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Postdoc Seminars

IMA Postdoc Seminars provide postdocs in residence the opportunity to give talks of interest to the mathematical community (though they may not necessarily be related to the annual theme). In addition, IMA visitors and University of Minnesota faculty also present on subjects of interest to the postdocs.

Elizabeth Munch of the University of Minnesota will be organizing the 2013-2014 seminar series.

Titles, abstracts, and speakers for each seminar will be posted as available.

## Cellular Cosheaves

**Amit Patel, University of Minnesota, Twin Cities**

September 24, 2013 1:30 PM - 2:30 PM

Lind Hall 409 [Map]### Abstract

A regular cell complex is a discretization of a space into simple pieces that glue together nicely. A cellular cosheaf assigns to each cell of a regular cell complex data and to each face relation a morphism between the data. A cellular cosheaf is a tame cosheaf in the same way a regular cell complex is a tame space. In this talk, I will slowly build up to the definition of a cellular cosheaf.## What is the Vietoris-Rips Complex for Evenly Spaced Points Around a Circle?

**Henry Adams, University of Minnesota, Twin Cities**

October 1, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

Consider the Vietoris-Rips complex for n evenly-spaced points around a circle. For small choices of the connectivity parameter, this complex is homotopy equivalent to a circle. For n even and for a connectivity parameter slightly less than the diameter of the circle, this complex is the boundary of a cross-polytope and is homeomorphic to the (n/2-1)-sphere. What happens in the intermediate range when the connectivity parameter is neither small nor large? We provide evidence for the following claim: for the correct choice of connectivity parameter and for t>1, the Vietoris-Rips complex on (2m+1)t evenly-spaced points around a circle is homotopy equivalent to the (t-1)-fold wedge sum of 2m-spheres. This is joint work-in-progress with Cory Previte, Chris Peterson, and Alexander Hulpke.## Computing Singular Vectors with Random Noise

**Ke Wang, University of Minnesota, Twin Cities**

October 15, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

Computing singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, the data is usually perturbed by noise. The following question is of importance: How much does the singular vector of data matrix change under a small perturbation? The classical perturbation results, i.e. Davis-Kahan theorem and Wedin sin theorem, give tight estimates for the worst-case scenario. We show that better estimates can be achieved if the data matrix is low rank and the perturbation is random. This is joint work with Sean O'Rourke and Van Vu.## Categorification of Reeb Graphs

**Elizabeth Munch, University of Minnesota Twin Cities**

October 22, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

In order to understand the properties of a real-valued function on a topological space, we can study the Reeb graph of that function. Since it is efficient to compute and is a useful descriptor for the function, it has found its place in many applications. As with many other constructions in computational topology, we are interested in how to deal with this construction in the context of noise. In particular, we would like a method to "smooth out" the topology to get rid of, for example, small loops in the Reeb graph. In this talk, we will define a generalization of a Reeb graph as a functor. Using the added structure given by category theory, we can define interleavings on Reeb graphs. This also gives an immediate method for topological smoothing and we will discuss an algorithm for computing this smoothed Reeb graph.## An Exploration of Tipping Points From the Viewpoint of Persistent Topology

**Jesse Berwald, University of Minnesota, Twin Cities**

November 5, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

A tipping point occurs when adjusting one or more parameters of a system causes it to transition abruptly to a new state. Examples include desertification due to grazing, financial crises, and drastic shifts observed in past climate states. Data from such systems, whether "real world" or model-derived, may provide interesting insight into the system. We review the mechanisms underpinning recent statistical work in predicting "tipping points" in 1D time series. We will then discuss analysis of such data, theoretically of any dimension, from the angle of persistent topology.## Induced Matchings of Barcodes and the Algebraic Stability of Persistence

**Michael Lesnick, University of Minnesota, Twin Cities**

November 12, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

We deﬁne a simple, explicit map sending a morphism f : M → N of pointwise ﬁnite dimensional persistence modules to a matching between the barcodes of M and N . Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker(f) and coker(f). As an immediate corollary, we obtain a new proof of the Algebraic Stability of Persistence, a fundamental result in the theory of persistent homology due originally to Chazal et al., building on work of Cohen-Steiner et al. In contrast to previous proofs of the Algebraic Stability of Persistence, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. This is joint work with Ulrich Bauer.## Cohomological Waves

