Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
|
|
|
Session Overview | |
|
Location: Unitobler, F006 30 seats, 57m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | MS143, part 1: Algebraic geometry in topological data analysis |
| Unitobler, F006 | |
|
|
10:00am - 12:00pm
Algebraic geometry in topological data analysis In the last 20 years methods from topology, the mathematical area that studies “shapes", have proven successful in studying data that is complex, and whose underlying shape is not known a priori. This practice has become known as topological data analysis (TDA). As additional methods from topology still find their application in the study of complex structure in data, the practice is evolving and expanding, and now moreover draws increasingly upon data science, computer science, computational algebra, computational topology, computational geometry, and statistics. While ideas from category theory, sheaf theory and representation theory of quivers have driven the theoretical development in the past decade, in the last years ideas from commutative algebra and algebraic geometry have started have started to be used to tackle some theoretical problems in TDA. The aim of the minisymposium is to seize this momentum and to bring together experts in algebraic geometry and researchers in topological data analysis to explore new avenues of research and foster research collaborations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Algebraic geometry in topological data analysis: an overview In this talk I will give an overview on how techniques and ideas from algebraic geometry have been used so far in topological data analysis, and discuss possible new avenues of research.
Applications of Groebner bases I will discuss the Groebner bases theory and its application to calculation of Hilbert series, and related invariants of quadratic algebras and operads. Examples where this technique can be used and on these bases, for example, Koszulity proved, will be given. It can be applied to persistent homology problems, to problems in neuroscience, etc. The Groebner technique itself originated in computer science, more precisely, in computer algebra, but as we will see, after proper algebraic formulation it can serve for proving some structural and homological results about algebras presented by generators and relations. This in turn can serve for the solution of applied problems, for example, by the study of neural codes as pseudo monomial ideals. Decomposition of 2-parameter persistence modules Decomposition theorems are one of the pillars of persistence theory. They yield complete discrete invariants for persistence modules, which can be used as descriptors for data in downstream applications. While the case of 1-parameter persistence modules is by now well-understood, the multi-parameter case remains mostly unexplored and appears to be much more complicated. In this talk I will focus on the 2-parameter case, and provide an overview of the current state of the art together with some perspectives. Classification of filtered chain complexes Persistent homology has proven to be a useful tool to extract information from data sets. Homology, however, is a drastic simplification and in certain situations might remove too much information. This prompts us to study filtered chain complexes. We prove a structure theorem for filtered chain complexes and list all possible indecomposables. We call these indecomposables interval spheres and classify them into three types. Two types correspond respectively to finite and infinite interval modules, while the third type is unseen by homology. The structure theorem states that any filtered chain complex can be written as the unique sum of interval spheres, up to isomorphism and permutation. The proof is based on a hierarchy of full subcategories of the category of filtered chain complexes. Such hierarchy suggests an algorithm for decomposing filtered chain complexes, which also retrieves the usual persistent barcodes. This approach offers a way to retrieve more geometrical aspects of data: while homology cannot tell the difference between a point and a disk, our decomposition provides a tool to count the contractible parts of the data, thus, we can obtain not only topological but also geometrical information.
