The classical isoperimetric inequality states that in Euclidean space spheres form the least perimeter enclosures for any give volume. We will review the historic development of this result in mathematics, and various approaches to proving it. Then we will discuss how one of these approaches, which is a variational argument, may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, in order to generalize the isoperimetric inequality.
An expectation-maximization (EM) algorithm is a powerful clustering method that was initially developed to fit Gaussian mixture distributions. In the absence of a particular probability density function, an EM algorithm aims to estimate the "best" function that maximizes the likelihood of data being generated by the model. We present an EM algorithm which addresses the problem of clustering "mutated" substrings of similar parent strings such that each substring is correctly assigned to its parent string. This problem is motivated by the process of simultaneously reading similar RNA sequences during which various substrings of the sequence are produced and could be mutated; that is, a substring may have some letters changed during the reading process. Because the original RNA sequences are similar, a substring is likely to be assigned to the wrong original sequence. We describe our EM algorithm and present a test on a simulated benchmark which shows that our method yields a better assignment of the substrings than what has been achieved by previous methods. We conclude by discussing how this assignment problem applies to RNA structure prediction.
We give a formula relating various notions of heights of abelian varieties. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener, and it extends the Faltings-Silverman formula for elliptic curves. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a key role. Based on joint works with Robin de Jong (Leiden).
We study the geometry of minimizers of the interaction energy functional. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to be resolved by the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique up to rotation and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.
Using what we have studied in the Brändén-Huh paper, we will go over the proof of the ultra-log-concavity version of Mason's conjecture.
In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale's $\alpha$-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.
We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity. This is a joint work with D. Borthwick and L. Corsi.
I will describe a few classical problems in capillarity and the associated classical variational framework. These problems include the well-known shape and rise height problems for the meniscus in a tube as well as the problems associated with sessile and pendent drops. I will briefly discuss elements of recent modifications of the variational theory allowing floating objects. Finally, I will describe a few open problems.