**Yiqing Cai, University of Minnesota, Twin Cities**

November 21, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

Waves are usually treated analytically. Another viewpoint to exam periodic waves is topologically, by looking at the map from the space which the waves lie on, to the period, which we think as a circle. Especially, if the space has non-trivial first homology, the degree (or winding number) of the map gives some information about the waves. One could extend the idea of degree to discrete time periodic waves, which has many applications in coverage problems in sensor networks. The results show that the cohomology class associated to every periodic wave by considering degrees indicates whether an evasion path exists or not.## Hadwiger and Lefschetz: Valuations on Simplicial Maps

**Matthew Wright, University of Minnesota, Twin Cities**

December 3, 2013 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

The intrinsic volumes generalize both Euler characteristic and volume, quantifying the "size" of a set in various ways. Hadwiger's Theorem says that any consistent notion of "size" for sets is a linear combination of the intrinsic volumes. In topological fixed-point theory, the Lefschetz number is a generalization of Euler characteristic to self-maps of simplicial complexes. It is possible to obtain analogues of the intrinsic volumes, called Lefschetz volumes, in the setting of simplicial self-maps. This talk will introduce Lefschetz volumes and present a recent version of Hadwiger's Theorem for simplicial self-maps.## Ergodic Properties of Randomly Forced Flows

**Juraj Foldes, University of Minnesota, Twin Cities**

February 4, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

By observing a turbulent flow, one realizes that it is unpredictable and seemingly chaotic what is caused by the sensitivity with respect to initial data and parameters. As early as in the 19th century, it was conjectured that turbulent flow cannot be solely described by deterministic methods, and it was indicated that a stochastic framework should be used. On the other hand, some statistical properties of these flows are very stable, and the invariant measures of the corresponding stochastic equations presumably contain the characteristics of the stable patterns posited by the basic theories of turbulence. In the talk, we will discuss the existence and uniqueness of statistically invariant state for the Boussinesq system and its mixing and ergodic properties. No previous knowledge of stochastic equations or fluid mechanics is required.## Keeping the TDA promise

**Jose Perea, Duke University**

February 18, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

Advocacy for the use of topological methods in the analysis of real-world data often comes as a two-part promise: 1. Topological methods provide ways of identifying hidden structures in data,which would otherwise be inaccessible. Moreover, finding such structures is not only interesting but really useful. 2. Persistence diagrams and bar codes yield rich and succinct representations of highly nontrivial objects. These representations, in turn, can be used to successfully attack challenging problems. In this talk I will present results geared toward keeping this two-part promise. I will show how having access to the Klein-bottle model for highly occurring image-patches can be used with high success in the problem of texture classification; and how understanding persistence diagrams from time-delay embeddings yields good methods for quantifying periodicity and quasiperiodicity in time-series data.## An Algorithm in Computational Geometry and an Exploration in Computational Topology

**Lori Ziegelmeier, Macalester College**

February 25, 2014 1:30 PM - 2:30 PM

Lind Hall 409 [Map]### Abstract

In this talk, a novel algorithm related to a fundamental problem in computational geometry as well as a new exploration in computational topology will be presented. The convex hull of a set of points, C, appears as a useful construct in a variety of contexts. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine and one should expand the list of high interest candidates to points lying near the boundary of the convex hull. In the first part of this talk, a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull is discussed. In the second part of this talk, a new exploration of applying topological data analysis techniques to biological aggregations, or swarming data, will be discussed. Standard metrics used in the swarming community require a priori knowledge to reveal structure from such data. However, an analysis of simulation data indicates that persistent homology barcodes may naturally reveal this structure.## Non-Compact Global Attractors and Infinite-Time Blow-Up

