Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the functional order parameter of the famous Sherrington-Kirkpatrick model and we will explain a proof of uniqueness of the Parisi functional. Based on joint works with Wei-Kuo Chen.
Strict convexity of the Parisi functional
Oct. 23, 2014 3pm (Weber 201…
Algebraic Geometry
Mark Walsh (Wichita State University)
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The moduli space of positive scalar curvature metrics
A central theme in modern geometry is the relationship between curvature and topology. Given a smooth manifold, we may ask whether it admits a geometric structure (Riemannian metric) which satisfies particular curvature constraints, say positive, constant or negative curvature. For example, we know that the 2-torus admits a constant zero-curvature geometry but cannot admit everywhere positive (or negative) curvature, whereas the 2-sphere admits positive curvature geometries but not flat or negative curvature ones.
A particular case of this problem, for which much is known, is the question of whether or not a smooth manifold admits metrics whose scalar curvature is strictly positive. Moreover, given a manifold which admits such metrics, we may wonder about the topology of the space of such metrics, or its corresponding moduli spaces. Are these spaces path connected? Do they have non-trivial homotopy or homology?
The first of my two talks will be a gentle introduction to the existence problem for positive scalar curvature metrics, beginning with the classical case in dimension 2. I will go on in the second talk, to discuss recent progress in understanding the topology of the space of positive scalar curvature metrics, with a particular emphasis on the moduli space obtained by the action of the automorphism group of the manifold.
Strict convexity of the Parisi functional