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Un dels problemes oberts més excitants en topologia quàntica es relaciona amb la categorificació de la teoria complexa de Chern-Simons. En aquest context, Gukov i Manolescu introdueixen una sèrie en dues variables $F_K(x,q)$ com la restricció dels invariants de 3-varietats $\hat{Z}$ a complements de nusos, la qual s'ha demostrat que té propietats sorprenents. En particular, aquesta sèrie es pot pensar com una continuació analítica del celebrat polinomi de Jones de teoria de nusos. Subjacent a aquesta construcció s'hi hagi l'elecció d'un grup de Lie $G=SU(2)$. Una pregunta natural concerneix l'estudi d'aquests invariants per a $SU(N)$ de major rang. Fins ara, han hagut intents de definir-los rigorosament, però una definició matemàtica de $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ per a nusos no torals encara manca en l'escenari general. Ens centrem en el grup de Lie $SU(3)$. Per a poder estudiar el comportament de l'invariant de nusos associat $F_K^{SU(3)}$, proporcionem una construcció a partir de $R$ matrius dels invariants de nusos cuántidos de $\sl_3$ per a qualsevol nus i representació irreductible finit-dimensional. El resultat principal d'aquesta tesi dona una construcció explícita de $F_K^{SU(3)}(x,y,q)$, que estén la seva prèvia definició a la família de major grandària nus de trena positiva. En particular, es mostra que $F_K^{LA SEVA(3)}(x,y,q)$ recupera el $\mathfrak{sl}_3$ invariant quàntic mitjançant especialitzacions en les variables $x$ e $y$., Uno de los problemas abiertos más excitantes en topología cuántica se relaciona con la categorificación de la teoría compleja de Chern-Simons. En este contexto, Gukov y Manolescu introducen una serie en dos variables $F_K(x,q)$ como la restricción de los invariantes de 3-variedades $\hat{Z}$ a complementos de nudos, la cual se ha demostrado que tiene propiedades sorprendentes. En particular, esta serie se puede pensar como una continuación analítica del celebrado polinomio de Jones de teoría de nudos. Subyacente a esta construcción se haya la elección de un grupo de Lie $G=SU(2)$. Una pregunta natural concierne el estudio de estos invariantes para $SU(N)$ de mayor rango. Hasta ahora, han habido intentos de definirlos rigurosamente, pero una definición matemática de $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ para nudos no torales todavía carece en el escenario general. Nos centramos en el grupo de Lie $SU(3)$. Para poder estudiar el comportamiento del invariante de nudos asociado $F_K^{SU(3)}$, proporcionamos una construcción a partir de $R$ matrices de los invariantes de nudos cuántidos de $\sl_3$ para cualquier nudo y representación irreducible finito-dimensional. El resultado principal de esta tesis da una construcción explícita de $F_K^{SU(3)}(x,y,q)$, que extiende su previa definición a la familia de mayor tamaño nudos de trenza positiva. En particular, se muestra que $F_K^{SU(3)}(x,y,q)$ recupera el $\mathfrak{sl}_3$ invariante cuántico mediante especializaciones en las variables $x$ e $y$., One of the most exciting open problems in quantum topology concerns the categorification of the complex Chern-Simons theory. In this context, Gukov and Manolescu introduce a two--variable series $F_K(x,q)$ as the restriction of the 3-manifold invariant $\hat{Z}$ to knot complements, which has been shown to exhibit surprising properties. In particular, this series can be thought as an analytical continuation of the celebrated colored Jones polynomial of knot theory. Underlying this construction there is the choice of a Lie group $G=SU(2)$. A natural question turns to the study of this invariants for higher rank $SU(N)$. So far, there have been attempts to define them rigorously, but a mathematical definition of $F_K^{SU(N)}(x_1,\dots,x_{N-1},q)$ for non-torus knots is still lacking in a general scenario. We focus on the Lie group $SU(3)$. In order to study the behaviour of the associated knot invariant $F_K^{SU(3)}$, we provide a construction via $R$-matrices of the $\mathfrak{sl}_3$ quantum knot invariants for any knot and finite-dimensional irreducible representation. The main result of this thesis gives an explicit construction of $F_K^{SU(3)}(x,y,q)$, which extends its former definition to the bigger family of positive braid knots. In particular, it is shown that $F_K^{SU(3)}(x,y,q)$ recovers the $\mathfrak{sl}_3$ quantum invariants via specializations on the $x$ and $y$ variables., Outgoing

