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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: 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: 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).

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: Combinatorics Seminar

An old question in additive number theory is determining the length of the longest progression in a sumset A+B = {a + b : a in A, b in B}, given that A and B are "large" subsets of {1,2,...,n}. I will survey some of the results on this problem, including a discussion of the methods, and also will discuss some open questions and conjectures.

Series: Algebra Seminar

According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

Friday, September 14, 2018 - 13:55 ,
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: ACO Student Seminar

We study the dynamic graph connectivity problem in the massively
parallel computation model. We give a data structure for maintaining a
dynamic undirected graph that handles batches of updates and
connectivity queries in constant rounds, as long as the queries fit on a
single machine. This assumption corresponds to the gradual buildup of
databases over time from sources such as log files and user
interactions. Our techniques combine a distributed data structure for
Euler Tour (ET) trees, a structural theorem about rapidly contracting
graphs via sampling n^{\epsilon} random neighbors, as well as
modifications to sketching based dynamic connectivity data structures.
Joint work with David Durfee, Janardhan Kulkarni, Richard Peng and Xiaorui Sun.

Series: Stochastics Seminar

Let (A_n) be a sequence of random matrices, such that for every n, A_n
is n by n with i.i.d. entries, and each entry is of the form b*x, where b
is a Bernoulli random variable with probability of success p_n, and x
is an independent random variable of unit variance. We show that, as
long as n*p_n converges to infinity, the appropriately rescaled spectral
distribution of A_n converges to the uniform measure on the unit disc
of complex plane. Based on joint work with Mark Rudelson.

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

Tropical geometry is a blend of algebraic geometry and polyhedral combinatorics that arises when one looks at algebraic varieties over a valued field. I will give a 50 minute introduction to the subject to highlight some of the key themes.