In recent years topological and geometric methods have been been availed as new promising tools for analyzing complex and diverse data. To broaden the scope of TGDA further we need stronger synergy among mathematical fields such as algebraic topology and algorithmic developments such as those in computational geometry. We also need tighter interaction with applied scientific domains. This conference aims to bring together researchers from mathematics and computational and applied sciences in order to exchange ideas and consolidate synergistic activities through fostering collaborations.

### Accepted Speakers

- Sun, May 15, 2016
- Mon, May 16, 2016
- Tue, May 17, 2016
- Wed, May 18, 2016
- Thu, May 19, 2016
- Fri, May 20, 2016
- Full Schedule

Sunday, May 15, 2016 | |
---|---|

Time | Session |

Monday, May 16, 2016 | |
---|---|

Time | Session |

08:30 AM 08:45 AM | Coffee |

08:45 AM 09:00 AM | Opening Remarks |

09:00 AM 10:00 AM | Gunnar Carlsson - Excision and product results for persistent homology Motivated by work on data sets coming from the biology and evolution, we have some results concerning the persistent homology of trees and also products of finite metric spaces. |

10:00 AM 10:30 AM | Break |

10:30 AM 11:30 AM | Don Sheehy - Some thoughts on Sampling In this talk I will discuss techniques and heuristics for subsampling metric data as well as a space of tree-like data structures that one might build on top of such samples, generalizing cover trees, net trees, navigating nets, deformable spanners, and some classes of hierarchical spanners. |

11:30 AM 01:30 PM | Lunch Break |

01:30 PM 02:30 PM | Benjamin Schweinhart - Topological Similarity of Cell Complexes Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. I'll introduce the method of swatches, which describes the local topology of a cell complex in terms of probability distributions of local configurations. It allows a distance to be defined which measures the similarity of the local topology of cell complexes. Convergence in this distance is related to the notion of a Benjamini Schramm graph limit. In my talk, I will use this to state universality conjectures about the long-term behavior of graphs evolving under curvature flow, and to test these conjectures computationally. This system is of both mathematical and physical interest. If time permits, I will discuss other applications of computationally topology to curvature flow on graphs, and describe recent work on a new notion of geometric graph limit. |

02:30 PM 03:30 PM | Vidit Nanda - The Discrete Flow Category Large-scale homology computations are often rendered tractable by discrete Morse theory. Every discrete Morse function on a given cell complex X produces a Morse chain complex whose chain groups are spanned by critical cells and whose homology is isomorphic to that of X. However, the space-level information is typically lost because very little is known about how critical cells are attached to each other. In this talk, we discretize a beautiful construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X from an overlaid discrete Morse function. |

03:30 PM 04:00 PM | Break |

04:00 PM 05:00 PM | Brittany Fasy - Statistics and Persistent Homology: A Sampling of Theory and Applications |

05:00 PM 07:00 PM | Poster session and Reception at MBI |

Tuesday, May 17, 2016 | |
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Time | Session |

08:30 AM 09:00 AM | Coffee |

09:00 AM 10:00 AM | TBD |

10:00 AM 10:30 AM | Break |

10:30 AM 11:30 AM | Carina Curto - Emergent dynamics from network connectivity: a minimal model Many networks in the brain display internally-generated patterns of activity -- that is, they exhibit emergent dynamics that are shaped by intrinsic properties of the network rather than inherited from an external input. While a common feature of these networks is an abundance of inhibition, the role of network connectivity in pattern generation remains unclear. In this talk I will introduce Combinatorial Threshold-Linear Networks (CTLNs), which are simple "toy models" of recurrent networks consisting of threshold-linear neurons with binary inhibitory interactions. The dynamics of CTLNs are controlled solely by the structure of an underlying directed graph. By varying the graph, we observe a rich variety of emergent patterns including: multistability, neuronal sequences, and complex rhythms. These patterns are reminiscent of population activity in cortex, hippocampus, and central pattern generators for locomotion. I will present some theorems about CTLNs, and explain how they allow us to predict features of the dynamics by examining properties of the underlying graph. Finally, I'll show examples illustrating how these mathematical results guide us to engineer complex networks with prescribed dynamic patterns. |