A configuration space is a space whose points represent the possible states of a given physical system. As such they appear naturally both in theoretical physics and technical applications. For an example of the former, in analytical mechanics, the Lagrangian and Hamiltonian formulations of classical mechanics depend heavily on the use of a physical system’s configuration space for the description of its kinematical and dynamical behavior, and importantly, its evolution in time. As an example of a technical application, consider robotics, where the space of possible configurations of the mechanical linkages that make up a robot is an important tool in motion planning. In this case it is of particular interest to study the singularities of these mechanical linkages, to see if a given configuration is singular or not. This can be done with the help of configuration spaces and their topological properties. Arguably, the simplest configuration space possible arises when the system is just a collection of point-like particles in a plane. Despite its simplicity, the corresponding configuration space has substantial complexity and is of great interest in mathematics, physics and technology: For instance, it arises naturally in the mathematical modelling of robots performing tasks in a warehouse. In this thesis we go through the mathematics necessary to study the behaviour of paths in this space, which corresponds to motions of the particles. We use the theory of groups, algebraic topology, and manifolds to examine the properties of the configuration space of point-like particles in a plane. An important role in the discussion will be played by braids, which are certain collections of curves, interlaced in three-space. They are connected to many different topics in algebra, geometry, and mathematical physics, such as representation theory, the Yang-Baxter equation and knot theory. They are also important in their own right. Here we focus on their relation to configurations

Given an E3-ring spectrum A and an E1-A-algebra B, we prove that the A-based Adams spectral sequences and B-based Adams spectral sequence are isomorphic for well-behaved spec- tra. As an application, we show that for the spectrum Z constructed by Bhattacharya and Egger [BE50], its tmf-based Adams spectral sequence and k(2)-based Adams spectral sequence are isomorphic.

Given an E3-ring spectrum A and an E1-A-algebra B, we prove that the A-based Adams spectral sequences and B-based Adams spectral sequence are isomorphic for well-behaved spec- tra. As an application, we show that for the spectrum Z constructed by Bhattacharya and Egger [BE50], its tmf-based Adams spectral sequence and k(2)-based Adams spectral sequence are isomorphic.

This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle. Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well., Applied Mathematics

This thesis is about homological algebra and singular (co)homology. In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups. In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.

Manifolds with boundary, with corners, b-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds were introduced in Nest and Tsygan (2001 Asian J. Math.5 599–635) and investigated in depth in Miranda and Scott (2021 Rev. Mat. Iberoam.37 1207–24). In this article we explore their physical facets by extending gauge theories to the E-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E-symplectic manifold. Following the scheme inaugurated in Weinstein (1978 Lett. Math. Phys.2 417–20), we show the existence of a universal model for a particle interacting with an E-gauge field. In addition, we generalise the description of phase spaces in Yang–Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in Montgomery (1986 PhD Thesis University of California, Berkeley), for base manifolds endowed with an E-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang–Mills field, become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole., Peer Reviewed, Postprint (author's final draft)

The two most prevalent classes of ordered states in quantum materials are those arising from spontaneous symmetry breaking (SSB) and from topological order. However, a systematic study for their coexistence in interacting systems is still lacking. In this thesis, we will investigate how the topological configuration in order parameter spaces from SSB interplays with the symmetry protected/enriched topological orders in two spatial dimensions. We start from a phenomenological model of the domain wall structure in chiral spin liquids with domains of opposite chiralities. Based on a standard model, we obtain a spatially varying, self-consistent mean-field solution for the spinons that describes both the gapless edge modes and the change of chirality at the domain wall. We derive the non-universal properties, such as the velocity of the topologically protected domain wall edge states and its modification to domain wall tension. Then we discuss the interplay between topological orders from a more formal point of view. We consider two-dimensional topologically ordered systems with coexisting long-range orders, where the only gapless excitations in the spectrum are Goldstone modes of spontaneously broken continuous symmetries. We show that the universal properties of point defects and textures are determined by the remnant symmetry enriched topological order in the symmetry-breaking ground state with non-fluctuating order parameters, and provide a classification for their properties using algebraic topology. We further investigate the quantum dynamics that can happen at the domain wall of topological orders, where particle-hole scattering leads to linear-in-temperature resistivity of edge modes. Finally, we introduce an effective edge network theory to characterize the boundary topology of coupled edge states generated from various types of topological insulators.

