Browsing by Subject "Topology"

Now showing 1 - 6 of 6
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    Applications of geometric techniques in Coxeter-Catalan combinatorics
    (2017-09) Douvropoulos, Theodosios
    In the seminal work [Bes 15], Bessis gave a geometric interpretation of the noncrossing lattice NC(W)​​ associated to a well-generated complex reflection group W​​. He used it as a combinatorial recipe to construct the universal covering space of the arrangement complement V⧵ ⋃ H​​, and to show that it is contractible, hence proving the K(π,1)​​ conjecture. Bessis' work however relies on a few properties of NC(W)​​ that are only known via case by case verification. In particular, it depends on the numerological coincidence between the number of chains in NC(W)​​ and the degree of a finite morphism, the LL​​ map. We propose a (partially conjectural) approach that refines Bessis' work and transforms the numerological coincidence into a corollary. Furthermore, we consider a variant of the LL​​ map and apply it to the study of finer enumerative properties of NC(W)​​. In particular, we extend work of Bessis and Ripoll and enumerate the so-called ``primitive factorizations" of the Coxeter element c​​. That is, length additive factorizations of the form c=w⋅ t1⋯ tk​​, where w​​ belongs to a prescribed conjugacy class and the ti​​'s are reflections.
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    Higher Picard groupoids and Dijkgraaf-Witten theory
    (2016-08) Sharma, Amit
    In the first part of this thesis we propose a model for additive $\infty$-categories based on $\gSs$ and construct the archetype example of an additive $\infty$-category, namely the (quasi)-category of higher Picard groupoids, $\pic$. The goal of the second part of this thesis is to categorify Dijkgraaf-Witten (DW) theory, aiming at providing foundation for a direct construction of DW theory as an Extended Topological Quantum Field Theory. The main tool is cohomology with coefficients in a Picard groupoid, namely the Picard groupoid of hermitian lines.
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    Highly Structured Multiplication & The Miller Spectral Sequence
    (2019-08) Hess, Daniel
    We examine the Miller spectral sequence for determining the mod-p homology of a connective spectrum X from the mod-p homology of its associated infinite loop space, Ω∞X, considered as an algebra over the mod-p Dyer-Lashof algebra. For each prime p, we give a Koszul complex for computing the E2 page of this spectral sequence, recovering a result of Miller (at p = 2) [35] and Kraines and Lada (at odd primes) [22]. As applications, we determine H∗(HZ; Fp) and H∗(HFp; Fp) at all primes, recovering well-known results. As an original application of the Miller spectral sequence, we study the relationship between H∗(Ω∞X; Fp) and H∗(X; Fp) when X is an E∞-ring spectrum. We show that the Miller spectral sequence can be used to detect nonzero “multiplicative” k-invariants of X at all primes. We also prove that for any integer n ≥ 1, the underlying spectrum of a commutative HFp-algebra R is equivalent to its strict unit spectrum, sl1(R), in a range that is wider than the stable range: [n, pn − 1]. This is a special case of a conjecture by Mathew and Stojanoska [29].
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    The homology of the kernel space of the Thom spectrum in low degrees
    (2024) Huttinga, Zane
    I present an investigation of the multiplicative structure of H_*(SL_1 MU; F_2) as an algebra over F_2. This includes a list of generators in low degrees and the multiplication table for these generators. This ring has applications in topology related to orientability of vector bundles, as it is closely tied to the homology of the spectrum sl_1 MU. The ring H_*(SL_1 MU) is closely tied to the Hopf ring H_*(MU_2*) introduced by Ravenel and Wilson in 1977. All of the available multiplicative information on H_*(SL_1 MU) descends from H_*(MU_2*), and accordingly, I present a complete algebraic description of this Hopf ring.
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    Nim in Topological Spaces
    (2019-10) Victorin-Vangerud, Aaron
    Nim is a mathematical game played with piles of rocks. Two players take turns choosing a pile and removing some of its rocks, and whoever takes the last rock wins. Strategy in nim is well understood, but what happens if we replace the rocks with sets in a space and say the game ends when the chosen sets make a cover? In a simple finite case, we present a winning strategy that evokes classic nim.
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    Simulation data for: "Two parameter scaling in the crossover from symmetry class BDI to AI"
    (2022-08-01) Kasturirangan, Saumitran; Kamenev, Alex; Burnell, Fiona J; kastu007@umn.edu; Kasturirangan, Saumitran
    The transport statistics at finite energies near a quantum critical point in the presence of disorder were not well understood analytically. This was approached by performing extensive simulations of transport using the package KWANT for python for disordered 1D quantum chains and metallic arm-chair graphene nanoribbons. This dataset contains the resulting data for several system sizes, strengths, and energies. This was used to establish two-parameter scaling and characterize the transport statistics.