11:30 AM 01:30 PM | Lunch Break |

01:30 PM 02:30 PM | Sayan Mukherjee |

02:30 PM 03:30 PM | Michael Lesnick - Algebraic Stability of Zigzag Persistence Modules The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistent homology of mathbb{R}- valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and interleavings. In this work, we establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions. As an application of our main theorem, we strengthen a result of Bauer, Munch, and Wang on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al. This is joint work with Magnus Botnan. |

03:30 PM 04:00 PM | Break |

04:00 PM 05:00 PM | Amit Patel - Semicontinuity of Persistence Diagrams The persistence diagram is very different in philosophy from the barcode. Suppose we have a constructible persistence module of vector spaces. Its barcode is its list of indecomposables. Its persistence diagram is an encoding of all persistent vector spaces. In the setting of vector spaces, we know that these two notions are equivalent. However, we quickly run into problems if we try to generalize the barcode beyond the setting of vector spaces. In this talk, I will generalize the persistence diagram to the setting of constructible persistence modules valued in any symmetric monoidal category. For example, the category of sets, the category of vector spaces, and the category of abelian groups are symmetric monoidal categories. As an immediate consequence, we can finally talk about persistent homology over integer coefficients! |

Wednesday, May 18, 2016 | |
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Time | Session |

08:30 AM 09:00 AM | Coffee |

09:00 AM 10:00 AM | Herbert Edelsbrunner - Expected sizes of Poisson--Delaunay mosaics and their discrete Morse functions Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in R^n, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we study general properties of the discrete gradient and obtain precise expressions for the expected numbers in low dimensions. In particular, we get the expected numbers of simplices in the Poisson--Delaunay mosaic in dimension n <= 4. |

10:00 AM 10:30 AM | Break |

10:30 AM 11:30 AM | Jose Perea - Sparse Cech filtrations, persistent cohomology and projective coordinates One of the main challenges in topological data analysis is to turn computed topological features, such as barcodes, into insights about the data set under analysis. We will show in this talk how the persistent cohomology of sparse Cech filtrations (introduced recently by D. Sheehy et. al.), in dimensions 1 and 2, can be used to construct robust representations of the data in the real and complex projective spaces. Examples will be presented in order to illustrate how projective coordinates provides a framework for topology-driven nonlinear dimensionality reduction, and geometric model generation. This work extends results of V. de Silva, D. Morozov and M. Vejdemo-Johansson on persistent cohomology and circular coordinates. |

11:30 AM 01:30 PM | Lunch Break |

01:30 PM 02:30 PM | Vin de Silva - Reeb cosheaves I shall talk about Reeb graphs, and the benefits of regarding them as set-valued cosheaves on the real line. For instance, there is a smoothing operation and an interleaving distance for these cosheaves, which can be interpreted as a smoothing operation and an interleaving distance for Reeb graphs. This is joint work with Elizabeth Munch and Amit Patel. If time allows, I will discuss various extensions of these ideas, worked out with the additional collaboration of Anastasios Stefanou; and also of Pomona College undergraduates Song Yu and Dmitriy Smirnov. |

02:30 PM 03:30 PM | Bei Wang - Categorical Representations of Reeb Space and Mapper: Convergence and Multivariate Data Analysis The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper (Singh et al.), and a special case of the mapper construction called the Joint Contour Net (Carr et al. ) have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit. We are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution. |

03:30 PM 04:00 PM | Break |

04:00 PM 05:00 PM | Justin Curry - Realization Problems in Persistence In this talk I will introduce the realization problem in persistence, which asks what isomorphism classes of diagrams of vector spaces can be realized by diagrams of topological spaces and continuous maps. In particular, one could ask what barcodes can be obtained by filtering a simplicial complex or by studying the level-set persistence of a map. I will review existing results for point clouds, and present some results of my own, obtained in collaboration with Ulrich Bauer and Hans Reiss. These results include studying the level-set barcodes realized by a stratified space and real-valued map, the space of barcodes realized by filtering a manifold by Gauss curvature, and the space of barcodes realized by Morse functions. To address which barcodes can be realized as the level-set barcodes of a Morse function $f$, I will present two constructions. One construction uses handlebody theory. The other construction is more novel and uses a cosheaf of spaces over the Reeb graph of $f$, which incidentally makes headway into a problem posed by Arnold. Additionally, this construction offers a vision for extending Mapper degree by degree (in analogy with Postnikov towers), offering a potentially powerful new tool in topological data analysis. |

