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The main goal of this article is to connect some recent perspectives in the study of 4 4 -manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4 4 -manifolds, which include broken Lefschetz fibrations and indefinite Morse 2 2 -functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1 1 -parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2 2 -functions on general 4 4 -manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4 4 -manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4 4 -manifolds, and show the existence and stable uniqueness of simplified trisections on all 4 4 -manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus- 3 3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4 4 -manifolds in the homeomorphism classes of complex rational surfaces.

In this Letter, the Nyquist plot for linear, continuous-time MIMO systems under odd frequency transformations is dealt with. Given a system $\mathrm {G}(s)$ , a frequency transformation of the type $s\leftarrow F(s)$ allows one to obtain a transformed system $\tilde {\mathrm {G}}(s)=\mathrm {G}(F(s))$ with useful properties. For instance, in analog filter design this operation allows to obtain multi-bandpass/bandstops filters from lossless frequency transformations applied to prototype lowpass filters. We prove that under these transformations the Nyquist plot is transformed into a locus having the same shape of that of the original system. As the number of encirclements of any point in the complex plane performed by the curves in the Nyquist plot can be also related to that of the original system, we conclude that closed-loop stability of the transformed system can be inferred from the original system.

The equidistance of the color space plays a significant role in determining the color difference in color transmission systems. Strictly equal contrasting color spaces can be considered only those color spaces in which equal changes in the visual perception of color are provided with an equal change in the color coordinates in this color space. Currently, the International Commission on Lighting (CIE) has adopted a number of color spaces called equal-contrast. The article presents the results of the study of color spaces adopted by CIE for equal contrast, i.e. on the differences in the thresholds of color differentiation in different areas of the color locus. The article investigated such color spaces as CIE 1931 (RGB), CIE 1931 (x, y), CIE 1960 (u, v), CIE 1976 (u*, v*), CIE LAB (a*, b*).

The Effect of Locus of External and Internal Control, Financial Attitude, Pocket Money and Lifestyle on The Use of E-Money. This study aims to determine the influence of Locus of external internal control, financial attitude, pocket money and lifestyle on the use of e-money in accounting students in Yogyakarta. The population is all students of accounting study programs that are still active in 4 universities in Yogyakarta with a total of 200 respondents and 134 is a sample of this research. This study was conducted by questionnaires using multiple liner regression tests. The results showed that the locus of external control and lifestyle positively influenced the locus of internal control, financial attitude and pocket money of students had no effect on the use of e-money in accounting students of the University in Yogyakarta. Keywords: Locus of control external and internal, Financial attitude, Pocket Money, Lifestyle, E- Money Abstrak : Pengaruh Locus Of Control Eksternal Dan Internal, Financial Attitude, Uang Saku , Dan Lifestyle Terhadap Penggunaan E-Money . Penelitian ini bertujuan untuk mengetahui pengaruhl Locus of control external dan internal, financial attitude , uang saku dan lifestyle terhadap penggunaan e-money pada mahasiswa akuntansi di Yogyakarta. Populasi dari penelitian ini adalah mahasiswa program studi akuntansi aktif di 4 universitas di Yogyakarta dengan total 200 orang dan 134 merupakan sample penelitian . Penelitian ini dilakukan dengan menyebarkan kuesioner online dengan uji regresi liner berganda. Hasil penelitian menunjukkan locus of control external dan lifestyle berpengaruh positif sedangkan locus of control internal, financial attitude dan uang saku tidak berpengaruh terhadap penggunaan e-money pada mahasiswa akuntansi Universitas di Yogyakarta. Kata Kunci : Locus of control external dan internal, Financial attitude, Uang Saku , Lifestyle, E- Money

The simplicity of the proportional-integral controller makes it very popular in many practical engineering applications. In the literature, several approaches have been introduced for tuning proportional-integral controllers by calculating the centroid of the stability region. However, all those approaches depend on graphical plottings which are time-consuming. Also, the design procedure has to be redone as the transfer function changes. Here, two new analytical methods are proposed to obtain the centroid of the stability region for the proportional-integral controllers to control a time delay process which can be modeled by a stable or unstable first-order plus dead-time model. The methods introduced eliminate the compulsory procedure of plotting the stability region. The efficiency of the suggested methods has been studied by conducting a robustness analysis and studying several simulation examples.