We report significant progress toward establishing an obstruction theory of equivariant Thom spectra with multiplicative structures arising from maps of $E_V$ spaces. Focusing on the Fujii-Landweber Real bordism spectrum, we explain an argument to show that a homotopy ring Real orientation of an $E_{\rho}$-spectrum satisfying strong equivariant evenness and mild additional commutativity condition lifts to an $E_{\rho}$ map.First, we address the foundational issues. We discuss model categories for equivariant $C_2$-spectra indexed on Real inner product spaces and comparisons among them. We explain which foundations are needed for our project and describe the parts that would be original even non-equivariantly.Next, we construct equivalences between several $E_V$ operads, including little disks and Steiner operads. We show that algebras over $E_{V \oplus \Rbb}$ operads can be strictified to monoids in the category of $E_V$-algebras.To compute obstruction groups for maps of $E_V$ algebras, we analyze the spectra of the derived indecomposibles of augmented $E_V$-algebras. We determine that such spectra are $V$-fold desuspensions of a lift of the May delooping machine to the augmented algebras.Next, we discuss the induced map on cohomology for a map between spaces whose $RO(C_2)$-graded cohomology carries obstruction classes for homotopy ring and $E_\rho$ ring maps out of the Real bordism spectrum.Finally, we review the strategies for the proofs of the goal results of our project, explaining how to use the technique of climbing the slice tower of the target to construct highly structured orientations out of Real bordism.

Homogeneity across lineages is a general assumption in phylogenetics according to which nucleotide substitution rates are common to all lineages. Many phylogenetic methods relax this hypothesis but keep a simple enough model to make the process of sequence evolution more tractable. On the other hand, dealing successfully with the general case (heterogeneity of rates across lineages) is one of the key features of phylogenetic reconstruction methods based on algebraic tools. The goal of this paper is twofold. First, we present a new weighting system for quartets (ASAQ) based on algebraic and semi-algebraic tools, thus especially indicated to deal with data evolving under heterogeneous rates. This method combines the weights of two previous methods by means of a test based on the positivity of the branch lengths estimated with the paralinear distance. ASAQ is statistically consistent when applied to data generated under the general Markov model, considers rate and base composition heterogeneity among lineages and does not assume stationarity nor time-reversibility. Second, we test and compare the performance of several quartet-based methods for phylogenetic tree reconstruction (namely QFM, wQFM, quartet puzzling, weight optimization and Willson’s method) in combination with several systems of weights, including ASAQ weights and other weights based on algebraic and semi-algebraic methods or on the paralinear distance. These tests are applied to both simulated and real data and support weight optimization with ASAQ weights as a reliable and successful reconstruction method that improves upon the accuracy of global methods (such as neighbor-joining or maximum likelihood) in the presence of long branches or on mixtures of distributions on trees., We would like to thank the reviewers of the paper for important contributions that improved the final version of the manuscript. MC, JFS and MGL were partially supported by Spanish State Research Agency grant PID2019-103849GB-I00. MC and JFS were also supported by AEI through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R &D (project CEX2020-001084-M) and by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP., Peer Reviewed, Postprint (published version)

We initiate the study of higher cyclic operads with involutive colour sets as cyclic dendroidal sets: presheaves on an appropriate category of trees. This theory lays the groundwork for an analogue of the Moerdijk-Cisinski program for cyclic operads. We introduce a homotopy coherent nerve from simplicial cyclic operads to cyclic dendoridal sets, and show it enjoys good homotopical properties. Finally, we introduce a new model for higher cyclic operads which is amenable to the construction of concrete examples.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes. First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes. Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode. Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which c