|
| 3:00pm - 5:00pm | MS165, part 1: Multiparameter persistence: algebra, algorithms, and applications |
| Unitobler, F006 | |
|
|
3:00pm - 5:00pm
Multiparameter persistence: algebra, algorithms, and applications Multiparameter persistent homology is an area of applied algebraic topology that studies topological spaces, often arising from complex data, simultaneously indexed by multiple parameters. In the usual setting, persistent homology studies a single-parameter filtration associated with a topological space. The homology of such a filtration is a persistence module, which can be conveniently described by its barcode decomposition. In many applications, however, a single-parameter filtration is not adequate to encode the structures of interest in complex data; two or more filtrations may be required. Multiparameter persistence studies the homology of spaces equipped with multiple filtrations. The homological invariants of these spaces are far more complicated than in the single-parameter setting, requiring new algebraic, computational, and statistical techniques. This work has deep connections to representation theory and commutative algebra, with compelling applications to data analysis. Recent years have seen considerable advances in multiparameter persistent homology, including algorithms for working with large multiparameter persistence modules, software for computing and visualizing invariants, statistical techniques, and applications. This minisymposium will highlight recent work in multiparameter persistence. Talks will include including theoretical results, algorithmic advances, and applications to data analysis. As many important questions remain to be answered in order to advance the theory and to increase the applicability of multiparameter persistence, this minisymposium seeks to cultivate discussion and collaboration that will lead to new results in the practical use of multiparameter persistent homology. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Multiparameter persistence: brief background and current challenges Persistent homology is a popular tool in topological data analysis, providing a method for discerning the shape of complex data. Applied in areas including computer graphics, biology, neuroscience, and signal processing, persistent homology produces easily-visualized algebraic invariants, called barcodes, which convey information about the topological structure of data. A multiparameter variant of persistent homology is particularly desirable for working with data simultaneously indexed by multiple parameters, but its algebraically complexity poses challenges in practice. This talk with introduce multiparameter persistent homology, with emphasis on mathematical foundations of this subject. We will see how multifiltered topological spaces arise from real-world data scenarios. We will introduce multiparameter persistence modules and see how their algebraic complexity poses challenges for their use in practice. We will briefly consider recent work, applications, and open questions in multiparameter persistent homology. Computing minimal presentations and bigraded Betti numbers of 2-parameter persistent homology Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm assumes that $M$ is given implicitly: It takes as input a short chain complex of free bipersistence modules [Faxrightarrow{ma} Fb xrightarrow{mb} Fc] such that $Mcong ker{mb}/im{ma}$. The algorithm runs in time $O(sum_i |F^i|^3)$ and requires $O(sum_i |F^i|^2)$ storage, where $|F^i|$ denotes the size of a basis of $F^i$. Given the presentation, the bigraded Betti numbers of the module are readily computed. We also present a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In preliminary experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin. A kernel for multi-parameter persistent homology and its computation Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. In this talk, we discuss a kernel construction for multi-parameter persistence and why this kernel can provably be useful in applications. Morse inequalities for multiparameter persistence Discrete Morse theory is a combinatorial version of Morse theory which has proved to be an incredibly useful tool with applications in a large variety contexts. Intuitively speaking, discrete Morse theory allows to reduce a combinatorial cell complex (for example, a simplicial or cubical complex) to a subset of its cells, called critical, that carry all the homological information. The relation between the number of critical cells and the Betti numbers is described by the so-called Morse inequalities. So far, the connection between persistent homology and discrete Morse theory has been studied mainly with the purpose of simplifying complexes and speeding up the algorithms that compute the persistence modules. In this sense, only the reduction aspect of discrete Morse theory has been leveraged in connection to persistence. In this talk we show the possibility of establishing Morse inequalities for persistence. To this aim, we consider a filtration of a cell complex that varies according to multiple parameters, the associated multiparameter persistence module, and the critical cells of a discrete gradient vector field compatible with the multifiltration. Our goal is to derive Morse inequalities relating the number of critical cells of the given vector field to the multigraded Betti numbers of the persistence module. Thisrequires the use of specific tools from homological algebra, which we briefly illustrate. |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS143, part 2: Algebraic geometry in topological data analysis |
| Unitobler, F006 | |
|
|
10:00am - 12:00pm
