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Monday, September 17, 2018 - 13:55 ,
Location: Skiles 005 ,
Professor Lourenco Beirao da Veiga ,
Università di Milano-Bicocca ,
Organizer: Haomin Zhou

This is a joint seminar by College of Engineering and School of Math.

The Virtual
Element Method (VEM), is a very recent technology introduced in [Beirao da
Veiga, Brezzi, Cangiani, Manzini, Marini, Russo, 2013, M3AS] for the
discretization of partial differential equations, that has shared a good
success in recent years. The VEM can be interpreted as a generalization of the
Finite Element Method that allows to use general polygonal and polyhedral
meshes, still keeping the same coding complexity and allowing for arbitrary
degree of accuracy. The Virtual Element Method makes use of local functions
that are not necessarily polynomials and are defined in an implicit way.
Nevertheless, by a wise choice of the degrees of freedom and introducing a novel
construction of the associated stiffness matrixes, the VEM avoids the explicit
integration of such shape functions.
In addition
to the possibility to handle general polytopal meshes, the flexibility of the
above construction yields other interesting properties with respect to more
standard Galerkin methods. For instance, the VEM easily allows to build discrete
spaces of arbitrary C^k regularity, or to satisfy exactly the divergence-free
constraint for incompressible fluids.
The present
talk is an introduction to the VEM, aiming at showing the main ideas of the
method. We consider for simplicity a simple elliptic model problem (that is
pure diffusion with variable coefficients) but set ourselves in the more
involved 3D setting. In the first part we introduce the adopted Virtual Element
space and the associated degrees of freedom, first by addressing the faces of
the polyhedrons (i.e. polygons) and then building the space in the full
volumes. We then describe the construction of the discrete bilinear form and
the ensuing discretization of the problem. Furthermore, we show a set of
theoretical and numerical results. In the very final part, we will give a
glance at more involved problems, such as magnetostatics (mixed problem with more
complex spaces interacting) and large deformation elasticity (nonlinear
problem).

Series: Geometry Topology Seminar

The study of transverse knots in dimension 3 has been instrumental in the development of 3 dimensional contact ge- ometry. One natural generalization of transverse knots to higher dimensions is contact submanifolds. Embeddings of one contact manifold into another satisfies an h-principle for codimension greater than 2, so we will discuss the case of codimension 2 contact embeddings. We will give the first pair of non-isotopic contact embeddings in all dimensions (that are formally isotopic).

Series: Research Horizons Seminar

In this talk, we describe transforming a theoretical algorithm from
structural graph theory into open-source software being actively used for
real-world data analysis in computational biology. Specifically, we apply
the r-dominating set algorithm for graph classes of bounded expansion in
the setting of metagenome analysis. We discuss algorithmic improvements
required for a practical implementation, alongside exciting preliminary
biological results (on real data!). Finally, we include a brief
retrospective on the collaboration process. No prior knowledge in
metagenomics or structural graph theory is assumed.
Based on joint work with T. Brown, D. Moritz, M. O’Brien, F. Reidl and T.
Reiter.

Series: High Dimensional Seminar

It is natural to question whether the center of mass of a convex body $K\subset \mathbb{R}^n$ lies in its John ellipsoid $B_K$, i.e., in the maximal volume ellipsoid contained in $K$. This question is relevant to the efficiency of many algorithms for convex bodies. We obtain an unexpected negative result. There exists a convex body $K\subset \mathbb{R}^n$ such that its center of mass does not lie in the John ellipsoid $B_K$ inflated $(1-o(1))n$ times about the center of $B_K$. (Yet, if one inflate $B_K$ by a factor $n$, it contains $K$.)Moreover, there exists a polytope $P \subset \mathbb{R}^n$ with $O(n^2)$ facets whose center of mass is not contained in the John ellipsoid $B_P$ inflated $O(\frac{n}{\log(n)})$ times about the center of $B_P$.

Series: Analysis Seminar

In this talk we shall explore some of the consequences of the solution
to the Kadison-Singer problem. In the first part of the talk we present
results from a joint work with Itay Londner. We show that every subset $S$ of the torus of positive Lebesgue measure admits a Riesz sequence of
exponentials $\{ e^{i\lambda x}\} _{\lambda \in \Lambda}$ in $L^2(S)$
such that $\Lambda\subset\mathbb{Z}$ is a set with gaps between
consecutive elements bounded by $C/|S|$. In the second part of the talk
we shall explore a higher rank extension of the main result of Marcus,
Spielman, and Srivastava, which was used in the solution of the
Kadison-Singer problem.

Wednesday, September 19, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyunki Min ,
Georgia Tech ,
Organizer: Hyun Ki Min

In 1957, Smale proved a striking result: we can turn a sphere inside out without any singularity. Gromov in his thesis, proved a generalized version of this theorem, which had been the starting point of the h-principle. In this talk, we will prove Gromov's theorem and see applications of it.

