This Week's Seminars and Colloquia

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).  
Monday, September 17, 2018 - 14:00 , Location: Skiles 006 , John Etnyre , Georgia Tech , Organizer: John Etnyre
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).
Wednesday, September 19, 2018 - 12:20 , Location: Skiles 005 , Blair Sullivan , North Carolina State University , , Organizer: Trevor Gunn
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.
Wednesday, September 19, 2018 - 12:55 , Location: Skiles 006 , Han Huang , University of Michigan , , Organizer: Konstantin Tikhomirov
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 $B_P$.
Wednesday, September 19, 2018 - 13:55 , Location: Skiles 005 , Marcin Bownik , University of Oregon , Organizer: Shahaf Nitzan
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.
Wednesday, September 19, 2018 - 16:00 , Location: Skiles 005 , Michael Loss , School of Mathematics, Georgia Tech , , Organizer: Michael Loss
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.
Wednesday, September 19, 2018 - 16:30 , Location: Skiles 006 , Dantong Zhu , Georgia Tech , Organizer: Xingxing Yu
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.
Thursday, September 20, 2018 - 11:00 , Location: Skiles 006 , Blair Sullivan , Department of Computer Science, NC State University , Organizer: Christine Heitsch
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.
Thursday, September 20, 2018 - 15:05 , Location: Skiles 006 , TBA , TBA , Organizer: Christian Houdre
Friday, September 21, 2018 - 13:05 , Location: Skiles 005 , Matthew Fahrbach , CS, Georgia Tech , , Organizer: He Guo
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 (arXiv:1807.07889).
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.
Friday, September 21, 2018 - 15:00 , Location: Skiles 005 , Yi Zhao , Georgia State University , Organizer: Lutz Warnke
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.