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Series: Geometry Topology Seminar

I will discuss joint work with Hutchings which gives a rigorousconstruction of cylindrical contact homology via geometric methods. Thistalk will highlight our use of non-equivariant constructions, automatictransversality, and obstruction bundle gluing. Together these yield anonequivariant homological contact invariant which is expected to beisomorphic to SH^+ under suitable assumptions. By making use of familyFloer theory we obtain an S^1-equivariant theory defined with coefficientsin Z, which when tensored with Q recovers the classical cylindrical contacthomology, now with the guarantee of well-definedness and invariance. Thisintegral lift of contact homology also contains interesting torsioninformation.

Series: Geometry Topology Seminar

I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit non-trivial L-space surgeries. A corollary of the classification is that 1-bridge braids in these manifolds admit non-trivial L-space surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.

Monday, April 3, 2017 - 14:00 ,
Location: Skiles 005 ,
Prof. Michael Muskulus ,
NTNU: Norwegian University of Science and Technology ,
michael.muskulus@ntnu.no ,
Organizer: Joseph Walsh

This talk addresses an important problem in arctic engineering due to interesting dynamic phenomena in a forced linear system. The nonautonomous system considered is representative of a whole class of engineering problems that are not approachable by standard techniques from dynamical system theory.The background are ice-induced vibrations of structures (e.g. wind turbines or measurement masts) in regions with active sea ice. Ice is a complex material and the mechanism for ice-induced vibrations is not fully clear at present. In particular, the conditions under which the observed, qualitatively different vibration regimes are active cannot be predicted accurately so far. A recent mathematical model developed by Delft University of Technology assumes that a number of parallel ice strips are pushing with a constant velocity against a flexible structure. The structure is modelled as a single degree of freedom harmonic oscillator. The contact force acts on the structure, but at the same time slows down the advancement of the ice, thereby introducing a dynamic nonlinearity in the otherwise linear system. When the local contact force becomes large enough, the ice crushes and the corresponding strip is reset to a random offset in front of the structure.This is the first mathematical model that exhibits all three different dynamic regimes that are observed in reality: for slow ice velocities the structure undergoes quasi-static sawtooth responses where all ice strips fail at the same time (a kind of synchronization phenomenon), for large ice velocities the structure response appears random, and for intermediate ice velocities the system exhibits vibrations at the structure eigenfrequency, commonly called frequency lock-in behavior. The latter type of vibrations causes a lot of damage to the structure and poses a safety and economic risk, so its occurrence needs to be predicted accurately.As I will show in this talk, the descriptive terms for the three vibration regimes are slightly misleading, as the mechanisms behind the observed behaviors are somewhat different than intuition suggests. I will present first results in analyzing the system and offer some explanations of the observed behaviors, as well as some simple criteria for the switch between the different vibration regimes.

Friday, March 31, 2017 - 15:05 ,
Location: Skiles 254 ,
Lei Zhang ,
School of Mathematics, GT ,
Organizer: Jiaqi Yang

In this talk, we will give an introduction to the variational approach to dynamical systems. Specifically, we will discuss twist maps and prove the classical results that area-preserving twist map has Birkhoff periodic orbits for each rational rotation number.

Series: Professional Development Seminar

A conversation with Dan Margalit, GT math professor and inaugural CoS Leddy Family Faculty Fellow, who was a tenure-track assistant professor at Tufts University for two years prior to coming to Tech.

Series: ACO Student Seminar

Using some classical results of invariant theory of finite reflection groups, and Lagrange multipliers, we prove that low degree or sparse real homogeneous polynomials
which are invariant under the action of a finite reflection group
$G$ are nonnegative if they are nonnegative on the hyperplane arrangement
$H$ associated to $G$. That makes $H$ a test set for the above kind of
polynomials. We also prove that under stronger sparsity conditions, for
the symmetric group and other reflection groups, the test set can
be much smaller. One of the main questions is deciding if certain
intersections of some simply constructed real $G$-invariant varieties are
empty or not.

Series: Algebra Seminar

Networks, or graphs, can represent a great variety of systems in the real world including neural networks, power grid, the Internet, and our social networks. Mathematical models for such systems naturally reflect the graph theoretical information of the underlying network. This talk explores some common themes in such models from the point of view of systems of nonlinear equations.

Series: Stochastics Seminar

We consider the problem of studying the limiting distribution of the number of monochromatic two stars and triangles for a growing sequence of graphs, where the vertices are colored uniformly at random. We show that the limit distribution of the number of monochromatic two stars is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree 1 or 2. Further, we show that any limit distribution for the number of monochromatic two stars has an expansion of this form. In the triangle case the problem is more challenging, as in this case the class of limit distributions can involve terms with products of Poisson random variables. In this case, we deduce a necessary and sufficient condition on the sequence of graphs such that the number of monochromatic triangles is asymptotically Poisson in distribution and in the first two moments. This work is joint with Bhaswar B. Bhattacharya at University of Pennsylvania.

Series: School of Mathematics Colloquium

Charles Stein brought the method that now bears his name to life in a 1972 Berkeley symposium paper that presented a new way to obtain information on the quality of the normal approximation, justified by the Central Limit Theorem asymptotic, by operating directly on random variables. At the heart of the method is the seemingly harmless characterization that a random variable $W$ has the standard normal ${\cal N}(0,1)$ distribution if and only if E[Wf(W)]=E[f'(W)] for all functions $f$ for which these expressions exist. From its inception, it was clear that Stein's approach had the power to provide non-asymptotic bounds, and to handle various dependency structures. In the near half century since the appearance of this work for the normal, the `characterizing equation' approach driving Stein's method has been applied to roughly thirty additional distributions using variations of the basic techniques, coupling and distributional transformations among them. Further offshoots are connections to Malliavin calculus and the concentration of measure phenomenon, and applications to random graphs and permutations, statistics, stochastic integrals, molecular biology and physics.

Series: Dissertation Defense

This thesis explores topics from two distinct fields of mathematics. The first part addresses a theme in abstract harmonic analysis, while the focus of the second part is a topic in compressive sensing. The first part of this dissertation explores the application of dominating operators in harmonic analysis by sparse operators. We make use of pointwise sparse dominations weighted inequalities for Calder\'on-Zygmund operators, Hardy-Littlewood maximal operator, and their fractional analogues. Dominating bilinear forms by sparse forms allows us to derive weighted inequalities for oscillatory integral operators (polynomially modulated CZOs) and random discrete Hilbert transforms. The later is defined on sets of initegers with asymptotic density zero, making these weighted inequalitites particulalry attractive. We also discuss a characterization of a certain weighted BMO space by commutators of multiplication operators with fractional integral operators. Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems. It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of slog(n/s) -- n is ambient dimension and s is sparsity threshhold.The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix. A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing. Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere. We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.