Today’s summaries are provided by Isabel Darcy.
Omer Bobrowski talked about Topological Estimation for Super Level Sets. The goal is to determine the homology of an unknown space from a sample of noisy data points. Super-level sets of a density function f correspond to dense regions: {x | f(x) > L}. In general, the density function is not known but can often be estimated. One can try to reconstruct the homology by looking at \(U_n(L, r)\) = the union of balls of fixed radius r around each point in a super level set, {x | f(x) > L}.
But if not enough points are chosen (i.e., L large), then the space may not be adequately covered by \(U_n(L, r)\). If too many points are chosen (i.e., L small), then more noise may be picked up. However one can obtain the correct homology with high probability by looking at how \(U_n(L_1, r)\) includes into \(U_n(L_2, r)\) for \(L_2 < L_1\). This induces a map on their homologies. The image of this map is the homology estimator, which equals the correct homology of the space of interest with high probability.
Elizabeth Munch talked about The Interleaving Distance for Reeb Graphs. Reeb graphs provide an efficient description to understand the properties of a real-valued function on a topological space and are useful in many applications. Thus it would be very useful to have a method for comparing two Reeb graphs. Interleavings (and interleaving distances) have been used to compare persistence modules. Interleavings can be applied to Reeb graphs by defining a generalization of a Reeb graph as a functor. One consequence is a concrete algorithm for smoothing a Reeb graph in order to remove noise.
Peter Bubenik talked about Generalized Persistence Modules and Stability. Generalized persistence modules is an abstract formulation of persistence modules using category theory which includes many forms of persistence modules that are currently studied. One consequence of this formulation is that one can give simpler common proofs for many standard results such as stability.
Yuliy Baryshnikov talked about Integral Operators in Euler World. One can integrate functions that take on a finite number of values using the Euler characteristic as the measure. For example if f(x) = 4 for x in [0, 1] and 0 elsewhere, then the integral of f with respect to the Euler characteristic = 4 times the Euler characteristic of [0, 1] = 4(-1 + 2) = 4. In applications, sometimes it is easier to solve a problem by transforming it into a simpler problem using an integral transform.
An example of an integral transform is convolution: Given functions f and g, one can create a new function, f*g, where the value f*g(x) is obtained from f and g by integrating the product f(t)g(x-t) with respect to the euler characteristic. Given f and f*g, one would like to be able to reconstruct g: that is one would like to calculate the inverse of the Euler integral transform. Cases where one can calculate the inverse transform were discussed.