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The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltr an and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that eciently implements a nonconstructive idea of

This large format book is described by the author as a coffee table art book; however, it is much more like a labor of love for Steve Smale. Smale not only describes the early difficulties of trying to publish such a mineral “art” volume, but also his efforts to develop the talent of photographing his own specimens, some of which are published here. You can understand his passion for mineral specimens and the development of this eye-candy book from the fact that he decided to enlist the talents of Jeff Scovil (a renowned mineral photographer) to provide the remainder of the color plate images. In total, there are 101 exquisite mineral color photographs. Beside the magnificent photos, Smale also provides a very valuable discourse on his (and his wife Clara’s) personal collecting criteria. Among the many characteristics he discusses, three elements are paramount: beauty, balance, and economy. Beauty is self evident to him; the specimen should be attractive from all angles and all components (matrix and main crystal) should enhance the …

This work gives a mathematical foundation for bifurcation from a stable equilibrium in the genome. We construct idealized dynamics associated with the genome. For this dynamics, we investigate the two main bifurcations from a stable equilibrium. Finally, we give mathematical proofs of existence and points of bifurcation for the repressilator and the toggle gene circuits.

SUMMARYGenome architecture is important in transcriptional regulation and study of its features is a critical part of fully understanding cell identity. Altering cell identity is possible through overexpression of transcription factors (TFs); for example, fibroblasts can be reprogrammed into muscle cells by introducing MYOD1. How TFs dynamically orchestrate genome architecture and transcription as a cell adopts a new identity during reprogramming is not well understood. Here we show that MYOD1-mediated reprogramming of human fibroblasts into the myogenic lineage undergoes a critical transition, which we refer to as a bifurcation point, where cell identity definitively changes. By integrating knowledge of genome-wide dynamical architecture and transcription, we found significant chromatin reorganization prior to transcriptional changes that marked activation of the myogenic program. We also found that the local architectural and transcriptional dynamics of endogenous MYOD1 and MYOG reflected the global genomic bifurcation event. These TFs additionally participate in entrainment of biological rhythms. Understanding the system-level genome dynamics underlying a cell fate decision is a step toward devising more sophisticated reprogramming strategies that could be used in cell therapies.

Significance A basic problem in biology is understanding how information from a single genome gives rise to function in a mature multicellular tissue. Genome dynamics stabilize to give rise to a protein distribution in a given cell type, which in turn gives rise to the identity of a cell. We build a highly idealized mathematical foundation that combines the genome (within cell) and the diffusion (between cell) dynamical forces. The trade-off between these forces gives rise to the emergence of function. We define emergence as the coordinated effect of individual components that establishes an objective not possible for an individual component. Our setting of emergence may further our understanding of normal tissue function and dysfunctional states such as cancer.

I first met Rene at the well-known 1956 meeting on topology in Mexico City. He then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my first years after my Ph.D. I studied his work in cobordism, singularities of maps, and transversality, gaining many insights. I also enjoyed listening to his provocations, for example his disparaging remarks on complex analysis, 19th century mathematics, and Bourbaki. There was also a stormy side in our relationship. Neither of us could hide the pain that our public conflicts over “catastrophe theory” caused. Rene Thom was a great mathematician, leaving his impact on a wide part of mathematics. I will always treasure my memories of him. Steve Smale

In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable data-derived simplicial complex.

Partial differential equations and the Laplacian operator on domains in Euclidean spaces have played a central role in understanding natural phenomena. However, this avenue has been limited in many areas where calculus is obstructed, as in singular spaces, and in function spaces of functions on a space X where X itself is a function space. Examples of the latter occur in vision and quantum field theory. In vision it would be useful to do analysis on the space of images and an image is a function on a patch. Moreover, in analysis and geometry, the Lebesgue measure and its counterpart on manifolds are central. These measures are unavailable in the vision example and even in learning theory in general.

