Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure

Spectral, Combinatorial, and Probabilistic Methods in Analyzing and Visualizing Vector Fields and Their Associated Flows

In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\'' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration

Asymptotic cohomological functions on projective varieties

In this paper we define certain analogues of the volume of a divisor - called asymptotic cohomological functions - and investigate their behaviour on the Neron--Severi space. We establish that asymptotic cohomological functions are invariant with respect to the numerical equivalence of divisors, and that they give rise to continuous functions on the real Neron--Severi space. To illustrate the theory, we work out these invariants for abelian varieties, smooth surfaces, and certain homogeneous spaces.Comment: 32 pages, 3 figure

Convex hypersurface theory in contact topology

We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one. We also prove that a $C^0$-generic family of mutually disjoint closed hypersurfaces parametrized by $t \in [0,1]$ is convex except at finitely many times $t_1, \dots, t_N$, and that crossing each $t_i$ corresponds to a bypass attachment. As applications of CHT, we prove the existence of compatible (relative) open book decompositions for contact manifolds and an existence h-principle for codimension 2 contact submanifolds.Comment: 93 pages, 27 figures; corrected a minor mistake in section 12 and a few typo

Poincar{\'e} series and linking of Legendrian knots

On a negatively curved surface, we show that the Poincar{\'e} series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincar{\'e} series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesic arcs orthogonal to both geodesics determine the linking number of the two geodesics.Comment: Minor modifications, 78

Getting a handle on contact manifolds

We develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with the key difference that the vector field does not necessarily have positive divergence everywhere. The surgery theory for contact manifolds contains the surgery theory for Weinstein manifolds via a sutured model for attaching critical points of low index. Using this sutured model, we show that the existence of convex structures on closed contact manifolds is guaranteed, a result equivalent to the existence of supporting Weinstein open book decompositions.Comment: 60 pages, 6 figures, comments welcome