# Read e-book Algebraic geometry for scientists and engineers

Local analytic geometry Shreeram Shankar Abhyankar E-bok. Ladda ned. Spara som favorit. Skickas inom vardagar. This book is intended as an algebraic geometry textbook for engineers and scientists. In addition to providing an elementary, manipulative, and algorithmic approach to the subject, the author also attempts to motivate and explain its link to more modern algebraic geometry based on abstract algebra.

It is not always enough to rely on the mathematics and statistics that is captured in textbooks or software, for two reasons: 1 progress is continually being made, and off-the-shelf techniques are unlikely to be cutting edge, and 2 solutions tailored to particular situations or questions can often be much more effective than more generic approaches.

These are the benefits to the nonmathematical sciences members of the team. For mathematical science collaborators, the benefits are likewise dual: 1 fresh challenges are uncovered that may stimulate new results of intrinsic importance to the mathematical sciences and 2 their mathematical science techniques and insights can have wider impact. In application areas with well-established mathematical models for phenomena of interest—such as physics and engineering—researchers are able to use the great advances in computing and data collection of recent decades to investigate more complex phenomena and undertake more precise analyses.

Conversely, where mathematical models are lacking, the growth in computing power and data now allow for computational simulations using alternative models and for empirically generated relationships as means of investigation. Computational simulation now guides researchers in deciding which experiments to perform, how to interpret experimental results, which prototypes to build, which medical treatments might work, and so on. Indeed, the ability to simulate a phenomenon is often regarded as a test of our ability to understand it.

Over the past years, computational capabilities reached a threshold at which statistical methods such as Markov chain Monte Carlo methods and large-scale data mining and analysis became feasible, and these methods have proved to be of great value in a wide range of situations.

## Shreeram Shankar Abhyankar - Wikipedia

For example, at one of its meetings the study committee saw a simulation of biochemical activity developed by Terrence Sejnowski of the Salk Institute for Biological Studies. It was a tour de force of computational simulation—based on cutting-edge mathematical sciences and computer science—that would not have been feasible until recently and that enables novel investigations into complex biological phenomena.

As another example, over the past 30 years or so, ultrasound has progressed from providing still images to dynamically showing a beating heart and, more recently, to showing the evolution of a full baby in the womb. The mathematical basis for ultrasound requires solving inverse problems and draws. As ultrasound technologies have improved, new mathematical challenges also need to be addressed.

The mathematical sciences contribute in essential ways to all the items on this list except the fourth. The great majority of computational science and engineering can be carried out well by investigators from the field of study: They know how to create a mathematical model of the phenomenon under study, and standard numerical solution tools are adequate.

However, as the phenomena being modeled become increasingly complex, perhaps requiring specialized articulation between models at different scales and of different mathematical types, specialized mathematical science skills become more and more important.

### School of Mathematical Sciences

Absent such skills and experience, computational models can be unstable or even produce unreliable results. Validation of such complex models requires very specialized experience, and the critical task of quantifying their uncertainties can be. The research teams must have strong statistical skills in order to create reliable knowledge in such cases.

In response to the need to harness this vast computational power, the community of mathematical scientists who are experts in scientific computation continues to expand. This cadre of researchers develops improved solution methods, algorithms for gridding schemes and computational graphics, and so on. Much more of that work will be stimulated by new computer architectures now emerging. And, a much broader range of mathematical science challenges stem from this trend.

The theory of differential equations, for example, is challenged to provide structures that enable us to analyze approximations to multiscale models; stronger methods of model validation are needed; algorithms need to be developed and characterized; theoretical questions in computer science need resolution; and so on.

High-throughput data in biology has been an important driver for new statistical research over the past years. Research in genomics and proteomics relies heavily on the mathematical sciences, often in challenging ways, and of disease, evolution, agriculture, and other topics have consequently become quantitative as genomic and proteomic information is incorporated as a foundation for research. Arguably, this development has placed statisticians as central players in one of the hottest fields of science.

Over the next years, acquiring genomic data will become fairly straightforward, and increasingly it will be available to illuminate biological processes. Biomedical Computation Review , September 1, gives an overview of sources of error and points to some striking published studies.