Spectral sequences were developed during the mid-twentieth century as a way of computing (co)homology, and have wide uses in both algebraic topology and algebraic geometry. Grothendieck introduced in his Tôhoku paper the Grothendieck spectral sequence, which given left exact functors $F$ and $G$ between abelian categories, uses the right-derived functors of $F$ and $G$ as initial data and converges to the right-derived functors of the composition $G\circ F.$ This thesis focuses on instead constructing a spectral sequence that uses the derived functors of $G$ and $G\circ F$ as initial data and converges to the derived functors of $F.$ Our approach takes inspiration from the construction of the Eilenberg-Moore spectral sequence, which given a fibration of topological spaces can calculate the singular cohomology of the fiber from the singular cohomology of the base space and total space. The Eilenberg-Moore spectral sequence can be constructed through the use of differential graded algebras and their bar construction, since this defines a double complex for which the column-wise filtration of the corresponding total complex induces the spectral sequence. The correct analogue of this with respect to composite functors is the bar construction for monads. Specifically, we let $G$ have an exact left adjoint $H$, which makes $G\circ H$ into a monad. Then, we extend our adjunction so that the derived functor $RG$ has left adjoint $RH$ in the corresponding derived categories, making $RG\circ RH$ into a monad. This allows us to apply the bar construction in the derived category, but we show that there emerge issues in obtaining a double complex and subsequent total complex from this construction. Additionally, we present the essential theory of spectral sequences in general, and of the Serre, Eilenberg-Moore and Grothendieck spectral sequences in particular., Spektralsekvenser utvecklades under mitten av 1900-talet som ett verktyg för att beräkna (ko)homologi, och har många användningsområden inom både algebraisk topologi och algebraisk geometri. Grothendieck introducerade i sin Tôhoku-artikel Grothendieck-spektralsekvensen, som givet vänsterexakta funktorer $F$ och $G$ mellan abelska kategorier använder de högerderiverade funktorerna av $F$ och $G$ som initialdata och som konvergerar till de högerderiverade funktorerna av kompositionen $G\circ F$. Denna masteruppsats fokuserar på att istället konstruera en spektralsekvens som använder de deriverade funktorerna av $G$ och $G\circ F$ som initialdata och konvergerar till de deriverade funktorerna av $F$. Vår metod tar inspiration från konstruktionen av Eilenberg-Moore-spektralsekvensen, som givet en fibrering av topologiska rum kan beräkna den singulära kohomologin av fibern från den singulära kohomologin av basrummet och totalrummet. Eilenberg-Moore spektralsekvensen kan konstrueras genom användningen av graderade differentialalgebror och deras bar-konstruktion, eftersom detta definierar ett dubbelkomplex vars kolumnvisa filtrering av det resulterande totalkomplexet inducerar spektralsekvensen. Vad gäller kompositioner av funktorer så är den korrekta analogin till detta bar-konstruktionen för monader. Specifikt så låter vi $G$ ha en exakt vänsteradjungerad funktor $H$, vilket gör $G\circ H$ till en monad. Sedan utvidgar vi denna adjunktion sådant att den deriverade funktorn $RG$ har vänsteradjunkt $RH$ i den deriverade kategorin, vilket gör $RG\circ RH$ till en monad. Detta ger oss möjligheten att använda bar-konstruktionen i den deriverade kategorin, men vi visar att det uppstår problem när vi ska definiera ett dubbelkomplex och resulterande totalkomplex från denna konstruktion. Utöver detta så innehåller denna uppsats en genomgång av den viktigaste teorin om spektralsekvenser i allmänhet, och om Serre-, Eilenberg-Moore- och Grothendieck-spektralsekvensen i synnerhet.

