 

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3. Materialistic: Selecting Similar Materials in Images

   https://doi.org/10.1145/3592390  

   Sharma, Prafull ; Philip, Julien ; Gharbi, Michaël ; Freeman, Bill ; Durand, Fredo ; Deschaintre, Valentin ( August 2023 , ACM Transactions on Graphics)

   Separating an image into meaningful underlying components is a crucial first step for both editing and understanding images. We present a method capable of selecting the regions of a photograph exhibiting the same material as an artist-chosen area. Our proposed approach is robust to shading, specular highlights, and cast shadows, enabling selection in real images. As we do not rely on semantic segmentation (different woods or metal should not be selected together), we formulate the problem as a similarity-based grouping problem based on a user-provided image location. In particular, we propose to leverage the unsupervised DINO [Caron et al. 2021] features coupled with a proposed Cross-Similarity Feature Weighting module and an MLP head to extract material similarities in an image. We train our model on a new synthetic image dataset, that we release. We show that our method generalizes well to real-world images. We carefully analyze our model's behavior on varying material properties and lighting. Additionally, we evaluate it against a hand-annotated benchmark of 50 real photographs. We further demonstrate our model on a set of applications, including material editing, in-video selection, and retrieval of object photographs with similar materials.

    

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4. Spectral Gaps and Incompressibility in a $${\varvec{\nu }}$$ = 1/3 Fractional Quantum Hall System

   https://doi.org/10.1007/s00220-021-03997-0  

   Nachtergaele, Bruno ; Warzel, Simone ; Young, Amanda ( April 2021 , Communications in Mathematical Physics)

   null (Ed.)

   Abstract We study an effective Hamiltonian for the standard $$\nu =1/3$$ ν = 1 / 3 fractional quantum Hall system in the thin cylinder regime. We give a complete description of its ground state space in terms of what we call Fragmented Matrix Product States, which are labeled by a certain family of tilings of the one-dimensional lattice. We then prove that the model has a spectral gap above the ground states for a range of coupling constants that includes physical values. As a consequence of the gap we establish the incompressibility of the fractional quantum Hall states. We also show that all the ground states labeled by a tiling have a finite correlation length, for which we give an upper bound. We demonstrate by example, however, that not all superpositions of tiling states have exponential decay of correlations. 

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5. The QuaSEFE Problem

   Angelini, P. ; Forster, H. ; Hoffmann, M. ; Kaufmann, M. ; Kobourov, S. ; Liotta, G. ; Patrigniani, M. ( January 2019 , International Symposium on Graph Drawing and Network Visualization)

   We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. %We call this problem the QSEFE problem. We show that a triple consisting of two planar graphs and a tree admit a QSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QSEFE for two matchings if the quasiplanar drawing of one matching is fixed. 

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