The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU(2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of 2-spheres or 2-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL(3,R)-representations

Euler number of Instanton Moduli space and Seiberg-Witten invariants

We show that a partition function of topological twisted N=4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of Euler number and signature of the four manifolds vanish. The partition function is the sum of Euler number of instanton moduli space when it is possible to apply the vanishing theorem. And we get a relation of Euler number labeled by the instanton number $k$ with Seiberg-Witten invariants, too. All calculation in this paper is done without assuming duality.Comment: LaTeX, 34 page

Lipid changes within the epidermis of living skin equivalents observed across a time-course by MALDI-MS imaging and profiling

© 2015 Mitchell et al. Abstract Background: Mass spectrometry imaging (MSI) is a powerful tool for the study of intact tissue sections. Here, its application to the study of the distribution of lipids in sections of reconstructed living skin equivalents during their development and maturation is described. Methods: Living skin equivalent (LSE) samples were obtained at 14 days development, re-suspended in maintenance medium and incubated for 24 h after delivery. The medium was then changed, the LSE re-incubated and samples taken at 4, 6 and 24 h time points. Mass spectra and mass spectral images were recorded from 12 μm sections of the LSE taken at each time point for comparison using matrix assisted laser desorption ionisation mass spectrometry. Results: A large number of lipid species were identified in the LSE via accurate mass-measurement MS and MSMS experiments carried out directly on the tissue sections. MS images acquired at a spatial resolution of 50 μm × 50 μm showed the distribution of identified lipids within the developing LSE and changes in their distribution with time. In particular development of an epidermal layer was observable as a compaction of the distribution of phosphatidylcholine species. Conclusions: MSI can be used to study changes in lipid composition in LSE. Determination of the changes in lipid distribution during the maturation of the LSE will assist in the identification of treatment responses in future investigations

Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K_X \otimes [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K_X \otimes [D] coincides with the degree with respect to the complete K\"ahler-Einstein metric g_{X \setminus D} on X \setminus D. For stable holomorphic vector bundles, we prove the existence of a Hermitian-Einstein metric with respect to g_{X \setminus D} and also the uniqueness in an adapted sense.Comment: 21 pages, International Journal of Mathematics (to appear

Pathways to Online Hate: Behavioural, Technical, Economic, Legal, Political & Ethical Analysis.

The Alfred Landecker Foundation seeks to create a safer digital space for all. The work of the Foundation helps to develop research, convene stakeholders to share valuable insights, and support entities that combat online harms, specifically online hate, extremism, and disinformation. Overall, the Foundation seeks to reduce hate and harm tangibly and measurably in the digital space by using its resources in the most impactful way. It also aims to assist in building an ecosystem that can prevent, minimise, and mitigate online harms while at the same time preserving open societies and healthy democracies. A non-exhaustive literature review was undertaken to explore the main facets of harm and hate speech in the evolving online landscape and to analyse behavioural, technical, economic, legal, political and ethical drivers; key findings are detailed in this report

Accountability

A week into the anti-racism protests that followed the chilling murder of George Floyd by a Minnesota police officer, the co-founder of Black Lives Matter (BLM), Patrisse Cullors, appeared on ABC’s Nightline. The show recapped the extraordinary events of that day: US President Donald Trump had announced that he was ‘dispatching thousands and thousands of heavily armed soldiers, military personnel, and law enforcement officers’ to quell the demonstrations. Federal police and military troops had used tear gas, rubber bullets and flash grenades on peaceful protestors gathered around the White House to clear the route to a cynical photo-op: the President posing before a church that he did not enter, holding aloft a Bible that he did not open

The ADHM Construction of Instantons on Noncommutative Spaces

We present an account of the ADHM construction of instantons on Euclidean space-time $\mathbb{R}^4$ from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parameterised by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal-Groenewold plane $\mathbb{R}^4_\hbar$ and the Connes-Landi plane $\mathbb{R}^4_\theta$.Comment: Latex, 40 page

Exotic Smoothness and Physics

The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space-time models, especially the simplest one, ${\bf R^4}$, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures available for space-time models. These can be thought of as source of new global, but not properly topological, features. This paper reviews some background differential topology together with a discussion of the role which a differentiable structure necessarily plays in the statement of any physical theory, recalling that diffeomorphisms are at the heart of the principle of general relativity. Some of the history of the discovery of exotic, i.e., non-standard, differentiable structures is reviewed. Some new results suggesting the spatial localization of such exotic structures are described and speculations are made on the possible opportunities that such structures present for the further development of physical theories.Comment: 13 pages, LaTe