Spectral sequences were developed during the mid-twentieth century as a way of computing (co)homology, and have wide uses in both algebraic topology and algebraic geometry. Grothendieck introduced in his Tôhoku paper the Grothendieck spectral sequence, which given left exact functors $F$ and $G$ between abelian categories, uses the right-derived functors of $F$ and $G$ as initial data and converges to the right-derived functors of the composition $G\circ F.$ This thesis focuses on instead constructing a spectral sequence that uses the derived functors of $G$ and $G\circ F$ as initial data and converges to the derived functors of $F.$ Our approach takes inspiration from the construction of the Eilenberg-Moore spectral sequence, which given a fibration of topological spaces can calculate the singular cohomology of the fiber from the singular cohomology of the base space and total space. The Eilenberg-Moore spectral sequence can be constructed through the use of differential graded algebras and their bar construction, since this defines a double complex for which the column-wise filtration of the corresponding total complex induces the spectral sequence. The correct analogue of this with respect to composite functors is the bar construction for monads. Specifically, we let $G$ have an exact left adjoint $H$, which makes $G\circ H$ into a monad. Then, we extend our adjunction so that the derived functor $RG$ has left adjoint $RH$ in the corresponding derived categories, making $RG\circ RH$ into a monad. This allows us to apply the bar construction in the derived category, but we show that there emerge issues in obtaining a double complex and subsequent total complex from this construction. Additionally, we present the essential theory of spectral sequences in general, and of the Serre, Eilenberg-Moore and Grothendieck spectral sequences in particular., Spektralsekvenser utvecklades under mitten av 1900-talet som ett verktyg för att beräkna (ko)homologi, och har många användningsområden inom både algebraisk topologi och algebraisk geometri. Grothendieck introducerade i sin Tôhoku-artikel Grothendieck-spektralsekvensen, som givet vänsterexakta funktorer $F$ och $G$ mellan abelska kategorier använder de högerderiverade funktorerna av $F$ och $G$ som initialdata och som konvergerar till de högerderiverade funktorerna av kompositionen $G\circ F$. Denna masteruppsats fokuserar på att istället konstruera en spektralsekvens som använder de deriverade funktorerna av $G$ och $G\circ F$ som initialdata och konvergerar till de deriverade funktorerna av $F$. Vår metod tar inspiration från konstruktionen av Eilenberg-Moore-spektralsekvensen, som givet en fibrering av topologiska rum kan beräkna den singulära kohomologin av fibern från den singulära kohomologin av basrummet och totalrummet. Eilenberg-Moore spektralsekvensen kan konstrueras genom användningen av graderade differentialalgebror och deras bar-konstruktion, eftersom detta definierar ett dubbelkomplex vars kolumnvisa filtrering av det resulterande totalkomplexet inducerar spektralsekvensen. Vad gäller kompositioner av funktorer så är den korrekta analogin till detta bar-konstruktionen för monader. Specifikt så låter vi $G$ ha en exakt vänsteradjungerad funktor $H$, vilket gör $G\circ H$ till en monad. Sedan utvidgar vi denna adjunktion sådant att den deriverade funktorn $RG$ har vänsteradjunkt $RH$ i den deriverade kategorin, vilket gör $RG\circ RH$ till en monad. Detta ger oss möjligheten att använda bar-konstruktionen i den deriverade kategorin, men vi visar att det uppstår problem när vi ska definiera ett dubbelkomplex och resulterande totalkomplex från denna konstruktion. Utöver detta så innehåller denna uppsats en genomgång av den viktigaste teorin om spektralsekvenser i allmänhet, och om Serre-, Eilenberg-Moore- och Grothendieck-spektralsekvensen i synnerhet.

Our discussion starts with the study of convergence and clustering of filters initiated in pointfree setting by Hong, and then characterize compact and almost compact frames in terms of these filters. We consider the strict extension and show that tQL is a zerodimensional compact frame, where Q denotes the set of filters in L. Furthermore, we study the notion of general filters introduced by Banaschewski and characterize compact frames and almost compact frames using them. For filter selections, we consider F−compact and strongly F−compact frames and show that lax retracts of strongly F−compact frames are also strongly F−compact. We study further the ideals Rs(L) and RK(L) of the ring of realvalued continuous functions on L, RL. We show that Rs(L) and RK(L) are improper ideals of RL if and only if L is compact. We consider also fixed ideals of RL and showthat L is compact if and only if every ideal of RL is fixed if and only if every maximalideal of RL is fixed. Of interest, we consider the class of isocompact locales, which is larger that the class of compact frames. We show that isocompactness is preserved by nearly perfect localic surjections. We study perfect compactifications and show that the Stone-Cˇech compactifications and Freudenthal compactifications of rim-compact frames are perfect. We close the discussion with a small section on Z−closed frames and show that a basically disconnected compact frame is Z−closed.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes. First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes. Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode. Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which c, Applied Mathematics | Applied Physics

We begin by investigating analogues of the Ravenel conjectures in chromatic homotopy in the setting of Real-oriented homotopy theory, where one carries the data of canonical group actions by the cyclic group of order 2 via complex conjugation. This analysis yields a formula for Bousfield localization of $C_2$-spectra at the Real Johnson-Wilson theories, $E_\R(n)$, from which follows a smash product theorem and a chromatic convergence theorem for cofree $C_2$-spectra. We turn to a systematic study of cofreeness in Real-oriented homotopy theory and establish the cofreeness of the norms of Real bordism theory, $N_{C_2}^{C_{2^n}}MU_\R$, for all $n\ge1$, recovering a result of Hu and Kriz at $n=1$. The method of proof establishes a connection to the Segal conjecture for $C_2$ - also known as Lin's theorem - and yields a new, conceptual proof of this classical result.We finish by bringing various equivariant spectra in Real-oriented homotopy theory into the world of stacks and chromatic homotopy by applying a method of Hopkins' to their fixed point spectra. We demonstrate this in detail for the Real Johnson-Wilson theories and give several modular descriptions of the stacks $\mc M_{ER(n)}$, recovering and generalizing Hopkins' description at $n=1$ of $\mc M_{KO}$ as the moduli stack of nonsingular quadratic equations.

