# The **Applied Topology** Seminar

All talks will be held **3pm–4pm** in the David Rittenhouse Laboratory. The organizers are Robert Ghrist and Brendan Fong. To schedule a talk, email fo at seas dot upenn dot edu. Here is a list of this year's seminar talks from the official Penn math calendar.

# Spring 2017

## 28 Mar 2017: **Dan Guralnik**, Penn

**Speaker**: Dan Guralnik, University of Pennsylvania

**Title**: Cubings in service of AI

**Location**: DRL 3C4

**Abstract**: The Sageev-Roller duality between poc-sets and median algebras may be applied for learning cubical representations (or products thereof) of an agent’s interactions with its environment. These product representations are interpreted — and, when implemented, are actually realized — as a network of agents with highly plastic memory structure, utilizing decision mechanisms based on the CAT(0) geometry of the corresponding evolving cubing, where SR-duality is used as a tool for lowering computational and representational costs. In this talk I will outline the UMA project (Universal Memory Architectures) at KodLab and some of the directions in which it is currently being developed.

## 14 Mar 2017: **Michael Robinson**, American University

**Speaker**: Michael Robinson, American University

**Title**: Analysis of sheaf encodings of complex models

**Location**: DRL 3C4

**Abstract**: Complex predictive models are notoriously hard to construct and to study. Without referring to the models directly -- only that a model consists of spaces and maps between them -- complex models can be assembled from smaller, easier-to-construct models. This talk will explain how a disciplined, diagrammatic process encodes continuous dynamical systems, partial differential equations, probabilistic graphical models, and discrete approximations of these models. Once encoded, a number of novel and powerful techniques are available; this talk will showcase several promising candidates.

## 13 Mar 2017: **Blair Sullivan**, North Carolina State University

**Speaker**: Blair Sullivan, North Carolina State University

**Title**: Tools for Tractability (in Topology?)

**Location**: DRL 4C2

**Abstract**: Rumor has it that one of the big challenges facing topological data analysis is the high computational complexity (and associated lack of scalability) of key methods. This talk will introduce the toolkit of parameterized algorithms for efficiently solving NP-hard problems, with examples drawn from network science. We will discuss how these approaches can exploit properties of the desired solution, structural features of the input, or even knowledge about solutions to other problems on the same network. Given the venue, we will go out of our way to define structural sparsity using topological minors and highlight some of our group's recent work on applications in computational biology. No background in parameterized complexity (or topology) will be assumed; bring your algorithmic challenges and come see if tractability is just a parameter away.

# Fall 2016

## 29 Nov 2016: **Steve Huntsman**, BAE Systems

**Speaker**: Steve Huntsman, BAE Systems

**Title**: Topological Density Estimation

**Abstract**: We discuss how the topological notions of unimodal category and persistence can be applied to the foundational statistical problem of estimating the probability density of a one-dimensional random variable from sample data. The resulting technique outperforms conventional methods on multimodal data, and can also indicate when a conventional method is likely to offer better performance. Prospects for applications to multidimensional problems will also be discussed.

## 15 Nov 2016: **Rachel Levanger**, Rutgers

**Speaker**: Rachel Levanger, Rutgers University

**Title**: New applications of persistent homology to image and time series analysis

**Abstract**: What do fluid flows and Picasso drawings have in common? It turns out they both provide fertile ground for new applications of persistent homology. In this talk, we'll first show how persistent homology helps us to see the dynamics of simulated fuel combustion. We'll then discuss an application of persistence to analyzing the complicated patterns seen in convection flows. Finally, we'll wrap things up with a sneak peek at how persistence might be able to help solve a computer vision problem in the computational study of fine art.

## 01 Nov 2016: **Michael Lesnick**, Princeton

**Speaker**: Michael Lesnick, Princeton University

**Title**: Universality of the Homotopy Interleaving Distance: Towards an "Approximate Homotopy Theory" Foundation for TDA

**Abstract**: We introduce and study *homotopy interleavings* between filtered topological spaces. These are homotopy-invariant analogues of *interleavings*, objects commonly used in topological data analysis to articulate stability and inference theorems. Whereas ordinary interleavings can be interpreted as pairs of “approximate isomorphisms” between filtered spaces, homotopy interleavings can be viewed as pairs of “approximate weak equivalences.”

Our main results are that homotopy interleavings induce an extended pseudometric d_{HI} on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also show that d_{HI} (or more generally, any pseudometric satisfying these two axioms and an additional “homology bounding” axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces.

This is joint work with Andrew Blumberg.

