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Consider the complex Monge-Amp`ere equation (i∂∂u¯ )n = dµ in a bounded domain in Cn, where u is a plurisubharmonic function and dµ is a positive Radon measure. We give a sharp condition on the right hand side to guarantee the existence of a H¨older continuous solution. Namely, there exists a H¨older continuous plurisubharmonic function ϕ such that dµ ≤ (i∂∂ϕ¯ )n in the sense of measures. In particular, the answer to a question of Ahmed Zeriahi is always positive. Then, we will discuss its counterparts on compact complex manifolds and some applications.
We survey recent applications of weak plurisubharmonic solutions to complex MongeAmp`ere equations, using pluripotential theory, for problems arising from complex geometry and algebraic geometry. On one hand it is used to construct singular K¨ahler-Einstein metrics, which may occur as the limits of the Kahler-Ricci flow or the limits of families of Calabi-Yau metrics when the K¨ahler class hits the boundary of the K¨ahler cone. On the other hand it played a crucial role in studying structure of the K¨ahler cone. In particular, we will discuss a weak form of a conjecture due to Demailly and Paun in 2004.
Over the past decade, electrical bioimpedance has been undergoing a rebirth as enhanced methodologies and new theories are greatly extending its use in the field of neuromuscular disease (NMD). Simply put, NMDs change the structure and internal composition of skeletal muscle which, in turn, alter the electrical properties muscle. Thus, the capability of measuring the electrical properties of muscle with accuracy has great potential to provide valuable new insights to inform medical assessment and diagnosis of NMDs. One technique well-suited for measuring the electrical properties of muscle is electrical bioimpedance, where an electrical current is applied to the muscle using two electrodes and the resultant voltage is measured using two additional electrodes. However, the accuracy to detect onset of disease, track disease progression and response to therapy using surface electrodes placed on the skin is limited: data are largely influenced by skin and subcutaneous fat (SF) overlying the muscle. Here, we will present a new source separation (SS) technique that, unlike existing blinded SS techniques principal component analysis (PCA) and independent component analysis (ICA), can distinguish muscle from SF with the accuracy being 99.2%.
However, the standard procedure of patient care for diagnosing NMDs consists of inserting needles electrodes into the muscle to measure the electrical activity at rest and during muscle contraction. To take advantage of this, we have designed an enhanced needle device also integrating impedance recording capabilities. Our new needle improves the accuracy measuring the electrical properties by recording these properties and their direction dependence directly in the muscle, the latter also known as anisotropy. Ongoing work in this area promises exciting and valuable new applications in the years to come.
(This is a reading seminar for graduate students.) Quillen's $K$-theory is defined for categories in which a suitable notion of exactness can be spoken. In much more generality, Waldhausen introduced the $S$-construction for categories equipped with suitable notions of cofibrations and weak equivalences. We will discuss the definition of S-construction, its dependence on the derived category, the cofinality theorem, and its comparison to Quillen's $K$-theory.
We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani.
Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via three different methods and demonstrate many interesting properties of the space:
- It is isomorphic to an open subscheme of a punctual Hilbert scheme.
- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.
- It is isomorphic to a bundle of regular jets.
- It gives examples of affine space bundles that are not vector bundles.
This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).
The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.
Since the early 2000s, it has turned out that well developed mathematical tools can play crucial roles in quantum information theory. One monumental work was made by Hastings in 2009. He disproved a long standing conjecture in quantum information theory, which is called the additivity conjecture of Holevo capacities. A natural way to prove this result will be covered in this talk based on the theory of i.i.d. random unitary matrices. Another outstanding application of random matrix theory in quantun information theory is to provide a systematic way to produce PPT entangled states in high dimensional tensor product spaces. This construction comes from i.i.d. random Gaussian matrices and I will try to explain why this application is important in view of quantun information theory.
Directed polymer models are well known Gibbs measures on random walk paths. Canonically they are defined so as to tilt the path distribution towards regions of space-time where an independent random field happens to be large, and as a result the paths tend to exhibit superdiffusive Kardar-Parisi-Zhang type fluctuation exponents, somehow betraying their random walk upbringing. Constructing these models on in the discrete space-time setting with a finite time horizon is straightforward, but extending them to infinite time horizons is difficult even in the fully discrete setting. I will review some relatively recent progress in the discrete and semi-discrete setting by myself and several other authors, some previous work of myself, Khanin, and Quastel on constructing continuous space-time models in the finite time horizon setting, and some attempts in progress to connect the two.
Let Qn be the n-dimensional Hamming cube (hypercube) and N=2n. We prove that the number of maximal independent sets in Qn is asymptotically 2n2N/4, as was conjectured by Ilinca and Kahn in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof of the upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in Qn. This is joint work with Jeff Kahn.