User:Tazerenix

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WikiProject Mathematics.

Pure mathematician, differential geometer, algebraic geometer. Primarily interested in complex geometry and its relations to mathematical physics. I try to create pictures for geometry pages which don't have many. See my Wikimedia profile.

The dx¹⊗σ₃ coefficient of a BPST instanton on the (x¹,x²)-slice of R⁴ where σ₃ is the third Pauli matrix (top left). The dx²⊗σ₃ coefficient (top right). These coefficients determine the restriction of the BPST instanton A with g=2,ρ=1,z=0 to this slice. The corresponding field strength centered around z=0 (bottom left). A visual representation of the field strength of a BPST instanton with center z on the compactification S⁴ of R⁴ (bottom right).

A principal bundle connection form $\omega$ may be thought of as a projection operator on the tangent bundle $TP$ of the principal bundle $P$. The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection.

Two trivial vector bundles over open sets $U_{\alpha }$ and $U_{\beta }$ may be glued over the intersection $U_{\alpha \beta }$ by transition functions $g_{\alpha \beta }$ which serve to stick the shaded grey regions together in a non-trivial way (note the transformation of the blue quadrilateral under the effect of $g_{\alpha \beta }$). Different choices of transition functions may result in different vector spaces which are non-trivial after gluing is complete.

Hilbert–Mumford Criterion

Articles created or made significant contributions to

Article	Contribution
K-stability	Created article.
Yang–Mills equations	Created article.
Hermitian Yang–Mills connection	Created article.
Gauge theory (mathematics)	Created article.
Complex geometry	Significantly improved exposition and discussion of the relevant topics. Essentially rewrote the whole article.
Donaldson's theorem	Added a summary of the proof with diagram of the cobordism between the moduli space of connections and the four-manifold.
Holomorphic vector bundle	Added a discussion of Dolbeault operators and their relation to holomorphic structures.
Holomorphic tangent bundle	Cleaned up the distinction between the holomorphic tangent bundle and the holomorphic tangent space.
Serre duality	Explained the differential-geometric statement of Serre duality.
Deformed Hermitian Yang–Mills equation	Created article.
Bridgeland stability condition	Created article.
Stable principal bundle	Created article.
Connection (vector bundle)	Added discussion of motivation, induced connections, gauge transformations, and expanded the sections on curvature and parallel transport with important formulae. Rewrote the section on local expressions and merged the majority of the redundant content of metric connections into the article. Standardised notation with gauge theory (mathematics).
SYZ conjecture	Added a mathematical statement and a diagram of a Lagrangian torus fibration. Still needs some discussion of the relation to homological mirror symmetry, and perhaps some discussion of the recent work on mirror symmetry in algebraic geometry and so on.
Nonabelian Hodge correspondence	Rewrote article. Added precise statement of definitions and theorem, and discussion of nonabelian Hodge theorem.
Kähler–Einstein metric	Significantly expanded article, including definition and discussion of existence. Included a discussion of the so-called Song--Tian program and the analytic minimal model program
Kobayashi–Hitchin correspondence	Rewrote article and massively expanded. Included precise statement of correspondence, detailed history and list of many important generalisations and other conjectures influenced by it.
Hitchin's equations	Created article.
Quillen metric	Wrote entire article (from one sentence to full article).
Differential topology	Replaced the old lead which was completely wrong, and wrote a new much broader lead summarizing the subject.
Mabuchi functional	Created article.
Constant scalar curvature Kähler metric/Extremal Kähler metric	Included definition and discussion of where extremal Kähler metrics come from. Still some work to do.
Manifold hypothesis	Created stub.
Differential geometry	New expanded lead. Fixed and greatly expanded the history section.
Vector bundle	Added a number of diagrams
Principal bundle	Added diagrams of the Mobius principal bundle and its description as the frame bundle of the Mobius strip.
Connection (principal bundle)	Added various diagrams describing principal bundle connections
K-stability of Fano varieties	Created article
Thomas–Yau conjecture	Created article
Stability (algebraic geometry)	Created article. Needs to be significantly expanded. Plans are at User:Tazerenix/Stability (algebraic geometry)
∂∂̅-lemma	Created article
Kähler identities	Created article. TODO: Populate CG wikipedia with more links to article.

Works in progress

Article	Plan
Stability (algebraic geometry)	Need to find some good sources in addition to summarising the main concepts of stability.
User:Tazerenix/Differential_geometry	Updating many sections
User:Tazerenix/de Rham cohomology	Adding section on cohomology twisted by flat vector bundle. Some of that content is included in flat vector bundle so should be put there. Including a section on twisted de Rham cohomology to lead to Twisted Poincare Duality.

Future projects

Article	Plan
All	Add at least one diagram or picture to every article mentioned on this page.
Constant scalar curvature Kähler metric	Expand the discussion on cscK metrics specifically, as page mostly focuses on extremal Kahler metrics.
Chern connection and Hermitian connection	Expand discussion about Chern connection. Possibly split into two pages.
Higgs bundle	Significantly expand the discussion of Higgs bundles and their stability. Include a definition of principal Higgs bundles. Perhaps add a discussion of the relevance of Higgs bundles to the geometric Langlands correspondence, and of the moduli space of G-Higgs bundles for Langlands dual structure groups to mirror symmetry.
Narasimhan–Seshadri theorem	Give a more precise statement of the theorem and an example in the case of line bundles, where the theorem follows from basic Hodge theory and essentially gives an isomorphism between the Jacobian and the complex torus.
Kempf–Ness theorem	Give a precise statement of the theorem and a summary of the proof, including the key details about density of rational one-parameter subgroups in the set of all one-parameter subgroups, following the summary proof given in the notes of Richard Thomas on GIT and symplectic reduction.
Geometric invariant theory	Improve the explanation of the Hilbert–Mumford criterion.
Stability (algebraic geometry) (see Draft:Stability (algebraic geometry))	Create an article explaining the many manifestations of the concept of stability in algebraic geometry, and how they all relate to each other and fit together. Explain the difference between a "GIT stability notion" and the more general picture.
Stable vector bundle	Clean up the page. Add some discussion of the derivation of slope stability from the Quot scheme and the relevant generalisation of Gieseker stability.
SYZ conjecture	Give discussion of relation between SYZ conjecture and homological mirror symmetry conjecture.
Complex geometry	Add further discussion of classification (e.g. algebraic surfaces, moduli spaces, minimal model program).
Kodaira embedding theorem	Give a precise statement of the theorem and discussion of what it means for positivity versus ampleness, and make some comments about the proof and its use of blowups.
Metric connection	Take the parts of this article that apply to any connection and merge them with Connection (vector bundle). If only a few statements are left over then merge this article into the vector bundles article.