## 06 Sep 2016: **Josh Tan**, MIT and Categorical Informatics

**Speaker**: Josh Tan, MIT and Categorical Informatics

**Title**: A Categorical Theory of Co-Design

**Abstract**: A design problem is a (non-convex, non-differentiable) optimization problem that involves finding a feasible solution that delivers some required functionality using minimal resources; a (co-)design problem is the composition of many such design problems by three operations: series, parallel, and feedback. I will (1) show that there is a category of design problems where the three operations above correspond to the composition, monoidal product, and trace, (2) illustrate the theory with several examples from robotics, and (3) explore some fun extensions suggested by the category theory. This is joint work with Andrea Censi and David Spivak. Click here for slides.

# Spring 2016

## 25 Apr 2016: **Pablo Camara**, Columbia

**Speaker**: Pablo Camara, Columbia University

**Title**: Topological Methods for Molecular Phylogenetics

**Abstract**: The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relations within and across species. Recombination, re-assortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Recently, topological data analysis methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. In this talk I will discuss recent developments in the study of recombination using persistent homology, and I will present several biological applications, including the construction of high-resolution whole-genome human recombination maps.

## 18 Apr 2016: **Rachel Levanger**, Rutgers

**Speaker**: Rachel Levanger, Rutgers University

**Title**: A Comparison Framework for Interleaved Persistence Modules

**Abstract**: While it is often desired to compute the persistence diagram of a filtration of a topological space precisely, this is routinely not possible for many reasons. First, it might be impossible to encode the exact filtration into a computer. Second, even if this step is accomplished, it might be computationally infeasible to compute the associated persistence diagram (e.g. nerve lemma and Čech complex). As a result, it is commonplace to substitute an approximation (e.g. Vietoris-Rips complex) and use its persistence diagram instead. But what, exactly, have we lost? Most computational topologists are aware of the bottleneck distance between two persistence diagrams, and the literature up until this point typically stops here in terms of error bounds of approximations (or perhaps goes one step further to log-bottleneck). In this talk, we propose a more rigorous framework for analyzing the approximations of persistence diagrams.

## 11 Apr 2016: **Omer Bobrowski**, Duke

**Speaker**: Omer Bobrowski, Duke University

**Title**: Topological Consistency via Kernel Estimation

**Abstract**: The level sets of probability density functions are of a considerable interest in many areas of statistics, and topological data analysis (TDA) in particular. In this talk we focus on the problem of recovering the homology of level sets from a finite sample. The main difficulty stems from the fact that even small perturbations to the estimated density function can generate a very large error in homology. In this talk we present an estimator that overcomes this difficulty and recovers the homology accurately (with a high probability). We discuss two possible applications of the proposed estimator. The first one is recovering the homology of a compact manifold from a noisy point cloud. The second application is recovering the persistent homology of the super level sets filtration. Finally, we show that similar methods can be used in the analysis of nonparametric regression models.

## 28 Mar 2016: **Jeffrey Seely**, Columbia

**Speaker**: Jeffrey Seely, Columbia University

**Title**: Neural Computation: Visual Cortex versus Motor Cortex

**Abstract**: Visual cortex and motor cortex lie at opposing ends of the brain’s processing pathway. The goal of visual cortex is to compute representations of its input. The goal of primary motor cortex is to generate movement—the brain’s output. Here, we analyze datasets from both areas. All data is arranged in a neuron by stimulus by time tensor. We show how the ‘tensor structure’ of the data can reveal fundamentally different computational strategies employed by both areas—namely, whether an area encodes external variables or acts as a dynamical system. I will also include my current attempts on applying persistent homology and mapper to the data, as well try to gear the talk toward the question, "How might TDA be helpful for this type of data?"

## 14 Mar 2016: **Robert Short**, Lehigh

**Speaker**: Robert Short, Lehigh University

**Title**: Where Motion Planning and Cohomology Collide

**Abstract**: Topological robotics is a subfield of applied algebraic topology that concerns itself with motion planning algorithms. It seeks to pin down a value known as the topological complexity for any given space. Loosely, this corresponds to the least number of rules needed to perform continuous motion planning on the space.

While the name ``topological complexity'' is recent, the ideas making the loose statement precise have been around for decades. Using techniques driven by work from the 1960s, along with more recent principles, it is possible to calculate the topological complexity exactly for many spaces.

In this talk, we will focus primarily on the work of Michael Farber in his book *Invitation to Topological Robotics* as well as the recent work of Don Davis studying the topological complexity of polygon spaces. In the process, we will examine some of the tools for finding upper and lower bounds for topological complexity, and see the application of them in some nice examples.

Previous talks can be found here.