The relevance of the article is based on the fact that over the past decades of the XXI century it has become the urgent need in philology to examine innovative methods, stylistic decisions, new narrative strategies used by the poet, writer, publicist, aesthete and philosopher Andrey Bely (1880–1934) in his last (Soviet) working period. The article considers the issue how in the 30s of the XX century Andrey Bely, being under the great influence of V. Meyerhold’s experimental theater, turned the first and second chapters of the novel “Moscow” (“The Moscow Eccentric” and “Moscow under Siege”) into the drama “Moscow” which wasn’t on the stage. A question is raised about Andrey Bely’s innovation as a playwright and some methods of the stage performance of the play “Moscow” are identified in the article. Special attention is paid to the locus of the house in the novel and drama, the question of transformation is raised: from the closed, protected, inner and personal space to the opened, hostile, unlocked one where hostile and mystic forces invade. The fight between two main antagonists — Korobkin, a scientist, and Mandro, a spy — is given in a wide mythological, christian and antroposophical context. The author of the article presents her many years experience on various approaches to understanding of not fully researched works (novel and play “Moscow”) by Andrey Bely.

The existing aircraft center track extraction methods only extract the position information of the trajectory, which cannot meet the requirements of abnormal trajectory detection and trajectory prediction. This paper innovatively proposes a center locus extraction algorithm based on multidimensional hierarchical clustering. Firstly, to solve the problem that trajectory resampling is easy to lose the original trajectory features, an equal arc length interpolation resampling method is proposed to process the original trajectory data. Then, the weighted Euclidean distance matrix of the trajectory set is calculated. The calculation model of the weighted Euclidean distance matrix is novel and takes into account the influence of multidimensional features. Finally, multidimensional hierarchical clustering is used to get the traffic flow distribution and output the center trajectory. 703 departure trajectory data from the terminal area of an airport are used for example verification. The results show that compared with the traditional hierarchical clustering, this method has a significant advantage in accurately dividing traffic flow. Moreover, the extracted center locus can retain the multidimensional features of locus, which has certain practical significance.

We study a family of surfaces of general type with $$p_g=q=2$$ p g = q = 2 and $$K^2=7$$ K 2 = 7 , originally constructed by C. Rito in [35]. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $$\mathcal {M}$$ M in the moduli space of surfaces of general type. In particular we prove that $$\mathcal {M}$$ M is an open subset, and it has three connected components, all of which are 2-dimensional, irreducible and generically smooth

We prove a conjecture of Koll\'ar stating that the local fundamental group of a klt singularity $x$ is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of $x$ is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair., Comment: minor corrections, added Corollary 5 and some references

For the Al-Mg-Li alloy sheet with significant anisotropy, the uniaxial and cruciform biaxial tension tests and the Nakazima test are firstly performed to systematically determine the mechanical properties (including strength coefficient, strain hardening exponent, normal anisotropy coefficient, experimental yielding points and forming limits) of this sheet. Afterward, the widely used Hill’s 48 and Yld2000-2d yield criteria are, respectively, identified to describe the yield locus of this sheet and compared with the experimental yielding points. Comparison shows that the identified Yld2000-2d yield function provides a better description of the yield locus for the Al-Mg-Li alloy sheet. Finally, the forming limits of this sheet are predicted by the modified M-K (Marciniak–Kuczynski) model with the Yld2000-2d yield criterion identified in this work. The results show that the predicted forming limits correspond well with the experimental ones within the whole range of strain paths.

Extending DiNezza-Lu's approach to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge-Amp{\`e}re equations on compact K{\"a}hler manifolds are continuous on a Zariski open set. This allows us to show that singular K{\"a}hler-Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part., Comment: to appear in Int. Math. Res. Not

Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2,Λ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ. We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦 k ⊂ M X (2,Λ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦 k in the Picard group of M X (2, Λ).

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in $${\mathbb {RP}}^2$$ of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in $${\mathbb {CP}}^2$$ of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.