## Algebraic geometry for scientists and engineers

See also National Research Council, As biology transitions from a descriptive science to a quantitative science, the mathematical sciences will play an enormous role. To different degrees, the social sciences are also embracing the tools of the mathematical sciences, especially statistics, data analytics, and mathematics embedded in simulations.

For example, statistical models of disease transmission have provided very valuable insights about patterns and pathways. Business, especially finance and marketing, is increasingly dependent on methods from the mathematical sciences. Some topics in the humanities have also benefited from mathematical science methods, primarily data mining, data analysis, and the emerging science of networks. The mathematical sciences are increasingly contributing to data-driven decision making in health care.

Operations research is being applied to model the processes of health care delivery so they can be methodically improved. The applications use different forms of simulation, discrete optimization, Markov decision processes, dynamic programming, network modeling, and stochastic control. As health care practices move to electronic health care records, enormous amounts of data are becoming available and in need of analysis; new methods are needed because these data are not the result of controlled trials.

The new field of comparative effectiveness research, which relies a great deal on statistics, aims to build on data of that sort to characterize the effectiveness of various medical interventions and their value to particular classes of patients.

Embedded in several places in this discussion is the observation that data volumes are exploding, placing commensurate demands on the mathematical sciences. This prospect has been mentioned in a large number of previous reports about the discipline, but it has become very real in the past 15 years or so. What really matters is our ability to derive from them new insights, to recognize relationships, to make increasingly accurate predictions.

Our ability, that is, to move from data, to knowledge, to action. Large, complex data sets and data streams play a significant role in stimulating new research applications across the mathematical sciences, and mathematical science advances are necessary to exploit the value in these data. However, the role of the mathematical sciences in this area is not always recognized.

Indeed, the stated goals for the OSTP initiative,. Multiple issues of fundamental methodology arise in the context of large data sets. Some arise from the basic issue of scalability—that techniques developed for small or moderate-sized data sets do not translate to modern massive data sets—or from problems of data streaming, where the data set is changing while the analysis goes on. Data that are high-dimensional pose new challenges: New paradigms of statistical inference arise from the exploratory nature of understanding large complex data sets, and issues arise of how best to model the processes by which large, complex data sets are formed.

Not all data are numerical—some are categorical, some are qualitative, and so on—and mathematical scientists contribute perspectives and techniques for dealing with both numerical and non-numerical data, and with their uncertainties. Noise in the data-gathering process needs to be modeled and then—where possible—minimized; a new algorithm can be as powerful an enhancement to resolution as a new instrument. Often, the data that can be measured are not the data that one ultimately wants.

This results in what is known as an inverse problem—the process of collecting data has imposed a very complicated transformation on the data one wants, and a computational algorithm is needed to invert the process.

The classic example is radar, where the shape of an object is reconstructed from how radio waves bounce off it. Simplifying the data so as to find its underlying structure is usually essential in large data sets. The general goal of dimensionality reduction—taking data with a large number of measurements and finding which combinations of the measurements are.

Various methods with their roots in linear algebra and statistics are used and continually being improved, and increasingly deep results from real analysis and probabilistic methods—such as random projections and diffusion geometry—are being brought to bear. Statisticians contribute a long history of experience in dealing with the intricacies of real-world data—how to detect when something is going wrong with the data-gathering process, how to distinguish between outliers that are important and outliers that come from measurement error, how to design the data-gathering process so as to maximize the value of the data collected, how to cleanse the data of inevitable errors and gaps.

As data sets grow into the terabyte and petabyte range, existing statistical tools may no longer suffice, and continuing innovation is necessary. In the realm of massive data, long-standing paradigms can break—for example, false positives can become the norm rather than the exception—and more research endeavors need strong statistical expertise. For example, in a large portion of data-intensive problems, observations are abundant and the challenge is not so much how to avoid being deceived by a small sample size as to be able to detect relevant patterns.