**Nitsan Ben-Gal, University of Minnesota, Twin Cities**

March 11, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

Global attractors and their deconstructions provide us with a versatile tool for studying the long-time behavior of solutions to reaction-diffusion equations. The majority of global attractor theory assumes that the equation in question is dissipative, but recent work has shown that similar results can be obtained for non-dissipative reaction-diffusion equations which exhibit infinite-time blow-up of solutions without any finite-time blow-up. Such phenomena appear in the analysis of various physical models ranging from suspension bridges to Rayleigh-Benard convection to tumor growth. In this talk I will discuss my recent results in this area and some of the geometric methods which yield them, including bifurcation diagrams, Fucik spectra, and nodal properties, as well as the relation of this work to the analysis of non-smooth dynamical systems. This is joint work with Kristen Moore and Juliette Hell.## The Truth About How to Get a Job

**Robert Ghrist, University of Pennsylvania**

March 18, 2014 1:00 PM - 2:00 PM

Lind Hall 409 [Map]Abstract forthcoming.

## Universality of the Homotopy Interleaving Distance

**Michael Lesnick, University of Minnesota, Twin Cities**

March 25, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

We observe that two key results about the persistence barcodes of point cloud data can be lifted to statements about filtrations, formulated directly on the topological level, given a choice of pseudometric on filtrations satisfying (i) a stability property and (ii) a homotopy invariance property. We introduce a pseudometric d_{HI}, the homotopy interleaving distance, satisfying these properties. We show that d_{HI} is universal in a sense analogous to which the bottleneck distance on persistence barcodes is universal: Namely, we show that if d is another distance on filtrations satisfying properties (i) and (ii) then d leq d_{HI}. This is joint work with Andrew Blumberg.## Intrinsic Volumes of Random Cubical Complexes

**Matthew Wright, University of Minnesota, Twin Cities**

April 1, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

The intrinsic volumes generalize both Euler characteristic and Lebesgue volume, quantifying the size of a set in various ways. A random cubical complex is a union of (possibly high-dimensional) unit cubes selected from a lattice according to some probability model. I will describe a simple model of random cubical complex and derive exact polynomial formulae, dependent on a probability, for the expected value and variance of the intrinsic volumes of the complex. I will also give a central limit theorem and an interleaving theorem about the roots of the expected intrinsic volumes -- that is, the values of the probability parameter at which an expected value is zero. Lastly, I will discuss connections to random fields and applications, especially image recognition and the study of noise in digital images. This work is in collaboration with Michael Werman of The Hebrew University of Jerusalem.## A Lattice Point Counting Problem relating to the Heisenberg Groups

**Krystal Taylor, University of Minnesota, Twin Cities**

April 15, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

The problem of counting integer lattice points inside, on, and near large dilates of convex surfaces is a classic and time-honored problem in number theory and related areas. Given a non-empty convex set B ⊂ Rn, one expects that the number of lattice points in RB is well approximated by the volume. We are interested in studying the error in this approximation in a particular setting. In particular, we estab- lish an error estimate for the number of points in a ball of large radius R described by the natural radial and Heisenberg-homogeneous norms on the Heisenberg groups given by Nα,A((z, t)) = |z|α + A|t|α/21/α, for α ≥ 2. Our method of bounding the error term involves obtain- ing decay estimates on the Euclidean Fourier transform of these balls. We comment on the extend to which our result is sharp and make comparison with an analogue of our method to some Euclidean lattice point counting results. This is joint work with Rahul Garg and Amos Nevo of the Israel Institute of Technology## Reduced-order Modeling of Complex Fluid Flows

**Zhu Wang, University of Minnesota, Twin Cities**

April 22, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]### Abstract

In many scientific and engineering applications of complex flows, computational efficiency is of paramount importance. Thus, model reduction techniques are frequently used. To achieve a balance between the low computational cost required by a reduced-order model and the complexity of the target turbulent flows, appropriate closure modeling strategies need to be employed. In this talk, we present reduced-order modeling strategies synthesizing ideas originating from proper orthogonal decomposition and large eddy simulation, develop rigorous error estimates and design efficient algorithms for the new reduced-order models.## Title Forthcoming

**Henry Adams, University of Minnesota, Twin Cities**

May 6, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]Abstract forthcoming.

**Ke Wang, University of Minnesota, Twin Cities**

May 13, 2014 1:30 PM - 2:30 PM

Lind Hall 305 [Map]Abstract forthcoming.