Let $\mathcal{KC}_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface and $C\subset S$ a genus $g$ curve with divisibility $k$ in $\mathrm{Pic}(S)$. In this article we study the forgetful map $c_g^k:(S,C) \mapsto C$ from $\mathcal{KC}_g ^k$ to $\mathcal{M}_g$ for $k>1$. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when $S$ is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending $C$ in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $c_g^k$ dominates the locus in $\mathcal{M}_g$ of $k$-spin curves with the appropriate number of independent sections. We are able to do this only when $S$ is a complete intersection, and obtain in these cases some classification results for spin curves., v2: post-final version. Various enhancements in Sec.4 (including new subsection 4.4 on maximal variation) that will not appear in the published version, to appear in Boll. Unione Mat. Ital

This article examines the diverse nature of Muslim interest in Christian monasteries during the medieval Islamic period. According to a variety of contemporary accounts, Muslim visitation to monasteries often involved wine consumption and licentious behavior on the part of the elites. While not dismissing this possibility, this research suggests that there was often a greater religious dimension to Muslim fascination with monastic sites. Sacred shrines throughout the late antique Levant had, after all, been held in esteem for their hospitality and miraculous powers long before the arrival of Islam. This examination contends that Muslim interest in such Christian shrines and monasteries represents a dynamic, flexible confessional environment at the dawning of Islam. The pious spirit of pilgrimage and ziyāra/visitation was simply transferred into a new religious context; one that was defined by its fluid character and amorphous sectarian lines.

A characterization of the Kadomtsev-Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic-geometrical solutions and in particular elliptic solutions are discussed in detail. The new identity for theta-functions of curves with holomorphic involution having fixed points is obtained., 44 pages, no figures

We provide a classification of developable cubic hypersurfaces in P 4 . Using the correspondence between forms of degree 3 on P 4 and Artinian Gorenstein K -algebras, given by Macaulay-Matlis duality, we describe the locus in GOR ( 1 , 5 , 5 , 1 ) corresponding to those algebras which satisfy the Strong Lefschetz property.

The article is dedicated to representations of the southern French village of La Favière in the literature of the first wave of Russian emigration. It aims to examine the representations of a colony founded on the Côte d’Azur by Russian refugees, scholars, writers and artists, which existed until the outbreak of World War II. The objects of study include Favière-inspired poetry by Sasha Chernyi, published in the Poslednie novosti journal between 1927 and 1932, Aleksandr Kuprin’s series of essays entitled Huron Headland (1929), and Marina Tsvetaeva’s poems written in the summer of 1935. Emphasis is also placed on the epistolography and memoirs of Russian emigrants visiting the “Russian village”: letters by Georgi Griebenshchikov, memoirs by Ludmila Wrangel, Ksenia Kuprina, and Galina Rodionova. The La Favière phenomenon in the works of Russian emigrants has not been discussed before, which is what makes this article timely. The analysis of the above-mentioned works covers primarily their subject matter, but their connections with composition, stylistics and the question of genre affiliation are also discussed. Research demonstrates that despite the different ways in which the countryside is portrayed (its idealization in the poetry of Chernyi, comparison with Crimea — a true but lost paradise in Kuprin’s prose, rejection of Provence in Tsvetaeva’s poems), the tendency in Russian expatriate literature to interpret La Favière as locus amoenus persisted. In the works of Russian emigrants, the countryside evokes positive connotations, enchants with its landscape and tranquility, the naturalness of life replete with manual labor and close contact with nature. As a result, in the system of local texts of Russian emigration, bound together by the motif of foreignness as locus horribilis, La Favière holds a unique place, representing a rare example of Arcadia found in exile.

We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between $${\text {GSp}}_4$$ GSp 4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $${\text {GSp}}_4$$ GSp 4 .

Implicit locus equations in GeoGebra allow the user to do experiments with generalization of the concept of ellipses, namely with $n$-ellipses. By experimenting we obtain a geometric object that is very similar to a set of two circles., Comment: 13 pages, 14 figures

Polarized rational surfaces $(X, \mathcal L)$ of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of $\mathbb F_1$ at some points lying on distinct fibers. Ampleness and very ampleness of $\mathcal L$ are studied in terms of their location. When $\mathcal L$ is very ample and there is a line contained in $X$ and transverse to the fibers, the conic fibrations $(X, \mathcal L)$ are classified and a related property concerned with the inflectional locus is discussed., Comment: To appear in Advances in Geometry

