The Hermitian Two Matrix Model With An Even Quartic Potential
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The Hermitian Two Matrix Model with an Even Quartic Potential
Author: Maurice Duits
language: en
Publisher: American Mathematical Soc.
Release Date: 2012
The authors consider the two matrix model with an even quartic potential $W(y)=y^4/4+\alpha y^2/2$ and an even polynomial potential $V(x)$. The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices $M_1$. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a $4\times4$ matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of $M_1$. The authors' results generalize earlier results for the case $\alpha=0$, where the external field on the third measure was not present.
Wave Front Set of Solutions to Sums of Squares of Vector Fields
Author: Paolo Albano
language: en
Publisher: American Mathematical Soc.
Release Date: 2013-01-25
The authors study the (micro)hypoanalyticity and the Gevrey hypoellipticity of sums of squares of vector fields in terms of the Poisson-Treves stratification. The FBI transform is used. They prove hypoanalyticity for several classes of sums of squares and show that their method, though not general, includes almost every known hypoanalyticity result. Examples are discussed.
A Theory of Generalized Donaldson-Thomas Invariants
Author: Dominic D. Joyce
language: en
Publisher: American Mathematical Soc.
Release Date: 2011
This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which `count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar{DT}{}^\alpha(\tau)$ are defined for all classes $\alpha$, and are equal to $DT^\alpha(\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. To prove all this, the authors study the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\mathfrak M$ may be written locally as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.