Algebraic geometry in topological data analysis In the last 20 years methods from topology, the mathematical area that studies “shapes", have proven successful in studying data that is complex, and whose underlying shape is not known a priori. This practice has become known as topological data analysis (TDA). As additional methods from topology still find their application in the study of complex structure in data, the practice is evolving and expanding, and now moreover draws increasingly upon data science, computer science, computational algebra, computational topology, computational geometry, and statistics. While ideas from category theory, sheaf theory and representation theory of quivers have driven the theoretical development in the past decade, in the last years ideas from commutative algebra and algebraic geometry have started to be used to tackle some theoretical problems in TDA. The aim of the minisymposium is to seize this momentum and to bring together experts in algebraic geometry and researchers in topological data analysis to explore new avenues of research and foster research collaborations. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) High-throughput topological screening of nanoporous materials Thanks to the Materials Genome Initiative, there is now a database of millions of different classes of nanoporous materials, in particular zeolites. In this talk I will sketch a computational approach to tackle high-throughput screening of this database to find the the best nano-porous materials for a given application, using a topological data analysis-based descriptor (TD) recognizing pore shapes. For methane storage and carbon capture applications, our method enables us to predict performance properties of zeolites. When some top-performing zeolites are known, TD can be used to efficiently detect other high-performing materials with high probability. We expect that this approach could easily be extended to other applications by simply adjusting one parameter: the size of the target gas molecule. Sampling real algebraic varieties for topological data analysis I will discuss an adaptive algorithm for finding provably dense samples of points on a real algebraic variety given the variety's defining polynomials as input. Our algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling and it also employs geometric heuristics to reduce the size of the sample. As persistent homology methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying these methods more feasible. I will also present results of applying persistent homology to point samples generated by an implementation of the algorithm. How wild is the homological clustering problem? Connected components form the basis of many clustering methods, often requiring a choice of two parameters (geometric scale and density). Applying 0th homology yields a diagram of vector spaces reflecting the connected components, with surjections in the scale parameter direction. This motivates the study of the parameter landscape by means of quiver representations: indecomposable summands can be interpreted as topological features. We identify all cases where the set of possible indecomposables has a simple classification (finite type or tame). The result is obtained using tilting theory and a novel equivalence theorem on cotorsion-torsion triples, whose development has been motivated by the clustering problem. Learning elliptic curves Elliptic curves are all homeomorphic as topological spaces, more precisely, they are all real tori of dimension two. However, they carry infinitely many different complex structures. The topological structure can be detected easily by the tools of persistent homology, but can we also recover the complex structure? In other words, can we "learn" an elliptic curve from data? In my talk I would like to address this question. |
| 3:00pm - 5:00pm | MS165, part 2: Multiparameter persistence: algebra, algorithms, and applications |
| Unitobler, F006 | |
|
|
3:00pm - 5:00pm
Multiparameter persistence: algebra, algorithms, and applications Multiparameter persistent homology is an area of applied algebraic topology that studies topological spaces, often arising from complex data, simultaneously indexed by multiple parameters. In the usual setting, persistent homology studies a single-parameter filtration associated with a topological space. The homology of such a filtration is a persistence module, which can be conveniently described by its barcode decomposition. In many applications, however, a single-parameter filtration is not adequate to encode the structures of interest in complex data; two or more filtrations may be required. Multiparameter persistence studies the homology of spaces equipped with multiple filtrations. The homological invariants of these spaces are far more complicated than in the single-parameter setting, requiring new algebraic, computational, and statistical techniques. This work has deep connections to representation theory and commutative algebra, with compelling applications to data analysis. Recent years have seen considerable advances in multiparameter persistent homology, including algorithms for working with large multiparameter persistence modules, software for computing and visualizing invariants, statistical techniques, and applications. This minisymposium will highlight recent work in multiparameter persistence. Talks will include including theoretical results, algorithmic advances, and applications to data analysis. As many important questions remain to be answered in order to advance the theory and to increase the applicability of multiparameter persistence, this minisymposium seeks to cultivate discussion and collaboration that will lead to new results in the practical use of multiparameter persistent homology. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Algebraic distances for persistent homology One of the main ideas in Topological Data Analysis is to convert application data into an algebraic object called a persistence module and to calculate distances between such modules. I will introduce these constructions and describe the main examples of such distances, called Wasserstein distances. The weakest of these distances, called the bottleneck distance, has previously been described algebraically (called interleaving distance). This has been useful in theory and in applications. I will give an algebraic description of all of the Wasserstein distances and discuss their generalizations to multiparameter persistence. This is joint work with Jonathan Scott and Don Stanley. Multiparameter persistence landscapes An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. For single parameter persistence modules Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this talk we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter persistence landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance. Moreover the multiparameter landscapes enjoy many more desirable properties: the collection of multiparameter landscapes associated to a module are interpretable, computable, amenable to statistical analysis, and faithfully represent the rank invariant. We shall provide example calculations to demonstrate potential applications and how one can interpret the multiparameter landscapes associated to a multiparameter module. Geometric perspectives on multiparameter persistence Using ideas inspired from geometric and differential topology, we introduce a version of multiparameter persistence, which combines sub-level and zig-zag persistence. Our construction arises from one-parameter families of smooth functions on compact manifolds. We show how to analyse this version of multiparameter persistence in geometric terms with several examples. Furthermore, we focus on practical aspects of this theory, with an emphasis on visualization and potential algorithm development. This is joint work with Peter Bubenik. Persistent homology of noise I will describe a sequence of fairly naive experiments and small observations, towards a characterization of the persistent homology of noise. This should be viewed as an attempt to quantify what it means for a bar to be "short", vs. "long" or "interesting". |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS146, part 1: Random geometry and topology |