In that approach, one uses a sample of the data to discover relationships between a quantity of interest and explanatory variables. Strong mathematical scientists who work in this area combine best practices in data modeling, uncertainty management, and statistics, with insight about the application area and the computing implementation. These prediction problems arise everywhere: in finance and medicine, of course, but they are also crucial to the modern economy so much so that businesses like Netflix, Google, and Facebook rely on progress in this area.

A recent trend is that statistics graduate students who in the past often ended up in pharmaceutical companies, where they would design clinical trials, are increasingly also being recruited by companies focused on Internet commerce. Finding what one is looking for in a vast sea of data depends on search algorithms. This is an expanding subject, because these algorithms need to search a database where the data may include words, numbers, images and video, sounds, answers to questionnaires, and other types of data, all linked.

New York Times , August 5. New techniques of machine learning continue to be developed to address this need. Another new consideration is that data often come in the form of a network; performing mathematical and statistical analyses on networks requires new methods.

Statistical decision theory is the branch of statistics specifically devoted to using data to enable optimal decisions. What it adds to classical statistics beyond inference of probabilities is that it integrates into the decision information about costs and the value of various outcomes. Ideas from statistics, theoretical computer science, and mathematics have provided a growing arsenal of methods for machine learning and statistical learning theory: principal component analysis, nearest neighbor techniques, support vector machines, Bayesian and sensor networks, regularized learning, reinforcement learning, sparse estimation, neural networks, kernel methods, tree-based methods, the bootstrap, boosting, association rules, hidden Markov models, and independent component analysis—and the list keeps growing.

This is a field where new ideas are introduced in rapid-fire succession, where the effectiveness of new methods often is markedly greater than existing ones, and where new classes of problems appear frequently. Large data sets require a high level of computational sophistication because operations that are easy at a small scale—such as moving data between machines or in and out of storage, visualizing the data, or displaying results—can all require substantial algorithmic ingenuity.

### Algebraic Geometry for Scientists and Engineers

As a data set becomes increasingly massive, it may be infeasible to gather it in one place and analyze it as a whole. Thus, there may be a need for algorithms that operate in a distributed fashion, analyzing subsets of the data and aggregating those results to understand the complete set. One aspect of this is the. This is essential when new waves of data continue to arrive, or subsets are analyzed in isolation of one another, and one aims to improve the model and inferences in an adaptive fashion—for example, with streaming algorithms.

The mathematical sciences contribute in important ways to the development of new algorithms and methods of analysis, as do other areas as well. Simplifying the data so as to find their underlying structure is usually essential in large data sets. The general goal of dimensionality taking data with a large number of measurements and finding which combinations of the measurements are sufficient to embody the essential features of the data set—is pervasive.

Various methods with their roots in linear algebra, statistics, and, increasingly, deep results from real analysis and probabilistic methods—such as random projections and diffusion geometry—are used in different circumstances, and improvements are still needed. Related to search and also to dimensionality reduction is the issue of anomaly detection—detecting which changes in a large system are abnormal or dangerous, often characterized as the needle-in-a-haystack problem.

The Defense Advanced Research Projects Agency DARPA has its Anomaly Detection at Multiple Scales program on anomaly-detection and characterization in massive data sets, with a particular focus on insider-threat detection, in which anomalous actions by an individual are detected against a background of routine network activity. A wide range of statistical and machine learning techniques can be brought to bear on this, some growing out of statistical techniques originally used for quality control, others pioneered by mathematicians in detecting credit card fraud.

Two types of data that are extraordinarily important yet exceptionally subtle to analyze are words and images. The fields of text mining and natural language processing deal with finding and extracting information and knowledge from a variety of textual sources, and creating probabilistic models of how language and grammatical structures are generated.

Image processing, machine vision, and image analysis attempt to restore noisy image data into a form that can be processed by the human eye, or to bypass the human eye altogether and understand and represent within a computer what is going on in an image without human intervention. Related to image analysis is the problem of finding an appropriate language for describing shape. As part of this problem, methods are needed to describe small deformations of shapes, usually using some aspect of the geometry of the space of.

Shape analysis also comes into play in virtual surgery, where surgical outcomes are simulated on the computer before being tried on a patient, and in remote surgery for the battlefield.