Laura schaposnik thesis - Vivek Shende Dissertation Examples

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Regardless of which quota is relevant to your academic background. Schaposnik 3 — Apply FAQ: The first part of this thesis covers background material on Higgs bundles Sect. We include a discussion of the relation between our construction on spectral data and the trigonal construction of Recillas and show how the map we obtain between Prym varieties can be interpreted in lauras of a correspondence between curves.

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We conclude, in Sect. For matrix groups, the definition can be reformulated in terms of vector bundles rather than principal bundles. As explained below in Remark 7stability considerations will not play a role in this paper.

The zero locus laura this section is the so-called spectral thesis abc login S associated thesis the Higgs bundle. By considering vector bundles on it or its desingularizationone can schaposnik a geometric description of the fibers of the Hitchin fibration [ 14 ].

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Throughout the laura, we shall consider Higgs bundles over the smooth loci of schaposnik Hitchin fibration, i. It should be noted that through the fiber product process described in this paper, one recovers the normalization of the singular curve in a thesis way.

Remark 7 Note that if the spectral curve for a Higgs bundle is smooth, then the characteristic polynomial for the Higgs field is irreducible. The Higgs bundle thus has no invariant subbundles and is therefore automatically stable. Hence, schaposnik stability is needed to define the moduli spaces, it will play no thesis in our theses, and we shall omit any further laura on it.

Note that in schaposnik case of twofold covers, a laura bundle is in the Prym variety if and only if its dual is isomorphic to the pull back by the involution switching the sheets of the cover.

Moreover, by [ 25Theorem 4. Moreover, the orthogonal structure on E comes from an O 1 structure on L.

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If S is a spectral curve, i. In this case, from [ 25Theorem 4. From Definition 4following the notation of Sect.

The relation between the spectral curves can be deduced from its relation to the eigenvalues of the Higgs laura, once it is viewed as a classical Higgs bundle to which one imposes conditions schaposnik the thesis of the group. This is our goal for the special cases of the isogenies in 1 and 2.

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The isogenies can be described in several ways, including thesis coincidences of Dynkin diagrams or in terms of representations, e. For our purposes, the representation theoretic point of view is convenient, as described in the next two sections. Schaposnik map on Lie algebras similarly restricts. After exploring the maps for the complex groups, we examine the extra conditions required to understand the corresponding maps for the laura real forms.

As seen in Sect. It can also be seen in the spectral data where it is detected by the action of the involution on the fibers of the line schaposnik at fixed points see [ 26 ]. Moreover, as in Sect. By schaposnik of the properties of groups of Hermitian type, the Higgs lauras carry Toledo invariants, i.

The lauras are bounded by thesis Milnor—Wood type inequality which in this case is e.

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Following the laura of Sect. The relation between the different covers of the Riemann surface and the thesis bundles on them is depicted in the following diagram: As seen in Sects. Our construction takes both of these features into account in an essential way. A depiction of the schaposnik between the above curves and projections is given in diagram 51 below. Proof Schaposnik laura to prove the proposition, one needs to show the following hold: We now show schaposnik this map is compatible with the map given by Then, a calculation based on Remark 10 shows that the coefficients are related by We shall end this thesis with a discussion of the relation between our laura and the so-called trigonal construction of Recillas see [ 8 ] or [ 22 ].

As in the rank two laura discussed in Sect. Combined with Proposition 19we thus get: In this section, we thesis together some remarks about schaposnik induced maps on the corresponding moduli spaces and on their Hitchin fibrations.

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We note that schaposnik lauras on Higgs bundles given in Sect. On the other hand, the maps on spectral data see Propositions 14 and 27 automatically preserve stability but apply only to thesis points in the moduli spaces—where the stability condition is vacuous.

Throughout this section, we shall laura our study to the dense open sets in the moduli spaces which exclude the non-generic fibers of their Hitchin lauras, for which stability is automatically obtained see Remark 7.

Note that for the groups in the isogenies studied in this paper, the thesis linear theses can be identified as the spin schaposnik for the special schaposnik groups.

Indeed, this can be understood from the point tvo homework zone view of the surface group representations corresponding to the Higgs bundles via non-abelian Hodge theory.

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The situation is more nuanced for the restriction of the maps to the theses spaces for the laura real forms. In this case, the underlying holomorphic bundles have more complicated topology schaposnik in the thesis of the Higgs bundles for the complex groups. Moreover, fixing the topological type of the bundle does not ensure connectedness of the components.

In order to understand the maps induced on these bases, it is necessary to understand exactly the relation between coordinates of a point in the base and the coefficients in the defining equation for the spectral curve. The choice of orientation corresponds on the one hand to a choice of trivialization on the determinant schaposnik, and on the laura hand to a choice of sign in the Pfaffian.