We provide a model (for both continuous and discrete time) describing the evolution of a flock. Our model is parameterized by a constant beta capturing the rate of decay-which in our model is polynomial-of the influence between birds in the flock as they separate in space. Our main result shows that when beta

We consider the regression problem and describe an algorithm approximating the regression function by estimators piecewise constant on the elements of an adaptive partition. The partitions are iteratively constructed by suitable random merges and splits, using cuts of arbitrary geometry. We give a risk bound under the assumption that a "weak learning hypothesis" holds, and characterize this hypothesis in terms of a suitable RKHS. Two examples illustrate the general results in two particularly interesting cases.

The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS). We follow our previous work and apply the sampling operator to the error analysis in both the RKHS norm and the L2 norm. The tool for estimating the sample error is a Bennet inequality for random variables with values in Hilbert spaces. By taking the Hilbert space to be the one consisting of Hilbert-Schmidt operators in the RKHS, we improve the error bounds in the L2 metric, motivated by an idea of Caponnetto and de Vito. The error bounds we derive in the RKHS norm, together with a Tsybakov function we discuss here, yield interesting applications to the error analysis of the (binary) classification problem, since the RKHS metric controls the one for the uniform convergence.

The 4D organization of the interphase nucleus, or the 4D Nucleome (4DN), reflects a dynamical interaction between 3D genome structure and function and its relationship to phenotype. We present initial analyses of the human 4DN, capturing genome-wide structure using chromosome conformation capture and 3D imaging, and function using RNA-sequencing. We introduce a quantitative index that measures underlying topological stability of a genomic region. Our results show that structural features of genomic regions correlate with function with surprising persistence over time. Furthermore, constructing genome-wide gene-level contact maps aided in identifying gene pairs with high potential for coregulation and colocalization in a manner consistent with expression via transcription factories. We additionally use 2D phase planes to visualize patterns in 4DN data. Finally, we evaluated gene pairs within a circadian gene module using 3D imaging, and found periodicity in the movement of clock circadian regulator and period circadian clock 2 relative to each other that followed a circadian rhythm and entrained with their expression.

We continue our study [S. Smale, D.X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. 41 (2004) 279–305] of Shannon sampling and function reconstruction. In this paper, the error analysis is improved. Then we show how our approach can be applied to learning theory: a functional analysis framework is presented; dimension independent probability estimates are given not only for the error in the L 2 spaces, but also for the error in the reproducing kernel Hilbert space where the learning algorithm is performed. Covering number arguments are replaced by estimates of integral operators.

In this paper, we study an online learning algorithm in Reproducing Kernel Hilbert Spaces (RKHSs) and general Hilbert spaces. We present a general form of the stochastic gradient method to minimize a quadratic potential function by an independent identically distributed (i.i.d.) sample sequence, and show a probabilistic upper bound for its convergence.

whereA is an differential operator and L2 is the Hilbert space of square integrable functions on X with measure ρX on X defined via ρ. This minimization is well-conditioned and solved by straightforward finite dimensional least squares linear algebra to yield fγ,z : X → Y . The problem is posed: How good an approximation is fγ,z to fρ, or measure the error ∫ X(fγ,z − fρ)? and What is the best choice of γ to minimize this error? Our goal in this talk is to give some answers to these questions.

We present the development of a new theory of the pitchfork bifurcation, which removes the perspective of the third derivative and a requirement of symmetry.

(1) A main theme of this report is the relationship of approximation to learning and the primary role of sampling (inductive inference). We try to emphasize relations of the theory of learning to the mainstream of mathematics. In particular, there are large roles for probability theory, for algorithms such as least squares, and for tools and ideas from linear algebra and linear analysis. An advantage of doing this is that communication is facilitated and the power of core mathematics is more easily brought to bear. We illustrate what we mean by learning theory by giving some instances. (a) The understanding of language acquisition by children or the emergence of languages in early human cultures. (b) In Manufacturing Engineering, the design of a new wave of machines is anticipated which uses sensors to sample properties of objects before, during, and after treatment. The information gathered from these samples is to be analyzed by the machine to decide how to better deal with new input objects (see [43]). (c) Pattern recognition of objects ranging from handwritten letters of the alphabet to pictures of animals, to the human voice. Understanding the laws of learning plays a large role in disciplines such as (Cognitive) Psychology, Animal Behavior, Economic Decision Making, all branches of Engineering, Computer Science, and especially the study of human thought processes (how the brain works). Mathematics has already played a big role towards the goal of giving a universal foundation of studies in these disciplines. We mention as examples the theory of Neural Networks going back to McCulloch and Pitts [25] and Minsky and Papert [27], the PAC learning of Valiant [40], Statistical Learning Theory as developed by Vapnik [42], and the use of reproducing kernels as in [17] among many other mathematical developments. We are heavily indebted to these developments. Recent discussions with a number of mathematicians have also been helpful. In

