Today's summaries are provided by Isabel Darcy.

**Omer Bobrowski** talked about Topological Estimation for Super Level Sets. Â The goal is to determine the homology of an unknownÂ space from a sample of noisy data points. Â Super-level sets of a density function f correspond to dense regions: {x | f(x)Â >Â L}. Â In general, the density function is not known but can often be estimated. Â One can try to reconstruct the homology by looking at = the union of balls of fixed radius r around each point in a super level set, {x | f(x) >Â L}.

But if not enough points are chosen (i.e., L large), then the space may not be adequately covered by . Â IfÂ too many points are chosen (i.e., L small), then more noise may be picked up. Â However one can obtain the correct homology with high probability by looking at how Â includes into Â forÂ . Â This induces a map on their homologies. Â 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 toÂ understand the properties of a real-valued function on a topological space and are useful in many applications. Â Thus it would be very useful to have a method for comparing two Reeb graphs. Â Interleavings (and interleaving distances) have been used to compare persistence modules. Â Interleavings can be applied to Reeb graphs by defining a generalization of a Reeb graph as a functor. Â 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. Â Generalized persistence modules is an abstract formulation of persistence modules using category theory which includes many forms of persistence modules that are currently studied. Â 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. Â 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. Â 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. Â 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. Â Cases where one can calculate the inverse transform were discussed.