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 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, Duke University
October 1, 2013 1:30 PM - 2:30 PM
Lind 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 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 at Albany (SUNY)
October 22, 2013 1:30 PM - 2:30 PM
Lind 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, Target Corporation
November 5, 2013 1:30 PM - 2:30 PM
Lind 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 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 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 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, Université Libre de Bruxelles
February 4, 2014 1:30 PM - 2:30 PM
Lind 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 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 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, 3M
March 11, 2014 1:30 PM - 2:30 PM
Lind 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 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 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 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 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 South Carolina
April 22, 2014 1:30 PM - 2:30 PM
Lind 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.
On Hybrid SPLS-Elastic Net Feature Selection and Prediction: Application to Quality Assurance in Industrial Manufacturing Processes
Guy-vanie Miakonkana, University of Minnesota, Twin Cities
May 5, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
In this work we investigate on hybrid statistical methods based on a combination of Sparse Partial Least Squares methodology and Elastic Net regression. Through simulations, we first observe that while the Elastic Net methodology struggles much more than the Sparse Partial Least Squares in identifying noise variables, the Sparse Partial Least Square retains less relevant predictors in the models than the Elastic Net. In addition, the Sparse Partial Least Squares regression coefficients estimates of relevant predictors are inflated proportionally to the variance of the corresponding predictor. In order to mitigate the shortfalls of each methodology, we explore combinations of both techniques to form a hybrid Sparse Partial Least Squares-Elastic Net statistical method. A two-step approach that combines the strengths of these techniques is proposed so as to reduce the prediction error.
The Vietoris-Rips Complex of the Circle
Henry Adams, Duke University
May 6, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
Consider a Vietoris-Rips complex of the circle with the geodesic metric. This simplicial complex has an infinite number of vertices, one for each point in the circle. A theorem of Jean-Claude Hausmann implies that for small connectivity parameter, the Vietoris-Rips complex is homotopy equivalent to a circle. What happens as the connectivity parameter increases? We show that the Vietoris-Rips complex obtains the homotopy type of the circle, the 3-sphere, the 5-sphere, the 7-sphere, …, until finally it is contractible.
In particular, we describe the persistent homology of the Vietoris-Rips complex of the circle of unit circumference. The persistence diagram for (2k+1)-dimensional homology consists of the open interval with birth time k/(2k+1) and death time (k+1)/(2k+3).
Joint work-in-progress with Michal Adamaszek, Christopher Peterson, and Corrine Previte.
A Multi-Time-Scale Analysis of Stochastic Chemical Reaction Networks
Xingye Kan, University of Minnesota, Twin Cities
May 13, 2014 1:30 PM - 2:30 PM
Lind 305 [Map]Abstract
We consider stochastic descriptions of reaction networks in
which there are both fast and slow reactions, and the time scales are
widely separated. We obtain a reduced equation on a slow time scale by
applying a state space decomposition method to the full governing equation
and describe our reduction method on the reaction simplex. Based on the
analytic results, we approximate reaction probabilities, or so-called
propensity functions and present an efficient stochastic simulation
algorithm for the slow time scale dynamics. We illustrate the numerical
accuracy of the approximation by simulating several motivating examples.
This is an ongoing joint project with Chang Hyeong Lee and Hans G. Othmer.
Previous Postdoc Seminars