| Unitobler, F006 | |
|
|
10:00am - 12:00pm
Random geometry and topology This minisymposium is meant to report on the recent activity in the field of random geometry and topology. The idea behind the field is summarized as follows: take a geometric or topological quantity associated to a set of instances, endow the space of instances with a probability distribution and compute the expected value, the variance or deviation inequalities of the quantity. The most prominent example of this is probably Kostlan, Shub and Smales celebrated result on the expected number of real zeros of a real polynomial. Random geometry and topology offers a fresh view on classical mathematical problems. At the same time, since randomness is inherent to models of the physical, biological, and social world, the field comes with a direct link to applications. More infos at: https://personal-homepages.mis.mpg.de/breiding/siam_ag_2019_RAG.html (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Zero-sets of 3D random waves We study Berry's model in the three-dimensional case. This model contains as particular cases the monochromatic random waves and the black-body radiation, which are isotropic Gaussian fields that a.s. solve the Helmholtz equation. We generalize it to include more general features as anisotropy. We are interested in the zero-sets of the random waves as they represent lines of darkness, threads of silence, etc. We compute moments and find limit distributions under mild hypothesis and compare it with the well studied 2D-case. This is a joint work with Anne Estrade and José R. León. Curvature and randomness The Euclidean Distance degree measures the algebraic complexity of writing the optimal solution to the best approximation problem to an algebraic variety as a function of the coordinates of the data point. The number of real-valued critical points of the distance function can be different for different data points. For randomly sampled data the expected number of real valued critical points is of high interest and it is called the average ED degree. In this talk we will see connections between the average ED degree, the ED discriminant and different curvatures of the underlying variety. Random sections of line bundles over real Riemann surfaces We will explain how to compute the higher moments of the random variable ”number of real zeros of a random polynomial”. More generally, given a line bundle L over a real Riemann surface, we explain how to compute all the moments of the random variable ”number of real zeros of a random section of L”. On the topology of real components of real sections of vector bundles This talk will present joint work with Tanner Strunk. At present, the talk will consist of a collection of methods and numerical data concerning the probability of various topologies that arise as the real zeros of real sections of vector bundles. Some of the methods utilized for collecting such data are interesting and might be useful in a general context. However, the main focus of the talk will be on the number of different topologies that can arise in various settings and conjectural relationships between their probabilities. While some initial results are rather intriguing, we are currently only able to provide statistical data rather than theory to support the data. It is the hope that additional insight into the results is obtained by the time of the conference. At the very least, perhaps the data will lead to some interesting discussions. |
| 3:00pm - 5:00pm | MS189, part 1: Geometry and topology in applications. |
| Unitobler, F006 | |
|
|
3:00pm - 5:00pm
Geometry and topology in applications. This symppsium will bring together leading practitioners, mid-carreer scientists as well as PhD students and postdoctoral fellows who are interested in the theory and practice of the applications of geometry and topology in real life problems. The spectrum of possible applications is very wide, and covers the sciences, biology, medicine, materials science, and many others. The talks will address the theoretical foundations of the methodology as well as the state of the art of geometric and topological modelling. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Topological data analysis in materials science Topological data analysis (TDA) is an emerging concept in applied mathematics in which we characterize “shape of data” using topological methods. In particular, the persistent homology and its persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In my talk, I will explain our recent activity of TDA on materials science, e.g. glass, polymer, granular system, iron ore sinter etc. By developing several new mathematical tools based on quiver representations, inverse analysis, and machine learnings, we can explicitly characterize significant geometric and topological (hierarchical) features embedded in those materials, which are practically important for materials properties. Optimal transport in tropical geometric phylogenetic tree space Recent results by Monod et al. (2018) establish that palm tree space---that is, the space of phylogenetic trees in the tropical geometric construction, and endowed with the tropical metric---is a metric measure space with well-defined properties for probability and statistics on sets of phylogenetic trees. With the tropical metric as ground metric, we construct foundations for optimal transport theory on palm tree space. In particular, we build the Wasserstein-p metric which allows for the comparison of probability distributions of different random variables on palm tree space. We study the cases where p = 1, which gives an efficient way to compute geodesics, and p = 2, which gives deeper insight into the geometry of palm tree space. This is joint work with Wuchen Li (UCLA) and Bo Lin (Georgia Tech).
