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Gunnar Carlsson

Gunnar Carlsson is one of the most renowned mathematicians in the world with his undergraduate degree from Harvard University and doctorate from Stanford. Over the past 35 years, Gunnar has taught at University of Chicago, University of California, Princeton University, and since 1991, Gunnar has been a professor of mathematics at Stanford University, where he has been a thought leader in a branch of mathematics called topology, the study of shape. In the theoretical sense, topology has been around since the 1700’s, but Gunnar pioneered the applied use of topology to solve complex real world problems starting in the late 1990’s. In the early 2000’s, this work led to $10M in research grants from the National Science Foundation (NSF) and DARPA to study the application of Topological Data Analysis (TDA) to problems of interest within the U.S. government. In 2008, based on the success of these efforts, Gunnar, along with two other Stanford mathematicians, co-founded Ayasdi. Gunnar is married, has 3 grown boys including two sons who are mathematicians, and lives in Palo Alto.

Posts by Gunnar Carlsson


Cluster analysis is a highly developed field within statistics. It takes as input a set X equipped with a distance function, and potentially some other information, and from that produces a partition of the set, i.e. a decomposition of the set into a collection of groups so that (a) the sets in the collection do […]

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In our earlier post, we saw that data can exhibit a great deal of complexity, and observed that complexity is often the most significant hurdle to overcome in the analysis process. In this post, we will show how topological data analysis (TDA) can overcome this hurdle, and deal with the great diversity of patterns that […]

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As we kick off 2016 and prepare for the Presidential stretch run, we thought it prudent to examine how Topological Data Analysis can find patterns and insights in survey data. In particular, we are focused on extracting subtle relationships that don’t present themselves willingly, but are valuable to understand. In this installment, we are going […]

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In earlier posts, we have talked about what TDA is, and why it is a powerful tool. In this post, we will talk about specific ways it is used to get useable information from data. We will use the specific part of TDA that constructs topological networks to reflect the shape of the data set. […]

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One of the key capabilities in topological data analysis is the identification of coherent subpopulations in complex data sets. Often these groups are based an outcome of interest. For example, in a genomic data set based on a disease, where one has a sample from a number of subjects, the outcome might be death of […]

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Everyone knows that there is currently a great deal of discussion about “Big Data”. It turns out, though, that complexity of the data is as much of a difficulty in making use of the data as is the sheer size. We have made enormous strides in handling size of data, to a point where the […]

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Topological data analysis has been very successful in discovering information in many large and complex data sets. In this post, I would like to discuss the reasons why it is an effective methodology. One of the key messages around topological data analysis is that data has shape and the shape matters. Although it may appear […]

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Topology is the study of shape, but it is in fact the study of shape from a particular point of view.  For example, it does not distinguish between a perfectly round circle and a circle which has been “squashed” into an ellipse. There are actually three important properties which topological analysis enjoys, and which are […]

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Unlike many numerical properties we deal with, shape is a somewhat nebulous concept.  We find that we can recognize similarities between shapes, but we are often unclear about how we recognize it.  Further, we are even more unclear about how we might instruct a machine to recognize and classify shapes.  One of the main tasks […]

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Machine learning is a collection of techniques for understanding data, including methods for visualization, prediction, classification and other tasks relevant for making sense of data.  The visualization techniques come under the heading of scatterplot methods, where one produces projections of the data points on two or sometimes three dimensions, and then plots the projections on […]

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The world of data analysis needs to change. Today we are living in darkness, only examining small data segments of interest as hypotheses are generated and then validated. This has worked reasonably well so far, but we are at an inflection point. Data has become too large and complex to be handled with these traditional […]

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The idea of visualizing data is a very appealing one. It provides a way to leverage our visual capacities to obtain information and understanding more quickly than by examining the data by query or algebraic methods. There are many ways to visualize data sets. Histograms, pie charts, bar graphs, heat maps, scatterplots, etc. are all […]

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Topology within mathematics can be characterized as that part of the subject which studies notions of shape.  It really consists of at least two separate threads, one in which one attempts to “measure” shape, and in the other in which one attempts to find compressed combinatorial representations of shape and analyze the degree to which […]

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