Although its uninterrupted supply is essential for everyday life, the electricity occasionally experiences disruptions and outages. The work presented in the current paper aims to initiate the research to design a strategy based on advanced approaches of algebraic topology to prevent such malfunctions in a power grid network. Simplicial complexes are constructed to identify higher-order structures embedded in a network and, alongside a new algorithm for identifying delegates of the simplicial complex, are intended to pinpoint each element of the power grid network to its natural layer. Results of this methodology for analysis of a power grid network can single out its elements that are at risk to cause cascade problems which can result in unintentional islanding and blackouts. Further development of the outcomes of research can find implementation in the algorithms of the energy informatics research applications.

Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size.

Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size.

In \cite{BS}, the authors constructa spectral sequence which converges to the homology groups of the graph configuration space. This construction requires a characteristic $0$ field to ensure a commutative model for the cochain algebra. For arbitrary coefficients a commutative model may not exist and we suggest a different approach. Each vertex of a graph $G$ is colored by a copy of $C_{N}^{*}(M;R)$, the normalized cochains, where $R$ is a commutative ring with unity of any characteristic. We construct a complex similar to the Bendersky-Gitler complex in \cite{BG}. Its differential involves sums over sequences of collapsing edges of the graph: for a single collapsed edge multiplication of tensor factors in $C_{N}^{*}(M;R)$ is used, while for general sequences one uses the sequence operations of McClure and Smith, cf. \cite{MS}. If the graph $G$ has at most 5 vertices with a planar-type labelling and $Z_{G} \subset M^{\times n}$ is the closed subspace of diagonals built from the graph $G$, we show this computes the relative cohomology groups $H^{*}(M^{\times n},Z_{G};R)$. These are isomorphic to the homology groups of the graph configuration space $H_{nm-*}(M^{G};R)$ if $M$ is a compact oriented manifold. When $G$ is the complete graph on $n$ vertices, these are the homology groups of the usual (labelled) configuration space.

Blumberg and Hill defined categories of equivariant spectra interpolating between the equivariant stable categories indexed by universes. Indexed by an N∞ operad O, these O-spectra are characterized by what transfers they admit between their fixed points. We study structural properties of the O-incomplete equvariant stable categories. We first show an analog of a theorem of Guillou and May, giving an equivalence between O-spectra and spectrally enriched presheaves on a spectral enhancement of the incomplete Burnside category. Using this, we define the smash product and geometric fixed points of O-spectra in terms of Kan extensions and show that O-spectra can be recovered from gluing diagrams between their geometric fixed points. Finally, we apply this to give Mayer-Vietoris sequences describing the Picard group of O-spectra. In the presence of an appropriate Segal conjecture, we compare this to the Picard group of invertible abelian Mackey functors, giving a partial generalization of a theorem by Fausk, Lewis, and May.

Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size.

Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size.

© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works., We present the phylogenetic quartet reconstruction method SAQ (Semi-Algebraic Quartet reconstruction). SAQ is consistent with the most general Markov model of nucleotide substitution and, in particular, it allows for rate heterogeneity across lineages. Based on the algebraic and semi-algebraic description of distributions that arise from the general Markov model on a quartet, the method outputs normalized weights for the three trivalent quartets (which can be used as input of quartet-based methods). We show that SAQ is a highly competitive method that outperforms most of the well known reconstruction methods on data simulated under the general Markov model on 4-taxon trees. Moreover, it also achieves a high performance on data that violates the underlying assumptions., The authors were partially supported by Spanish government Secretar´ıa de Estado de Investigaci´on, Desarrollo e Innovaci´on [MTM2015-69135-P (MINECO/FEDER)] and [PID2019- 103849GB-I00 (MINECO)]; Generalitat de Catalunya [2014 SGR-634]. M. Garrote-L´opez was also funded by Spanish government, Ministerio de Econom´ıa y Competitividad research project Maria de Maeztu [MDM-2014-0445]., Peer Reviewed, Postprint (author's final draft)