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Series: High Dimensional Seminar

Note the special time!

TBA

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.

Series: High Dimensional Seminar

We already know that the Euclidean unit ball is at the center of the Banach-Mazur compactum, however its structure is still being explored to this day. In 1987, Szarek and Talagrand proved that the maximum distance $R_{\infty} ^n$ between an arbitrary $n$-dimensional normed space and $\ell _{\infty} ^n$, or equivalently the maximum distance between a symmetric convex body in $\mathbb{R} ^n$ and the $n$-dimensional unit cube is bounded above by $c n^{7/8}$. In this talk, we will discuss a related paper by A. Giannopoulos, "A note to the Banach-Mazur distance to the cube", where he proves that $R_{\infty} ^n < c n^{5/6}$.

Series: High Dimensional Seminar

It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd (real or complex), for p > 2 the p-frame potential ∑i ≠ j | < φj, φk > |p achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2 ∑i, j=1 | < φj, φk > |p . In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.

Series: High Dimensional Seminar

The n-dimensional L^p Brunn-Minkowski inequality for p<1 , in particular the log-Brunn-Minkowski inequality, is proposed by Boroczky-Lutwak-Yang-Zhang in 2013, based on previous work of Firey and Lutwak . When it came out, it promptly became the major problem in convex geometry. Although some partial results on some specific convex sets are shown to be true, the general case stays wide open. In this talk I will present a breakthrough on this conjecture due to E. Milman and A Kolesnikov, where we can obeserve a beautiful interaction of PDE, operator theory, Riemannian geometry and all sorts of best constant estimates. They showed the validity of the local version of this inequality for orgin-symmtric convex sets with a C^{2} smooth boundary and strictly postive mean curvature, and for p between 1-c/(n^{3/2}) and 1. Their infinitesimal formulation of this inequality reveals the deep connection with the poincare-type inequalities. It turns out, after a sophisticated transformation, the desired inequality follows from an estimate of the universal constant in Poincare inequality on convex sets.

Series: High Dimensional Seminar

I shall tell about some background and known results in regards to the celebrated and fascinating Log-Brunn-Minkowski inequality, setting the stage for Xingyu to discuss connections with elliptiic operators a week later.