Primary distance for multipersistence When persistent homology is used to summarize data objects, distances between the resulting persistence modules serve as proxies for distances between the data objects themselves. In the presence of more than one parameter, module distances are complicated by the rich algebraic structure of multipersistence. In particular, unboundedness of the set of parameters presents problems with integration, interleaving, and other measures. Primary decomposition and algebraic operations related to it provide canonical (functorial) ways to extract bounded parameter sets, yielding convergence for existing measures that are based on integration. In addition, primary distances isolate from mixtures of multipersistence types pure contributions that would, in many existing measures, otherwise introduce bias when truncation or enforced decay are used without taking into account the algebraic structure. Outlier robust subsampling techniques for persistent homology The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS146, part 2: Random geometry and topology |
| Unitobler, F006 | |
|
|
10:00am - 12:00pm
Random geometry and topology This minisymposium is meant to report on the recent activity in the field of random geometry and topology. The idea behind the field is summarized as follows: take a geometric or topological quantity associated to a set of instances, endow the space of instances with a probability distribution and compute the expected value, the variance or deviation inequalities of the quantity. The most prominent example of this is probably Kostlan, Shub and Smales celebrated result on the expected number of real zeros of a real polynomial. Random geometry and topology offers a fresh view on classical mathematical problems. At the same time, since randomness is inherent to models of the physical, biological, and social world, the field comes with a direct link to applications. More infos at: https://personal-homepages.mis.mpg.de/breiding/siam_ag_2019_RAG.html (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Grassmann Integral Geometry In this talk, I will give an overview on the probabilistic intersection theory on the real Grassmanniann. In particular I will focus on Schubert cells. We will show how Young Tableaux can be used to describe not only the cell itself but also its tangent and normal at one point. This description allows us to use the integral geometry formula to compute the average volume of intersection of Schubert cells. Moreover we will see how one can associate to any Schubert cell - and more generally to any submanifold of a homogeneous space - a particular convex body (zonoid) on the cotangent bundle together with a law of multiplication that form a probabilistic graded ring. I will present all the numerous questions that arise with this new point of view on integral geometry in homogeneous spaces. Topology of Gaussian Random Fields We present the space of smooth Gaussian Random Fields on a smooth manifold and discuss the notion of narrow convergence (or convergence in law) for sequences of such fields, which provides a good language to investigate a class of common problems in stochastic geometry. We will describe how narrow convergence is equivalent to smooth convergence of the covariance functions and see an application of this result to Kostlan polynomials. (This is a joint work with Antonio Lerario) Spectrum of the Laplace Operator for Random Geometric Graphs We investigate some properties of the spectrum of the normalized Laplace operator for random geometric graphs in the thermodynamic regime. This is a joint work with Antonio Lerario. Sampling from the uniform distribution on a variety This talk presents the problem of sampling from the uniform distribution on a real affine variety with finite volume, given just its defining polynomials. We can choose a point on such a variety by choosing first a hyperplane of the right codimension and then one of its intersection points with the variety. In this talk, I explain how to do this such that the chosen point is uniformly distributed. I show examples of the corresponding algorithm for sampling in action and highlight a connection to topological data analysis. This is joint work with Paul Breiding. |
| 3:00pm - 5:00pm | MS189, part 2: Geometry and topology in applications. |
| Unitobler, F006 | |
|
|
3:00pm - 5:00pm
