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| Author | : Atanas Atanasov |
| Publisher | : |
| Total Pages | : |
| Release | : 2018 |
| Genre | : Curves, Algebraic |
| ISBN | : 9781470449513 |
| Author | : Atanas Atanasov |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 105 |
| Release | : 2019-02-21 |
| Genre | : Curves, Algebraic |
| ISBN | : 147043489X |
Given n general points p1,p2,…,pn∈Pr, it is natural to ask when there exists a curve C⊂Pr, of degree d and genus g, passing through p1,p2,…,pn. In this paper, the authors give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle NC of a general nonspecial curve of degree d and genus g in Pr (with d≥g+r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H0(NC(−D))=0 or H1(NC(−D))=0), with exactly three exceptions.
| Author | : Atanas Valeryev Atanasov |
| Publisher | : |
| Total Pages | : |
| Release | : 2015 |
| Genre | : |
| ISBN | : |
Interpolation is a property of vector bundles on curves closely related to slope stability. The notion is motivated by the deformation theory of curves in projective space incident to given fixed subvarieties. If the normal bundle of a projective curve satisfies interpolation, then curves in the same component of the Hilbert scheme exhibit normal behavior with respect to incident problems.
| Author | : Dominic Joyce |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 139 |
| Release | : 2019-09-05 |
| Genre | : |
| ISBN | : 1470436450 |
If X is a manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φf:C∞(X)n→C∞(X) acting by Φf:(c1,…,cn)↦f(c1,…,cn), and these operations Φf satisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obvious R-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C∞-schemes, a category of geometric objects which generalize manifolds and whose morphisms generalize smooth maps. The author also studies quasicoherent sheaves on C∞-schemes, and C∞-stacks, in particular Deligne-Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C∞-rings and C∞ -schemes have long been part of synthetic differential geometry. But the author develops them in new directions. In earlier publications, the author used these tools to define d-manifolds and d-orbifolds, “derived” versions of manifolds and orbifolds related to Spivak's “derived manifolds”.
| Author | : Ines Kath |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 84 |
| Release | : 2020-02-13 |
| Genre | : Education |
| ISBN | : 1470441039 |
Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. The authors derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these parameters.
| Author | : Chen Wan |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 90 |
| Release | : 2019-12-02 |
| Genre | : Education |
| ISBN | : 1470436868 |
Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.
| Author | : Charles Collot |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 93 |
| Release | : 2019-09-05 |
| Genre | : |
| ISBN | : 1470436264 |
The authors consider the energy super critical semilinear heat equation The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.
| Author | : Tvrtko Tadić |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 170 |
| Release | : 2019-12-02 |
| Genre | : Education |
| ISBN | : 147043685X |
The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure— so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.
| Author | : Oskari Ajanki |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 133 |
| Release | : 2019-12-02 |
| Genre | : Education |
| ISBN | : 1470436833 |
The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z∈H, including the vicinity of the singularities.
| Author | : J. I. Hall |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 186 |
| Release | : 2019-09-05 |
| Genre | : Categories (Mathematics) |
| ISBN | : 1470436221 |
In 1925 Élie Cartan introduced the principal of triality specifically for the Lie groups of type D4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title in 1978 was made by Stephen Doro, who was in turn motivated by the work of George Glauberman from 1968. Here the author makes the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.”