06:00 PM 08:30 PM | Banquet |

Thursday, May 19, 2016 | |
---|---|

Time | Session |

08:30 AM 09:00 AM | Coffee |

09:00 AM 10:00 AM | Jeff Erickson - Untangling Planar Curves and Planar Graphs Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. The best bounds previously known were a 100-year-old O(n^2) upper bound due to Steinitz and the trivial Omega(n) lower bound. Our lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to an embedding. |

10:00 AM 10:30 AM | Break |

10:30 AM 11:30 AM | Dan Burghelea - From Computer and Data back to Topology and Geometry It is well known how basic Algebraic Topology and geometrization of Large Data led to Persistence Theory, a useful tool in Data Analysis. In this talk I will explore the other direction; how Persistence Theory suggests and motivates refinements of some basic topological invariants, like homology and Betti numbers, and suggests alternative descriptions of others invariants, like monodromy, of mathematical relevance and with computational implications. The mathematics described is a part of what I refer to as an ALTERNATIVE to MORSE-NOVIKOV theory. The refinements proposed are in terms of configurations of vector spaces for the relevant homologies, and in terms of polynomials for Betti numbers. The alternative description of monodromy is computer friendly, hence without the need of infinite objects (infinite cyclic cover). A few applications of these refinements in topology, geometric analysis and dynamics might be indicated. |

11:30 AM 01:30 PM | Lunch Break |

01:30 PM 02:30 PM | Jenn Gamble - TDA in Healthcare This talk will give some examples of applications of topological data analysis to the field of healthcare. In addition to the foundational 'TDA' algorithms of persistent homology and mapper, this includes TDA in the broader sense: explicitly taking a topological/geometric approach to a data analysis problem. Examples will include understanding and managing clinical variation in a hospital system, and describing complicated patterns of denials of medical claims. General approaches, including the incorporation of alternative metrics or application-relevant filtrations or mapper functions will also be discussed. |

02:30 PM 03:30 PM | Isabel Darcy |

03:30 PM 04:00 PM | Break |

04:00 PM 05:00 PM | Sanjeevi Krishnan - Derived Group Completion Every connected, finite CW complex is homotopy equivalent to the classifying space of a monoid [McDuff, 1979], a set with an associative and unital multiplication. Thus group completions of monoids correspond to fundamental groups of based spaces. We discuss concrete algorithms for computing (co)homology and higher homotopy as algebraic constructions on monoids. |

Friday, May 20, 2016 | |
---|---|

Time | Session |

08:30 AM 09:00 AM | Coffee |

09:00 AM 10:00 AM | Robert Ghrist - Sheaf Theory: Applications & Computations |

10:00 AM 10:30 AM | Break |

10:30 AM 11:30 AM | Elizabeth Munch - Applications of Persistence to Time Series Analysis |

11:30 AM 12:30 PM | Peter Bubenik - Higher Interpolation and Extension for Persistence Modules Persistence modules are the central algebraic object in topological data analysis. This motivates the study of the geometry of the space of persistence modules. We isolate an elegant coherence condition that guarantees the interpolation and extension of sets of persistence modules. This "higher interpolation" is a consequence of the existence of certain universal constructions. As an application, it allows one to compare Vietoris-Rips and Cech complexes built within the space of persistence modules. This is joint work with Vin de Silva and Vidit Nanda. |

Name | Affiliation |
---|

motivates the study of the geometry of the space of persistence modules. We isolate an elegant

coherence condition that guarantees the interpolation and extension of sets of persistence

modules. This "higher interpolation" is a consequence of the existence of certain universal

constructions. As an application, it allows one to compare Vietoris-Rips and Cech complexes

built within the space of persistence modules. This is joint work with Vin de Silva and Vidit

Nanda.

to Persistence Theory, a useful tool in Data Analysis.

In this talk I will explore the other direction; how Persistence Theory suggests and motivates

refinements of some basic topological invariants, like homology and Betti numbers, and suggests

alternative descriptions of others invariants, like monodromy, of mathematical relevance and

with computational implications. The mathematics described is a part of what I refer to as an

ALTERNATIVE to MORSE-NOVIKOV theory.