By way of the projected phase-space (PPS), we investigate the relation between galaxy properties and cluster environment in a subsample of groups from the Yang Catalog. The sample is split according to the gaussianity of the velocity distribution in the group into gaussian (G) and non-gaussian (NG). Our sample is limited to massive clusters with $\rm M_{200} \geq 10^{14} M_{\odot}$ and $\rm 0.03\leq z \leq 0.1$. NG clusters are more massive, less concentrated and have an excess of faint galaxies compared to G clusters. NG clusters show mixed distributions of galaxy properties in the PPS compared to the G case. Using the relation between infall time and locus on the PPS, we find that, on average, NG clusters accreted $\rm \sim 10^{11}\,M_{\odot}$ more stellar mass in the last $\sim 5$ Gyr than G clusters. The relation between galaxy properties and infall time is significantly different for galaxies in G and NG systems. The more mixed distribution in the PPS of NG clusters translates into shallower relations with infall time. Faint galaxies whose first crossing of the cluster virial radius happened 2-4 Gyr ago in NG clusters are older and more metal-rich than in G systems. All these results suggest that NG clusters experience a higher accretion of pre-processed galaxies, which characterizes G and NG clusters as different environments to study galaxy evolution., Comment: Accepted for publication in MNRAS, 17 pages, 9 Figures

We study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { x } - { 0 A } is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

Introduction: The present paper is the outcome of the research “Locus Recommendation using Probabilistic Matrix Factorization Techniques” carried out in Manav Rachna International Institute of Research and Studies, India in the year 2019-20. Methodology: Matrix factorization is a model-based collaborative technique for recommending new items to the users. Results: Experimental results on two real-world LBSNs showed that PFM consistently outperforms PMF. This is because the technique is based on gamma distribution to the model user and item matrix. Using gamma distribution is reasonable for check-in frequencies which are all positive in real datasets. However, PMF is based on Gaussian distribution that can allow negative frequency values as well. Conclusion: The motive of the work is to identify the best technique for recommending locations with the highest accuracy and allow users to choose from a plethora of available locations; the best and interesting location based on the individual’s profile. Originality: A rigorous analysis of Probabilistic Matrix Factorization techniques has been performed on popular LBSNs and the best technique for location recommendation has been identified by comparing the accuracy viz RMSE, Precision@N, Recall@N, F1@N of different models. Limitations: User’s contextual information like demographics, social and geographical preferences have not been considered while evaluating the efficiency of probabilistic matrix factorization techniques for POI Recommendations.

In this paper, we continue the study, began in Chen et al. (Slices of parameter space for meromorphic maps with two asymptotic values, arXiv:1908.06028 , 2019), of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values, and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In Fagella and Keen (Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563 , 2017) and Chen and Keen (Discrete and Continuous Dynamical Systems 39(10):5659–5681, 2019), it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here, we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle are also accessible from the shift locus.

Asymmetric self-excited periodic motions or periodic solutions which are produced by relay feedback systems that have symmetric characteristics are studied in the paper. Two different mechanisms of producing an asymmetric oscillation by a system with symmetric properties are noted and analyzed by the locus of a perturbed relay system (LPRS) method. Bifurcation between the ability to excite symmetric and asymmetric oscillation with variation of system parameters is analyzed. An algorithm of finding asymmetric solutions is proposed.

The generally greater degree of thermal motion of guest molecule(s) relative to the host often impedes their accurate modelling in crystal structures. We propose a 'rule-of-thumb' for estimating the maximum number of guest molecules that can be accommodated in a given amount of accessible space in an adequately modelled host structure. A survey of the Cambridge Structural Database was carried out to evaluate the fractional occupancy θ of the accessible space for almost 40 000 solvates involving 20 common solvents. Using widely accessible software tools, the volume of a guest is estimated as its van der Waals surface, while the guest-occupiable space of a potentially porous host is determined as that available to a virtual spherical probe. We propose terminology more appropriate to the supramolecular interpretation of surface typology: the probe-traversable and probe-accessible boundaries as traced out by the locus and surface of a spherical probe, respectively. High-throughput analysis using commercial and free software packages yielded a mean θ = 51.1(4)%, ranging from 45.3(6)% for hexane to 60(1)% for acetic acid.

We develop Hida theory for Shimura varieties of type A without ordinary locus. In particular we show that the dimension of the space of ordinary forms is bounded independently of the weight and that there is a module of $\Lambda$-adic cuspidal ordinary forms which is of finite type over $\Lambda$, where $\Lambda$ is a twisted Iwasawa algebra.

