Georgia Institute of Technology College of Sciences

Meeting link: https://bluejeans.com/722836372/4781?src=join_info

Anosov flows are an important class of dynamical systems due to their ergodic properties and structural stability. Geometrically, they are defined by two transverse invariant foliations with expanding and contracting behaviors. Much of our understanding of the structure of an Anosov flow relies on the study of the leaves space of the invariant foliations. In this talk we adopt a different approach: in the early 90s Mitsumatsu first noticed that and Anosov vector field also belongs to the intersection of two transverse contact structures rotating towards each other. After giving the necessary background I will use this point of view to address questions in surgery theory on Anosov flows and contact structures.

This paper proposes a model-free and data-adaptive feature screening method for ultra-high dimensional data. The proposed method is based on the projection correlation which measures the dependence between two random vectors. This projection correlation based method does not require specifying a regression model, and applies to data in the presence of heavy tails and multivariate responses. It enjoys both sure screening and rank consistency properties under weak assumptions.  A two-step approach, with the help of knockoff features, is advocated to specify the threshold for feature screening  such that the false discovery rate (FDR) is controlled under a pre-specified level. The proposed two-step approach enjoys both sure screening and FDR control simultaneously if the pre-specified FDR level is greater or equal to 1/s, where s is the number of active features.  The superior empirical performance of the proposed method is illustrated by simulation examples and real data applications. This is a joint work with Wanjun Liu, Jingyuan Liu and Runze Li.

If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.

Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk I will give a basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the "game" of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.

The main goal of manifold theory is to classify all n-dimensional topological manifolds. For a smooth 4-manifold X, we aim to understand all of the exotic smooth structures there are to the smooth structure on X. Exotic smooth structures are homeomorphic but not diffeomorphic. Cork twists, Gluck twists, and Log transforms are all ways to construct possible exotic pairs by re-gluing embedded surfaces in the 4-manifold. In this talk, we define these three constructions.  

Using techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers' type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of traveling waves mediating the transition to turbulence, with a focus on an intermediate Reynolds number regime.

This is joint work with Björn de Rijk and Christian Kuehn.

Dynamical systems model the way that real-world systems evolve in time. While the time-asymptotic behavior of many systems can be characterized by “simple” dynamical features such as equilibria and periodic orbits, some systems evolve in a chaotic, seemingly random way. For such systems it is no longer meaningful to track one trajectory at a time individually- instead, a natural approach is to treat the initial condition as random and to observe how its probabilistic law evolves in time. This is the core idea of ergodic theory, the topic of this talk. I will not assume much beyond some basics of probability theory, e.g., random variables. 

Structural graph theory has usually focused on classes of graphs that are 'sparse' rather than 'dense' (that is, have few edges rather than many edges). We discuss this paradigm, focusing on classes with a forbidden vertex-minor. In particular, we discuss progress on a conjecture of Geelen that would totally characterize classes with a forbidden vertex-minor. This is joint work with Jim Geelen and Paul Wollan.