Supported Blow-Up and Prescribed Scalar Curvature on $S^n$

by Man Chun Leung

Author	: Man Chun Leung
Publisher	: American Mathematical Soc.
Total Pages	: 112
Release	: 2011
Genre	: Mathematics
ISBN	: 0821853376

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The author expounds the notion of supported blow-up and applies it to study the renowned Nirenberg/Kazdan-Warner problem on $S^n$. When $n \ge 5$ and under some mild conditions, he shows that blow-up at a point with positive definite Hessian has to be a supported isolated blow-up, which, when combined with a uniform volume bound, is a removable singularity. A new asymmetric condition is introduced to exclude single simple blow-up. These enable the author to obtain a general existence theorem for $n \ge 5$ with rather natural condition.

Dimer Models and Calabi-Yau Algebras

by Nathan Broomhead

Author	: Nathan Broomhead
Publisher	: American Mathematical Soc.
Total Pages	: 101
Release	: 2012-01-23
Genre	: Mathematics
ISBN	: 0821853082

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In this article the author uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. He further shows that the algebras constructed are examples of non-commutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds. Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a `superpotential'. Some examples are Calabi-Yau and some are not. The author considers two types of `consistency' conditions on dimer models, and shows that a `geometrically consistent' dimer model is `algebraically consistent'. He proves that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows him to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.

The Lin-Ni's Problem for Mean Convex Domains

by Olivier Druet

Author	: Olivier Druet
Publisher	: American Mathematical Soc.
Total Pages	: 118
Release	: 2012
Genre	: Mathematics
ISBN	: 0821869094

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The authors prove some refined asymptotic estimates for positive blow-up solutions to $\Delta u+\epsilon u=n(n-2)u^{\frac{n+2}{n-2}}$ on $\Omega$, $\partial_\nu u=0$ on $\partial\Omega$, $\Omega$ being a smooth bounded domain of $\mathbb{R}^n$, $n\geq 3$. In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, they prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.

$n$-Harmonic Mappings between Annuli

by Tadeusz Iwaniec

Author	: Tadeusz Iwaniec
Publisher	: American Mathematical Soc.
Total Pages	: 120
Release	: 2012
Genre	: Mathematics
ISBN	: 0821853570

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Iwaniec and Onninen (both mathematics, Syracuse U., US) address concrete questions regarding energy minimal deformations of annuli in Rn. One novelty of their approach is that they allow the mappings to slip freely along the boundaries of the domains, where it is most difficult to establish the existence, uniqueness, and invertibility properties of the extremal mappings. At the core of the matter, they say, is the underlying concept of free Lagrangians. After an introduction, they cover in turn principal radial n-harmonics, and the n-harmonic energy. There is no index. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).

The Goodwillie Tower and the EHP Sequence

by Mark Behrens

Author	: Mark Behrens
Publisher	: American Mathematical Soc.
Total Pages	: 109
Release	: 2012
Genre	: Mathematics
ISBN	: 0821869027

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The author studies the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime $2$. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. He relates the Goodwillie filtration to the $P$ map, and the Goodwillie differentials to the $H$ map. Furthermore, he studies an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. He shows that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. He uses his theory to recompute the $2$-primary unstable stems through the Toda range (up to the $19$-stem). He also studies the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod $2$ stable homology of the Goodwillie layers of any functor from spaces to spaces.

On the Shape of a Pure $O$-Sequence

by Mats Boij

Author	: Mats Boij
Publisher	: American Mathematical Soc.
Total Pages	: 93
Release	: 2012
Genre	: Mathematics
ISBN	: 0821869108

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A monomial order ideal is a finite collection X of (monic) monomials such that, whenever M∈X and N divides M, then N∈X. Hence X is a poset, where the partial order is given by divisibility. If all, say t t, maximal monomials of X have the same degree, then X is pure (of type t). A pure O-sequence is the vector, h_=(h0=1,h1,...,he), counting the monomials of X in each degree. Equivalently, pure O-sequences can be characterized as the f-vectors of pure multicomplexes, or, in the language of commutative algebra, as the h h-vectors of monomial Artinian level algebras. Pure O-sequences had their origin in one of the early works of Stanley's in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their f f-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure O-sequences.

Second Order Analysis on $(\mathscr {P}_2(M),W_2)$

by Nicola Gigli

$Second Order Analysis on $(\mathscr {P}_2(M),W_2)$$

Author	: Nicola Gigli
Publisher	: American Mathematical Soc.
Total Pages	: 173
Release	: 2012-02-22
Genre	: Mathematics
ISBN	: 0821853090

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The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

Towards a Modulo $p$ Langlands Correspondence for GL$_2$

by Christophe Breuil

Author	: Christophe Breuil
Publisher	: American Mathematical Soc.
Total Pages	: 127
Release	: 2012-02-22
Genre	: Mathematics
ISBN	: 0821852272

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The authors construct new families of smooth admissible $\overline{\mathbb{F}}_p$-representations of $\mathrm{GL}_2(F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. When $F$ is unramified, these representations have the $\mathrm{GL}_2({\mathcal O}_F)$-socle predicted by the recent generalizations of Serre's modularity conjecture. The authors' motivation is a hypothetical mod $p$ Langlands correspondence.

Reifenberg Parameterizations for Sets with Holes

by Guy David

Author	: Guy David
Publisher	: American Mathematical Soc.
Total Pages	: 114
Release	: 2012
Genre	: Mathematics
ISBN	: 0821853104

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The authors extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers $\beta_1(x,r)$. In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of $\mathbb R^d$.