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

In this paper, we study the Hessian map $h_{d,r}$, which associates to any hypersurface of degree $d$ in ${{\mathbb{P}}}^r$ its Hessian hypersurface, and the general properties of this map and prove that $h_{d,1}$ is birational onto its image if $d\geqslant 5.$ We also study in detail the maps $h_{3,1}$, $h_{4,1}$, and $h_{3,2}$ and the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, proving that this restriction is injective as soon as $r\geqslant 2$ and $d\geqslant 3$, which implies that $h_{3,3}$ is birational onto its image. We also prove that the differential of the Hessian map is of maximal rank on the generic hypersurfaces of degree $d$ with Waring rank $r+2$ in ${{\mathbb{P}}}^r$, as soon as $r\geqslant 2$ and $d\geqslant 3$.

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).

In this article, we investigate Muirhead's classical system of differential operators for the hypergeometric function 1F1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system., Comment: 26 pages. v2 changes: included Appendix discussing the singular locus for degenerate parameters

By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338, 2010), which states that if $$\gamma $$ is a convex curve with length L and enclosed area A, then the best constant $$\varepsilon $$ in the inequality $$\begin{aligned} L^2\le 4\pi A+\varepsilon |{\tilde{A}}| \end{aligned}$$ is $$\pi $$ , where $${\tilde{A}}$$ denotes the oriented area of the locus of its curvature centers.

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a degeneration of a curve of class $(3,3)$. We prove that the compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of $\mathbb{P}^1$ by genus 4 curves., 49 pages, 9 figures. Added more details and improved the introduction. To appear in Transactions of the American Mathematical Society

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve $y^n=f(x)$ over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when $n$ is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus $g$., 31 pages, 8 figures

We give new examples of compact, negatively curved Einstein manifolds of dimension $4$. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of 4-manifolds $(X_k)$ previously considered by Gromov and Thurston. The construction begins with a certain sequence $(M_k)$ of hyperbolic 4-manifolds, each containing a totally geodesic surface $\Sigma_k$ which is nullhomologous and whose normal injectivity radius tends to infinity with $k$. For a fixed choice of natural number $l$, we consider the $l$-fold cover $X_k \to M_k$ branched along $\Sigma_k$. We prove that for any choice of $l$ and all large enough $k$ (depending on $l$), $X_k$ carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on $X_k$, which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from $M_k$. The second step in the proof is to perturb this to a genuine solution to Einstein's equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on $L^2$ coercivity estimates., Comment: 53 pages. v2 small modifications to exposition. v3 typos corrected. Same text as published version, to appear in the Journal of the American Mathematical Society

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of $X$ with branch locus $B$ depends on the ramification invariants of the cover. When $X$ is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal., 8 pages, comments welcome

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

Using the duality of positive cones, we show that applying the polar transform from convex analysis to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants have nice properties similar to those for divisors. In particular, this enables us to give a characterization of the divisorial components of the non-Kahler locus of a big class., 14 pages

We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a "homogeneous lamina") for degenerate Poncelet polygons. We also provide a proof for one of Dan Reznik invariants for billiard trajectories. Plus, we take a look at how the scene looks like when we shift to spherical geometry., Comment: 21 pages, 14 figures

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$., Comment: Final version. To appear in Canadian Jour. Math

We study the formal neighbourhood of a point in the $$\mu $$ -ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a “shifted cascade”. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms, analogous to the Serre–Tate canonical lift of an ordinary abelian variety.

We find an asymptotic description of the period locus of simply connected Jacobian elliptic surfaces and of the period locus of hyperelliptic curves. The two descriptions are essentially the same, and are given by the alkanes of organic chemistry., The use of the Minimal Model Program has been clarified, as has the Torelli theorem for special elliptic surfaces

We consider the family of irreducible crystalline representations of dimension $2$ of ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ given by the $V_{k,a_p}$ for a fixed weight integer $k\geq 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give qualitative results on this locus and show that for a fixed $p$ and $k$ it can be computed by determining the reduction modulo $p$ of $V_{k,a_p}$ for a finite number of values of the parameter $a_p$. We also generalize these results to other Galois types., to appear in Algebra & Number Theory