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Monday, April 22, 2019 - 13:55 ,
Location: Skiles 005 ,
Peter Binev ,
University of South Carolina ,
binev@math.sc.edu ,
Organizer: Wenjing Liao

The talk presents an extension for high dimensions of an idea from a recent result concerning near optimal adaptive finite element methods (AFEM). The usual adaptive strategy for finding conforming partitions in AFEM is ”mark → subdivide → complete”. In this strategy any element can be marked for subdivision but since the resulting partition often contains hanging nodes, additional elements have to be subdivided in the completion step to get a conforming partition. This process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This motivated us [B., Fierro, Veeser, in preparation] to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step is necessary. We also proved that this procedure is near best in terms of both error of approximation and complexity. This result is formulated in terms of tree approximations and opens the possibility to design similar algorithms in high dimensions using sparse occupancy trees introduced in [B., Dahmen, Lamby, 2011]. The talk describes the framework of approximating high dimensional data using conforming sparse occupancy trees.

Series: Algebra Seminar

This talk will be about polynomial decompositions that are relevant in machine learning. I will start with the well-known low-rank symmetric tensor decomposition, and present a simple new algorithm with local convergence guarantees, which seems to handily outperform the state-of-the-art in experiments. Next I will consider a particular generalization of symmetric tensor decomposition, and apply this to estimate subspace arrangements from very many, very noisy samples (a regime in which current subspace clustering algorithms break down). Finally I will switch gears and discuss representability of polynomials by deep neural networks with polynomial activations. The various polynomial decompositions in this talk motivate questions in commutative algebra, computational algebraic geometry and optimization. The first part of this talk is joint with Emmanuel Abbe, Tamir Bendory, Joao Pereira and Amit Singer, while the latter part is joint with Matthew Trager.

Series: Math Physics Seminar

I will talk about a conjecture that in Gibbs states of one-dimensional spin chains with short-ranged gapped Hamiltonians the quantum conditional mutual information (QCMI) between the parts of the chain decays exponentially with the length of separation between said parts. The smallness of QCMI enables efficient representation of these states as tensor networks, which allows their efficient construction and fast computation of global quantities, such as entropy. I will present the known partial results on the way of proving of the conjecture and discuss the probable approaches to the proof and the obstacles that are encountered.

Friday, April 19, 2019 - 14:00 ,
Location: Skiles 006 ,
Arash Yavari and Fabio Sozio, School of Civil and Environmental Engineering ,
Georgia Tech ,
Organizer: Igor Belegradek

We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material isadded to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial-boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.

Friday, April 19, 2019 - 12:00 ,
Location: Skiles 006 ,
Marc Härkönen ,
Georgia Tech ,
harkonen@gatech.edu ,
Organizer:

Series: Stochastics Seminar

Series: Graph Theory Seminar

To

any finite real sequence, we can associate a permutation $\pi$, via:

$\pi(k)$ is the index of the $k$th smallest element of the sequence.

This association was introduced in a 1987 paper

of Alavi, Malde, Schwenk and Erd\H{o}s, where they used it to study the

possible patterns of rises and falls that can occur in the matching

sequence of a graph (the sequence whose $k$th term is the number of

matchings of size $k$), and in the independent set

sequence.

The

main result of their paper was that {\em every} permutation can arise

as the ``independent set permutation'' of some graph. They left open the

following extremal question: for each $n$, what is

the smallest order $m$ such that every permutation of $[n]$ can be

realized as the independent set permutation of some graph of order at

most $m$?

We

answer this question. We also improve Alavi et al.'s upper bound on the

number of permutations that can be realized as the matching permutation

of some graph. There are still many open questions

in this area.

This is joint work with T. Ball, K. Hyry and K. Weingartner, all at Notre Dame.

Series: Graph Theory Working Seminar

The well known Erdos-Hajnal Conjecture states that every graph has the Erdos-Hajnal (EH) property. That is, for every $H$, there exists a $c=c(H)>0$ such that every graph $G$ with no induced copy of $H$ has the property $hom(G):=max\{\alpha(G),\omega(G)\}\geq |V(G)|^{c}$. Let $H,J$ be subdivisions of caterpillar graphs. Liebenau, Pilipczuk, Seymour and Spirkl proved that the EH property holds if we forbid both $H$ and $\overline{J}.$ We will discuss the proof of this result.

Series: High Dimensional Seminar

We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for a large class of probability distributions; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by myself in 2014). We discuss another class of measures for which this bound is sharp. For isotropic log-concave measures, the value of the lower bound is at least n^{1/8}.

In addition, we show a uniform upper bound of Cn||f||^{1/n}_{\infty} for all log-concave measures in a special position, which is attained for the uniform distribution on the cube. We further estimate the maximal perimeter of isotropic log-concave measures by n^2.

Wednesday, April 17, 2019 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

We will see some instances of swindles in mathematics, primarily focusing on some in geometric topology due to Barry Mazur.