This paper has two agendas. One is to develop the foundations of round-off in computation. The other is to describe an algorithm for deciding feasibility for polynomial systems of equations and inequalities together with its complexity analysis and its round-off properties. Each role reinforces the other.

We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in $n + 1$ complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed.

We continue the study of complexity classes over the weak model introduced by P. Koiran. In particular we provide several separations of complexity classes, the most remarkable being the strict inclusion of P in NP. Other separations concern classes defined by weak polynomial time over parallel or alternating machines as well as over nondeterministic machines whose guesses are required to be 0 or 1.

We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure between functions which we call the derived kernel. Based on a hierarchical architecture, we give a recursive definition of the neural response and associated derived kernel. The derived kernel can be used in a variety of application domains such as classification of images, strings of text and genomics data.

The problem of increasing the understanding of algorithms by considering the foundations of numerical analysis and computer science is considered. The schism between scientific computing and computer science is discussed from a theoretical perspective. These theoretical considerations have an intellectual importance when viewing the computer. In particular, the legitimacy and importance of models of machines that accept real numbers is considered.

Learning is key to developing systems tailored to a broad range of data analysis and information extraction tasks. We outline the mathematical foundations of learning theory and describe a key algorithm of it.

We describe a model for the evolution of the languages used by the agents of a society. Our main result proves convergence of these languages to a common one under certain conditions. A few special cases are elaborated in more depth.

90 Classical problems in economics are concerned with the solutions of several simultaneous nonlinear optimization problems, one for each consumer or producer, all facing constraints posed by the scarcity of resources. Often their interests conflict, and it is generally impossible to find a single real-valued function representing the interests of the whole of society. To deal with this problem, John Von Neumann introduces the theory of games. He also defined and established the existence of a general economic equilibrium, using topological tools [Von Neumann, 1938]. The work of Stephan Smale follows this tradition. He uses topological tools to deepen and refine the results on existence and other properties of another type of economic equilibrium, the Walrasian equilibrium (Walrus [1874-77]), as formalized by Kenneth J. Arrow and Gerard Debreu [1954], and of non-cooperative equilibrium in game theory as formalized by Nash (1950). This article aim to show that topology is intrinsically necessary for the understanding of the fundamental problem of conflict resolution in economics in its various forms and to situate Smale's contribution within this perspective. The study of conflicts of interests between individuals is what makes economics interesting and mathematically complex. Indeed, we now know that the space of all individual preferences, which define the individual optimization problems, is topologically nontrivial, and that its topological complexity is responsible for the impossibility of treating several individual preferences as if they were one, i.e., aggregating them (Chichilnisky, 1980; Chichilnisky and Heal, 1983). Because it is not possible, in general, to define a single optimization problem, other solutions are sought. This article will develop three solutions, discussed below. Because of the complexity arising from simultaneous optimization problems, economics differs from physics where many of the fundamental relations derive from a single optimization problem. The attempts to find solutions to conflicts among individual interests led to three different theories about how economies are organized and how they behave. These are general equilibrium theory, the theory of games, and social choice theory. Each of these theories leads naturally to mathematical problems of topological nature. Steve Smale has contributed fruitfully to the first two theories: general equilibrium theory and the theory of games. I will argue that his work is connected also with the third approach, social choice theory, by presenting in Section 4 results which link closely, and in unexpected ways, two seemingly different problems: the existence of a general equilibrium and the resolution by social choice of the resource allocation conflict in economics (Chichilnisky, 1991).