The refinements proposed are in terms of configurations of vector spaces for the relevant

homologies, and in terms of polynomials for Betti numbers. The alternative description of

monodromy is computer friendly, hence without the need of infinite objects (infinite cyclic

cover). A few applications of these refinements in topology, geometric analysis and dynamics

might be indicated.

results concerning the persistent homology of trees and also products of finite metric spaces.

they exhibit emergent dynamics that are shaped by intrinsic properties of the network rather than

inherited from an external input. While a common feature of these networks is an abundance of

inhibition, the role of network connectivity in pattern generation remains unclear.

In this talk I will introduce Combinatorial Threshold-Linear Networks (CTLNs), which are

simple "toy models" of recurrent networks consisting of threshold-linear neurons with binary

inhibitory interactions. The dynamics of CTLNs are controlled solely by the structure of an

underlying directed graph. By varying the graph, we observe a rich variety of emergent patterns

including: multistability, neuronal sequences, and complex rhythms. These patterns are

reminiscent of population activity in cortex, hippocampus, and central pattern generators for

locomotion. I will present some theorems about CTLNs, and explain how they allow us to

predict features of the dynamics by examining properties of the underlying graph. Finally, I'll

show examples illustrating how these mathematical results guide us to engineer complex

networks with prescribed dynamic patterns.

a finite sequence of local transformations called homotopy moves. We prove that simplifying a

planar curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case.

The best bounds previously known were a 100-year-old O(n^2) upper bound due to Steinitz and

the trivial Omega(n) lower bound. Our lower bound also implies that Omega(n^{3/2}) degree-1

reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any

planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for

rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are

required in the worst case to transform one non-contractible closed curve on the torus to another;

this lower bound is tight if the curve is homotopic to an embedding.

field of healthcare. In addition to the foundational 'TDA' algorithms of persistent homology and

mapper, this includes TDA in the broader sense: explicitly taking a topological/geometric

approach to a data analysis problem. Examples will include understanding and managing clinical

variation in a hospital system, and describing complicated patterns of denials of medical claims.

General approaches, including the incorporation of alternative metrics or application-relevant

filtrations or mapper functions will also be discussed.

of a monoid [McDuff, 1979], a set with an associative and unital multiplication. Thus group

completions of monoids correspond to fundamental groups of based spaces. We discuss concrete

algorithms for computing (co)homology and higher homotopy as algebraic constructions on

monoids.

While the original formulation of the result concerns the persistent homology of mathbb{R}-

valued functions, the result was later cast in a more general algebraic form, in the language of

persistence modules and interleavings. In this work, we establish an analogue of this algebraic

stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag

persistence module to a two-dimensional persistence module, and establish an algebraic stability

theorem for these extensions. As an application of our main theorem, we strengthen a result of

Bauer, Munch, and Wang on the stability of the persistent homology of Reeb graphs. Our main

result also yields an alternative proof of the stability theorem for level set persistent homology of

Carlsson et al.

This is joint work with Magnus Botnan.

theory. Every discrete Morse function on a given cell complex X produces a Morse chain

complex whose chain groups are spanned by critical cells and whose homology is isomorphic to

that of X. However, the space-level information is typically lost because very little is known

about how critical cells are attached to each other. In this talk, we discretize a beautiful

construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X

from an overlaid discrete Morse function.

We will show in this talk how the persistent cohomology of sparse Cech filtrations (introduced recently by D. Sheehy et. al.), in dimensions 1 and 2, can be used to construct robust representations of the data in the real and complex projective spaces. Examples will be presented in order to illustrate how projective coordinates provides a framework for topology-driven nonlinear dimensionality reduction, and geometric model generation.

This work extends results of V. de Silva, D. Morozov and M. Vejdemo-Johansson on persistent cohomology and circular coordinates.

not appear to be a standard way to quantify their statistical similarities and differences. I'll

introduce the method of swatches, which describes the local topology of a cell complex in terms

of probability distributions of local configurations. It allows a distance to be defined which

measures the similarity of the local topology of cell complexes. Convergence in this distance is

related to the notion of a Benjamini Schramm graph limit. In my talk, I will use this to state

universality conjectures about the long-term behavior of graphs evolving under curvature flow,

and to test these conjectures computationally. This system is of both mathematical and physical

interest.

