Tropical Derivation of Cohomology Ring of Heavy/Light Hassett Spaces

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as \calm_{g, w} for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of \calm_{0, w}, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of \overline{\calm}_{0,w} as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology

Equivariant log-concavity of graph matchings

For any graph, we show that the graded permutation representation of the graph automorphism group given by matchings is strongly equivariantly log-concave. The proof gives a family of equivariant injections inspired by a combinatorial map of Kratthenthaler and reduces to the hard Lefschetz theorem.Comment: 8 pages; v2, improved and corrected, final version accepted at AlC

KK-rings of wonderful varieties and matroids

We study the $K$-ring of the wonderful variety of a hyperplane arrangement and give a combinatorial presentation that depends only on the underlying matroid. We use this combinatorial presentation to define the $K$-ring of an arbitrary loopless matroid. We construct an exceptional isomorphism, with integer coefficients, to the Chow ring of the matroid that satisfies a Hirzebruch--Riemann--Roch-type formula, generalizing a recent construction of Berget, Eur, Spink, and Tseng for the permutohedral variety (the wonderful variety of a Boolean arrangement). As an application, we give combinatorial formulas for Euler characteristics of arbitrary line bundles on wonderful varieties. We give analogous constructions and results for augmented wonderful varieties, and for Deligne--Mumford--Knudsen moduli spaces of stable rational curves with marked points.Comment: 36 pages. Comments welcome

Kapranov degrees

The moduli space of stable rational curves with marked points has two distinguished families of maps: the forgetful maps, given by forgetting some of the markings, and the Kapranov maps, given by complete linear series of $\psi$-classes. The collection of all these maps embeds the moduli space into a product of projective spaces. We call the multidegrees of this embedding ``Kapranov degrees,'' which include as special cases the work of Witten, Silversmith, Gallet--Grasegger--Schicho, Castravet--Tevelev, Postnikov, Cavalieri--Gillespie--Monin, and Gillespie--Griffins--Levinson. We establish, in terms of a combinatorial matching condition, upper bounds for Kapranov degrees and a characterization of their positivity. The positivity characterization answers a question of Silversmith and gives a new proof of Laman's theorem characterizing generically rigid graphs in the plane. We achieve this by proving a recursive formula for Kapranov degrees and by using tools from the theory of error correcting codes.Comment: Added proof of Laman's theore

Permutohedral complexes and rational curves with cyclic action

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.Comment: 47 pages, 12 figure

Myrothecium-like new species from turfgrasses and associated rhizosphere

Myrothecium sensu lato includes a group of fungal saprophytes and weak pathogens with a worldwide distribution. Myrothecium s.l. includes 18 genera, such as Myrothecium, Septomyrothecium, Myxospora, all currently included in the family Stachybotryaceae. In this study, we identified 84 myrothecium-like strains isolated from turfgrasses and their rhizosphere. Five new species, i.e., Alfaria poae, Alf. humicola, Dimorphiseta acuta, D. obtusa, and Paramyrothecium sinense, are described based on their morphological and phylogenetic distinctions. Phylogenies were inferred based on the analyses of sequences from four DNA loci (ITS, cmdA, rpb2 and tub2). The generic concept of Dimorphiseta is broadened to include a third type of seta, i.e. thin-walled, straight with obtuse apices

CSI-PPPNet: A One-Sided One-for-All Deep Learning Framework for Massive MIMO CSI Feedback

To reduce multiuser interference and maximize the spectrum efficiency in orthogonal frequency division duplexing massive multiple-input multiple-output (MIMO) systems, the downlink channel state information (CSI) estimated at the user equipment (UE) is required at the base station (BS). This paper presents a novel method for massive MIMO CSI feedback via a one-sided one-for-all deep learning framework. The CSI is compressed via linear projections at the UE, and is recovered at the BS using deep learning (DL) with plug-and-play priors (PPP). Instead of using handcrafted regularizers for the wireless channel responses, the proposed approach, namely CSI-PPPNet, exploits a DL based denoisor in place of the proximal operator of the prior in an alternating optimization scheme. In this way, a DL model trained once for denoising can be repurposed for CSI recovery tasks with arbitrary compression ratio. The one-sided one-for-all framework reduces model storage space, relieves the burden of joint model training and model delivery, and could be applied at UEs with limited device memories and computation power. Extensive experiments over the open indoor and urban macro scenarios show the effectiveness and advantages of the proposed method

Tumor promoter TPA activates Wnt/β-catenin signaling in a casein kinase 1-dependent manner.

The tumor promoter 12-O-tetra-decanoylphorbol-13-acetate (TPA) has been defined by its ability to promote tumorigenesis on carcinogen-initiated mouse skin. Activation of Wnt/β-catenin signaling has a decisive role in mouse skin carcinogenesis, but it remains unclear how TPA activates Wnt/β-catenin signaling in mouse skin carcinogenesis. Here, we found that TPA could enhance Wnt/β-catenin signaling in a casein kinase 1 (CK1) ε/δ-dependent manner. TPA stabilized CK1ε and enhanced its kinase activity. TPA further induced the phosphorylation of LRP6 at Thr1479 and Ser1490 and the formation of a CK1ε-LRP6-axin1 complex, leading to an increase in cytosolic β-catenin. Moreover, TPA increased the association of β-catenin with TCF4E in a CK1ε/δ-dependent way, resulting in the activation of Wnt target genes. Consistently, treatment with a selective CK1ε/δ inhibitor SR3029 suppressed TPA-induced skin tumor formation in vivo, probably through blocking Wnt/β-catenin signaling. Taken together, our study has identified a pathway by which TPA activates Wnt/β-catenin signaling