Geometry and topology in applications. This symppsium will bring together leading practitioners, mid-carreer scientists as well as PhD students and postdoctoral fellows who are interested in the theory and practice of the applications of geometry and topology in real life problems. The spectrum of possible applications is very wide, and covers the sciences, biology, medicine, materials science, and many others. The talks will address the theoretical foundations of the methodology as well as the state of the art of geometric and topological modelling. (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) Persistent Betti numbers of random Cech complexes We study the persistent homology of random Cech complexes. Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions. Then we define the k-th persistent Betti number of a random Cech complex and determine its asymptotic order in the subcritical regime. This extends a result of Kahle on the asymptotic order of the ordinary k-th Betti number of such complexes to the persistent setting. This is joint work with Ulrich Bauer (TU Munich). Topological Analyses of Time Series Measurements from real-world systems produce time series, or sequential scalar-valued data, that contain information about complicated higher dimensional dynamics of the underlying system. Extracting this information from time series is often done by frequency analyses and statistics which demand linearity and stationarity. We present topological methods for investigating dynamics from nonlinear, non-stationary time series in application to TMS-EEG data. On the Robustness of the Homological Scaffold Abstract: Homological Scaffold has been firstly introduced by Petri, Expert, Vaccarino et al. in 2014 in studying the effects on the functional connectome of the human brain under the effect of psilocybin. At that time it was defined empirically by using javaplex. In this talk, we will present two new principled definitions of the scaffold and the results of a comparison of the three scaffolds on simulated and real data. Joint work with A. De Gregorio, M.Guerra and G.Petri Stable and discriminative topological invariants In this talk I will describe a framework that allows to compute a new class of stable invariants for multi-parameter persistence. The key element of our approach is defining metrics induced by so called ‘noise systems’. Such metrics generalize the classical notion of interleaving distance. At the same time, in the one parameter case, they allow to overcome the usual dichotomy interpreting short bars in a barcode as noise and long bars as relevant information. I will then focus on one of the proposed invariants, the stable rank, address its statistical properties and show how we can improve classification by adapting the noise system to the task. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS146, part 3: Random geometry and topology |
| Unitobler, F006 | |
|
|
10:00am - 12:00pm
Random geometry and topology This minisymposium is meant to report on the recent activity in the field of random geometry and topology. The idea behind the field is summarized as follows: take a geometric or topological quantity associated to a set of instances, endow the space of instances with a probability distribution and compute the expected value, the variance or deviation inequalities of the quantity. The most prominent example of this is probably Kostlan, Shub and Smales celebrated result on the expected number of real zeros of a real polynomial. Random geometry and topology offers a fresh view on classical mathematical problems. At the same time, since randomness is inherent to models of the physical, biological, and social world, the field comes with a direct link to applications. More infos at: https://personal-homepages.mis.mpg.de/breiding/siam_ag_2019_RAG.html (25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise) The integer homology threshold for random simplicial complexes The very first problem considered in the now-classic Linial-Meshulam model was to generalize the connectivity threshold from Erdős-Rényi random graphs to higher dimensions as homological connectivity. Early work by Linial, Meshulam, and Wallach had established this homology-vanishing threshold for finite field coefficients, however this a priori does not establish the threshold for integer coefficients. In joint work with Elliot Paquette discussed here, we establish this threshold for homology with integer coefficients to vanish. The real tau-conjecture is true on average Koiran's real tau-conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of f are independent standard Gaussian random variables, then the expected number of real zeros of f is O(mkt), which is linear in the number of parameters. Geometric limit theorems in topological data analysis In a joint work with V. Limic and S. Kalisnik Verosek we generalize the notion of barcodes in topological data analysis in order to prove limit theorems for point clouds sampled from an unknown distribution as the number of points goes to infinity. We also investigate rate of convergence questions for these limiting processes. Quantitative Singularity theory for Random Polynomials In this talk, based on a joint work with A. Lerario and P. Breiding, I will present some probabilistic approximations of singularity type of a polynomial. The case of special interest is the zero set of a polynomial. We will show with an overwhelming probability, the set of real zeros of a polynomial of degree d can be realized as the zero set of a polynomial of degree sqrt{d log(d)}. |
| 3:00pm - 5:00pm | Room free |
| Unitobler, F006 | |
