In this talk, we connect the allocation of risk capital to a general class of Dirichlet distributions defined on the n-dimensional simplex. The mixed-scaled Dirichlet distributions proposed herein contain the classical Dirichlet distribution as a special case, exhibit a multitude of desirable closure properties, and emerge naturally within the multivariate risk analysis context. As a by-product, our invention revisits the proportional allocation rule that is often used in applications.

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Tensors (aka multiway arrays) can be instrumental in revealing latent correlations residing in high dimensional spaces. Despite their applicability to a broad range of applications in machine learning, speech recognition, and imaging, inconsistencies between tensor and matrix algebra have been complicating their broader utility. Researchers seeking to overcome those discrepancies have introduced several different candidate extensions, each introducing unique advantages and challenges. In this talk, we review some of the common tensor definitions, discuss their limitations, and introduce our tensor product framework which permits the elegant extension of linear algebraic concepts and algorithms to tensors. Following introduction of fundamental tensor operations, we discuss tensor decompositions in further depth, focusing on tensor SVDs based on the tensor product framework which can be computed efficiently in parallel. We present theoretical results regarding compressibility and provide numerical results that show the promise of our approach for compression/approximation of operators and datasets, highlighting examples such as facial recognition and model reduction.

Abstract: Phase transitions in liquid crystals are often characterized by islands of one phase surrounded by the other with the phase boundary itself sometimes possessing corners or cusps. In this talk I will introduce a model problem that seeks to describe these transitions and has a mechanism for phase boundary singularity formation. The study is part rigorous, part formal, and part computation. This is joint work with D. Golovaty, P. Sternberg, and R. Venkatraman.

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The Bartnik boundary conditions are very well-known in general relativity. They play a key role when measuring the quasi-local mass of a bounded manifold with boundary. When the spacetime is static, Bartnik boundary conditions consists of the induced metric g and the mean curvature H on the boundary of the manifold. This pair (g,H) of boundary data is much studied by mathematicians, as they are closely related to not only Bartnik mass, but also various notions of quasi-local mass such as the Brown-York mass and the Hawking mass.

In general relativity, the existence of extensions of Bartnik boundary data which are Einstein manifolds is a very interesting and open problem. A fundamental question to approach this problem is whether the Bartnik boundary conditions are elliptic for the static/stationary vacuum spacetime.

In this mini course, I will start with an introduction to the ADM mass and the quasi-local mass in the first lecture. The second lecture will be devoted to discussions of Bartnik boundary data in the static spacetime, including the ellipticity and a local existence result. In the last lecture we will generalize these results from static to stationary spacetime.

Let $(M, \omega)$ be a compact symplectic manifold. If we choose a compatible almost complex structure $J$ (which in general is not integrable) then we can study the space of $J$-holomorphic maps $u : \Sigma \to (M, J)$ from a compact Riemann surface into $M$. By appropriately “compactifying” the space of such maps, one can obtain powerful global symplectic invariants of $M$. At the heart of such a compactification procedure is understanding the ways in which sequences of such maps can degenerate, or develop singularities. Crucial ingredients are conformal invariance and an energy identity, which lead to to a plethora of analytic consequences, including: (i) a mean value inequality, (ii) interior regularity, (iii) a removable singularity theorem, (iv) an energy gap, and (v) compactness modulo bubbling.

Riemannian manifolds with closed $G_2$ or $\mathrm{Spin}(7)$ structures share many similar properties to such almost Kahler manifolds. In particular, they admit analogues of $J$-holomorphic curves, called associative and Cayley submanifolds, respectively, which are calibrated and hence homologically volume-minimizing. A programme initiated by Donaldson-Thomas and Donaldson-Segal aims to construct similar such “counting invariants” in these cases. In 2011, a somewhat overlooked preprint of Aaron Smith demonstrated that such submanifolds can be exhibited as images of a class of maps $u : \Sigma \to M$ satisfying a conformally invariant first order nonlinear PDE analogous to the Cauchy-Riemann equation, which admits an energy identity involving the integral of higher powers of the pointwise norm $|du|$. I will discuss joint work with Da Rong Cheng (Chicago) and Jesse Madnick (McMaster) in which we establish the analogous analytic results of (i)-(v) in this setting. arXiv:1909.03512

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