Series: Math Physics Seminar

We introduce a quantum version of the Kac Master equation,and we explain issues like equilibria, propagation of chaos and the corresponding quantum Boltzmann equation. This is joint work with Eric Carlen and Maria Carvalho.

Series: Graph Theory Working Seminar

An $r$-cut of a $k$-uniform hypergraph $H$ is a partition of the vertex set of $H$ into $r$ parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every $m$-edge graph has a 2-cut of size $m/2+\Omega(\sqrt{m})$, and this is best possible. In this talk we will discuss recent results on analogues of Edwards’ result and related problems in hypergraphs.

Series: School of Mathematics Colloquium

Techniques from structural graph theory hold significant promise for
designing efficient algorithms for network science. However, their
real-world application has been hampered by the challenges of unrealistic
structural assumptions, hidden costs in big-O notation, and
non-constructive proofs. In this talk, I will survey recent results which
address many of these concerns through an algorithmic pipeline for
structurally sparse networks, highlighting the crucial role of certain
graph colorings, along with several open problems. For example, we give
empirical and model-based evidence that real-world networks exhibit a form
of structural sparsity known as "bounded expansion,'' and discuss
properties of several low-treedepth colorings used in efficient algorithms
for this class.
Based on joint works with E. Demaine, J. Kun, M. O'Brien, M. Pilipczuk, F.
Reidl, P. Rossmanith, F. Sanchez Villaamil, and S. Sikdar.

Thursday, September 20, 2018 - 13:30 ,
Location: Skiles 006 ,
Trevor Gunn ,
Georgia Tech ,
Organizer: Trevor Gunn

We will give a brief introduction to matroids with a focus on representable matroids. We will also discuss the Plücker embedding of the Grassmannian.

Series: Stochastics Seminar

Series: ACO Student Seminar

As a generalization of many classic problems in combinatorial optimization, submodular optimization has found a wide range of applications in machine learning (e.g., in feature engineering and active learning). For many large-scale optimization problems,
we are often concerned with the adaptivity complexity of an algorithm, which quantifies the number of sequential rounds where polynomially-many independent function evaluations can be executed in parallel. While low adaptivity is ideal, it is not sufficient
for a (distributed) algorithm to be efficient, since in many practical applications of submodular optimization the number of function evaluations becomes prohibitively expensive. Motivated by such applications, we study the adaptivity and query complexity
of adaptive submodular optimization.
Our main result is a distributed algorithm for maximizing a monotone submodular function with cardinality constraint $k$ that achieves a $(1-1/e-\varepsilon)$-approximation in expectation. Furthermore, this algorithm runs in $O(\log(n))$ adaptive rounds
and makes $O(n)$ calls to the function evaluation oracle in expectation. All three of these guarantees are optimal, and the query complexity is substantially less than in previous works. Finally, to show the generality of our simple algorithm and techniques,
we extend our results to the submodular cover problem.
Joint work with Vahab Mirrokni and Morteza Zadimoghaddam (<a href="https://arxiv.org/abs/1807.07889">arXiv:1807.07889</a>).

Friday, September 21, 2018 - 14:00 ,
Location: Skiles 006 ,
Peter Lambert-Cole ,
Georgia Insitute of Technology ,
Organizer: Peter Lambert-Cole

The Oka-Grauert principle is one of the first examples of an
h-principle. It states that for a Stein domain X and a complex Lie
group G, the topological and holomorphic classifications of principal
G-bundles over X agree. In particular, a complex vector bundle over X
has a holomorphic trivialization if and only if it has a continuous
trivialization. In these talks, we will discuss the complex geometry of
Stein domains, including various characterizations of Stein domains,
the classical Theorems A and B, and the Oka-Grauert principle.

Series: Combinatorics Seminar

For integers k>2 and \ell0, there exist \epsilon>0 and C>0 such that for sufficiently large n that is divisible by k-\ell,
the union of a k-uniform hypergraph with minimum vertex degree \alpha n^{k-1} and a binomial random k-uniform hypergraph G^{k}(n,p) on the same n-vertex set
with p\ge n^{-(k-\ell)-\epsilon} for \ell\ge 2 and p\ge C n^{-(k-1)} for \ell=1 contains a Hamiltonian \ell-cycle with high probability. Our result is best possible up to
the values of \epsilon and C and completely answers a question of Krivelevich, Kwan and Sudakov.
This is a joint work with Jie Han.

Friday, September 21, 2018 - 15:05 ,
Location: Skiles 156 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk I will present a proof of a generalization of a theorem by Siegel, about the existence of an analytic conjugation between an analytic map, $f(z)=\Lambda z +\hat{f}(z)$, and a linear map, $\Lambda z$, in $\mathbb{C}^n$. This proof illustrates a standar technique used to deal with small divisors problems. I will be following the work of E. Zehnder.