If time permits, I will discuss other applications of computationally topology to curvature flow

on graphs, and describe recent work on a new notion of geometric graph limit.

of tree-like data structures that one might build on top of such samples, generalizing cover trees,

net trees, navigating nets, deformable spanners, and some classes of hierarchical spanners.

We are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.

**Algebraic Stability of Zigzag Persistence Modules**

Michael Lesnick The stability theorem for persistent homology is a central result in topological data analysis.

While the original formulation of the result concerns the persistent homology of mathbb{R}-

valued functions, the result was later cast in a

**Semicontinuity of Persistence Diagrams**

Amit Patel The persistence diagram is very different in philosophy from the barcode. Suppose we have a constructible persistence module of vector spaces. Its barcode is its list of indecomposables. Its persistence diagram is an encoding of all persistent vector

**Expected sizes of Poisson--Delaunay mosaics and their discrete Morse functions**

Herbert Edelsbrunner Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in R^n, we study the expected number o

**Sparse Cech filtrations, persistent cohomology and projective coordinates**

Jose Perea One of the main challenges in topological data analysis is to turn computed topological features, such as barcodes, into insights about the data set under analysis.

We will show in this talk how the persistent cohomology of sparse Cech filtrat

**Reeb cosheaves**

Vin de Silva I shall talk about Reeb graphs, and the benefits of regarding them as set-valued cosheaves on the real line. For instance, there is a smoothing operation and an interleaving distance for these cosheaves, which can be interpreted as a smoothing operat

**Categorical Representations of Reeb Space and Mapper: Convergence and Multivariate Data Analysis**

Bei Wang The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses th

**Realization Problems in Persistence**

Justin Curry In this talk I will introduce the realization problem in persistence, which asks what isomorphism classes of diagrams of vector spaces can be realized by diagrams of topological spaces and continuous maps. In particular, one could ask what barcodes c

**Untangling Planar Curves and Planar Graphs**

Jeff Erickson Any generic closed curve in the plane can be transformed into a simple closed curve by

a finite sequence of local transformations called homotopy moves. We prove that simplifying a

planar curve with n self-crossings requires Theta(n^{3/

**From Computer and Data back to Topology and Geometry**

Dan Burghelea It is well known how basic Algebraic Topology and geometrization of Large Data led

to Persistence Theory, a useful tool in Data Analysis.

In this talk I will explore the other direction; how Persistence Theory suggests and motivates

**Derived Group Completion**

Sanjeevi Krishnan Every connected, finite CW complex is homotopy equivalent to the classifying space

of a monoid [McDuff, 1979], a set with an associative and unital multiplication. Thus group

completions of monoids correspond to fundamental groups of ba

**Sheaf Theory: Applications & Computations**

Robert Ghrist

**Excision and product results for persistent homology**

Gunnar Carlsson Motivated by work on data sets coming from the biology and evolution, we have some

results concerning the persistent homology of trees and also products of finite metric spaces.

**Applications of Persistence to Time Series Analysis**

Elizabeth Munch

**Some thoughts on Sampling**

Don Sheehy In this talk I will discuss techniques and heuristics for subsampling metric data as well as a space

of tree-like data structures that one might build on top of such samples, generalizing cover trees,

net trees, navigating nets, deforma

**Higher Interpolation and Extension for Persistence Modules**

Peter Bubenik Persistence modules are the central algebraic object in topological data analysis. This

motivates the study of the geometry of the space of persistence modules. We isolate an elegant

coherence condition that guarantees the interpolation

**Topological Similarity of Cell Complexes**

Benjamin Schweinhart Although random cell complexes occur throughout the physical sciences, there does

not appear to be a standard way to quantify their statistical similarities and differences. I'll

introduce the method of swatches, which describes th

**The Discrete Flow Category**

Vidit Nanda Large-scale homology computations are often rendered tractable by discrete Morse

theory. Every discrete Morse function on a given cell complex X produces a Morse chain

complex whose chain groups are spanned by critical cells and whose