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Algebraic quotients and Geometric Invariant Theory
(Universidad Nacional de Trujillo, 2020-06-30)
The quotient of an algebraic variety by action of an algebraic group does not always has a variety structure. The aim of this work is to describe a methodfor constructing good quotients, in the sense of Geometric invariant theory, in algebraicgeometry.
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On Nörlund–Voronoi Summability and Instability of Rational Maps
(Springer, 2020-12-01)
We investigate the connection between the instability of rational maps and summability methods applied to the spectrum of a critical point belonging to the Julia set of a rational map.
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Geometry of Horospherical Varieties of Picard Rank One
(Oxford University Press, 2020-10-29)
Abstract We study the geometry of smooth non-homogeneous horospherical varieties of Picard rank one. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin’s conjecture. We start with describing the cohomology groups of smooth horospherical varieties of Picard rank one. We show a Chevalley formula for these and establish that many Gromov–Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. In the last sections, we turn to derived categories of coherent sheaves. We first discuss a general construction of exceptional bundles on horospherical varieties. We work out in detail the case of the horospherical variety associated to the exceptional group $G_2$ and construct a full rectangular Lefschetz exceptional collection in the derived category.
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Foliations on the projective plane with finite group of symmetries
(Mattioli 1885, 2020-03-09)
Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.
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Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory
(Mathematical Sciences Publishers, 2021-01-01)
We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X whose equivariant K-theory of vector bundles does not surject onto its ordinary K-theory, and describe the operational K-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational K-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic K-theory of relatively perfect complexes to bivariant operational K-theory.
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Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture
(Cornell University, 2021-01-19)
Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.
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Singular integral operators, a brief historical overview of its evolution
(Universidad Nacional de Trujillo, 2022-06-30)
In this work we give an analytical-historical view of the classical theory of singular integrals introduced by A.P. Calderón-A. Zygmund. Emphasis is given to their applications to PDEs and to some projections of the theory.
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Bivariant K-theory of locally convex Z-graded algebras
(Universidad Nacional de Trujillo, 2022-06-30)
In the present work, we describe some results about the K-theory of Z-graded algebras. First, in the context of C* algebras, we begin with the Pimsner-Voiculescu sequence for crossed products and its generalizations. We will see that there are results analog to these in the context of locally convex algebras and we conclude with results for generalized Weyl algebras.
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Statistical tables in Primary Education textbooks of Peru
(Facultad de Ciencias de la Educacion, 2022-01-01)
In recent years, the literature shows an increase in research analyzing statistical representation of data in textbooks, but in the Peruvian context is still scarce. Therefore, this research aims to analyze activities on statistical tables present in mathematics textbooks, published by the Ministry of Education and distributed free of charge to teachers and students of Primary Education in Peru. The methodology is qualitative and descriptive. For data analysis, the content analysis technique was used by analyzing types of statistical table, types of task, reading levels and semiotic complexity levels as units of analysis in a complete series of mathematics textbooks of Primary Education, from first to sixth grade, one per level, due to their wide national coverage. The results allow us to observe predominance of tally charts and data tables, completion and comparison tasks, reading level 2 (reading within data) and semiotic complexity level 3 (representation of data distribution). It is concluded that it is necessary to increase the number and variety of tasks related to statistical tables proposed in mathematics textbooks, as well as to reinforce the presence of the highest levels of reading and semiotic complexity in the last years of Primary Education in Peru.
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The development of the notion of “half” in children from two Ucayali’s Shipibo-Konibo communities
(Centro Interamericano de Investigaciones Psicologicas y Ciencias Afines, 2022-01-01)
This research aims to identify and describe developmental levels of the notion of “half” in a group of children from two Shipibo-Konibo native communities of the Ucayali region, in the Peruvian Amazon rainforest. Fourteen students aged between 7 and 16 participated, six from Bethel community and eight from Bena Jema community. The community of Bethel is located approximately 6 hours by river from the city of Pucallpa, while Bena Jema is located within Pucallpa’s boundaries, in the Yarinacocha district. All participants were assessed using Jean Piaget’s clinical-critical method, with two tasks. In the first task, focusing on discrete quantities: different sets of cards with the picture of a fish were presented sequentially, some with an even number of units and some with an odd number. Participants were asked to choose and present back half of each set. In the second task, focusing on continuous quantities: participants were asked to choose and present half of a single raw spaghetti, which they needed to break with their hands. If the resulting pieces were unequal, they were asked to break them again and redistribute the results. Students from Bena Jema community were assessed with both tasks, while students from Bethel community were assessed only with the discrete quantities task. The results showed four developmental levels in the discrete quantities task. In the first level, participants took any one of two parts of the set to be “half”, without checking if they were equal or whether putting them back together reconstituted the original whole. In the second level, participants took any one of two equal parts to be "half", but were inconsistent and accepted the possibility of the parts being unequal. In the third level, participants only took any one of two equal parts to be “half” but had difficulties splitting odd-numbered sets, including sets with fewer elements than the even-numbered ones they had previously split correctly. In the fourth level, the task was solved successfully with both even-numbered and odd-numbered sets. For the continuous quantities task, three levels were found. In the first level, any one of two raw spaghetti pieces was taken to be “half”, without checking their evenness. In the second level, participants took any two equal pieces of the raw spaghetti to be “half”, but without checking whether putting them back together reconstituted the original whole, with no extra parts remaining. In the third level, participants took any one of two equal pieces to be “half”, checking whether putting them back together reconstituted the original whole. These results are consistent with reports from previous research assessing the notion of “half” in Western contexts. Results support universality in the development of the notion of “half”, but show a delay in the participants’ level of acquisition, taking Peru’s national curriculum as reference. This study is a contribution to the understanding of the development of the notion of “half” in indigenous children living in Amazonian native communities, and shows the relevance of the Piagetian clinical-critical interview in these sociocultural contexts. The implications of these findings are discussed in relation to the universality of logical-mathematical knowledge, as well as the need to rethink the timing and manner in which the notion of “half” appears in the curriculum. https://doi.org/10.16888/interd.2022.39.2.17

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