I will describe some topological obstructions to the existence of complete Riemannian metrics with (nonuniformly) positive scalar curvature. The motivation comes from conjectures about the simplicial volume of closed manifolds.
Scalar curvature on noncompact manifolds Sponsored by the Meyer Fund
Given two left modules for a meromorphic open-string vertex algebra , we will first use Huang's cohomology to describe the equivalence classes of modules fitting in the exact sequence while satisfying a technical convergence condition. Then, we will explain that the technical convergence condition automatically holds if contains a nice subalgebra , such that and are semisimple -modules, and the products of intertwining operators converge. Here is not required to be conformally embedded into . Nor will we need -modules to form a tensor category. The result leads to a new method for computing .
In areas a diverse as extremal combinatorics, signal processing, and quantum information, one is interested in finding subspaces with certain mathematical properties, called frames or fusion frames with additional adjectives depending on the desired outcomes. Often such optimal configurations have remarkable symmetry and are sometimes even orbits under group actions. Finally, when one constructs such a configuration, one might ask whether it is indeed "new." H* algebras, which may be thought loosely of as the Hilbert space analog of C* algebras, may be leveraged to answer this question. No prior knowledge of frames will be assumed.
Subspace Configurations, Group Representations, and H* Algebras
Sep. 28, 2023 3:35pm (MATH 3…
Probability
Shih Yu Chang (San Jose State University)
X
A concentration tail bound serves as a probabilistic inequality that establishes an upper limit on the extent to which a random variable is expected to deviate from its mean. These inequalities find valuable applications in probability theory, statistics, and machine learning, aiding in the comprehension of random variable behaviors and the management of risks associated with extreme outcomes. Among the frequently utilized concentration tail bounds are the Chernoff bound and the Hoeffding bound. Conversely, a tensor is a mathematical concept employed for the representation of multi-dimensional arrays of data. Tensors have a presence across various domains within mathematics and physics, and they find extensive utility in disciplines like physics, engineering, computer science, and machine learning. In this discussion, we will explore tail bounds pertaining to random tensors from five distinct viewpoints: (1) Tail bounds for the summation of random tensors, approached through Laplace transforms and Lieb's concavity principles; (2) General tail bounds derived through a majorization approach; (3) The consideration of T-product random tensors; (4) Examination of random tensor means; (5) Exploration of random double tensor integrals. In conclusion, we will deliberate on potential applications for tail bounds concerning random tensors, elucidating their relevance in diverse problem-solving scenarios.
Scalar curvature on noncompact manifolds