Quadratic Forms With Applns to Algebraic Geometry and Topology

Quadratic Forms and Algebraic Cobordism
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  • Submission history
  • London Mathematical Society Lecture Note Series
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    Add to cart. Be the first to write a review About this product. About this product Product Information This volume discusses results about quadratic forms that give rise to interconnections among number theory, algebra, algebraic geometry, and topology.

    Submission history

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    Ongoing projects

    Online Price 1 Label: List. Online Price 1: Print Price 1 Label: List. Print Price 1: Online Price 2: Print Price 2: Online Price 3: Print Price 3: Dual Price 1 Label: List. Dual Price 1: Dual Price 2: One direction is learning how to draw computer images of the Mandelbrot set, and of Julia sets of polynomials. We would also learn the mathematics of why our programs work. These sets have an interesting fractal structure. Students could also learn about iterating rational or general entire functions, or learn the proofs and the complex analysis behind some of the several important tools we use in this area.

    Exposure to real analysis e. MAT is a bonus. How should rational agents plan for the future? Each of these problems features agents interacting with uncertain, evolving environments. Agents are tasked with finding policies that optimize their outcomes with respect to particular performance criteria.

    Bernd Sturmfels (8/28/18): Learning algebraic varieties from samples

    In this project, the mentee will explore the theory of Markov Decision Processes, the mathematical model at the heart of this class of optimization problems. They will prove the existence of optimal policies for finite and infinite horizon MDPs under the expected total discounted reward criterion, and study algorithms which directly compute these policies. In the 's, Hodgkin and Huxley experimentally derived a system of non-linear ordinary differential equations that, to this day, people use to understand the behaviors of neurons. The goals of this DRP will be to understand how these equations come about and the ways in which people understand the solutions to this equation.

    The former will take us through a tour of how differential equations are used in chemistry and physics. The latter will take us on a tour of some concepts in dynamical systems. These are covered in Chapters 1 and 3 in the book. Time permitting, based on student interest, additional chapters from the book can be explored. Basic knowledge of probability will be very useful. Machine Learning ML is foremost known as a tool for the industry; however, the theoretical aspects behind the tools of ML also encompass a very active area of research.

    Most of these foundations are better understood under the lens of probability. In this project, we will learn enough probability to understand the whys and hows behind some of the basic algorithms of ML such as k-Nearest neighbors, Gaussian models, or Neural networks. Specific topics will depend on the interest of the student. If time permits it, and the student wishes to pursue doing so, we will implement i. MAT , probability and statistics e. AMS This project can be tailored to applications to physics, in which case statistical mechanics PHY is necessary. Random Matrix Theory frequently abbreviated as RMT is an active research area of mathematics with numerous applications to theoretical physics, number theory, combinatorics, statistics, financial mathematics, mathematical biology, engineering telecommunications, and many other fields.

    Random matrices are matrices with their entries drawn from various probability distributions, which are called random matrix ensembles. Our goal is to study the eigenvalues of such matrices, which oftentimes have a rich mathematical structure when the matrices are large. For example, the spacing between eigenvalues are described by universal laws which are also found to describe the nontrivial zeroes of the Riemann Zeta Function.

    During this project, we will cover a number of applications of RMT to mathematics or physics, depending on the interests of the student. MAT , real analysis e. A student interested in studying Brownian motion as part of this project must have taken or be enrolled in a course in measure theory e.

    There are two models for looking at the diffusion of heat inside of some medium. The heat equation is a deterministic partial differential equation which describes this diffusion. The path of one such particle can be thought of as a random walk or Brownian motion. Such random processes have applications in several areas of science, engineering, and finance. A discrete random walk can be thought of as a particle moving in the integer lattice, choosing a direction randomly at each stage. In this project, we will study discrete random walks in the plane and general euclidean space, answering questions such as how many times does a random walker return back to their starting point?

    We will see how this probabilistic point of view can enrich the study of a discrete version of the heat equation along with other areas of analysis. Prerequisites: A background in partial differential equations e. When we imagine a particle in motion, we envision a single object with zero size which does not change as it moves, but rather holds itself together in a consistent way.

    However, when a wave hits the shoreline, it is not a single particle that arrives. Waves have nonzero size.

    London Mathematical Society Lecture Note Series

    Moreover, they constantly encounter disturbances around them, from uneven ground below to animals swimming, yet still hold their fixed shape without error. However, for a certain class of nonlinear partial differential equations, these solitons are known to exist. Most famous among these is the Korteweg-de Vries equation. The explanation for this curious stability of solutions comes down to properties of this equation related to algebraic geometry. This project will investigate this shocking relationship between nonlinear PDE and algebraic geometry. To get to the core of this, we will introduce elliptic curves and the algebra of differential operators.

    This will bring us to a discussion of the Lax pair corresponding to this equation and, time permitting, the geometry of Grassmannian spaces. We will simultaneously learn how to use Mathematica to visualize and explore this subject. Graphs are mathematical objects used to represent societies and networks.

    The aim of spectral graph theory is to understand certain properties of graphs, by examining certain related matrices and their eigenvalues. For example, a classic theorem proved using spectral graph theory is the following: In a society where every two people have exactly one mutual friend, there is one person who is friends with everybody.

    This project can take on a more algebraic or a more analytic flavor depending on the interests of the student. Prerequisites: A course in multivariable calculus e. In a standard course in calculus in multiple variables one will encounter three theorems: Green's theorem, relating the flow of a vector field in a planar region to the circulation on the boundary; Stokes' theorem, relating the curl of a vector field on a surface to the circulation on the boundary; and the Divergence theorem, relating the divergence of a vector field in a 3-D region to the flux on the boundary. These are all special cases of what is known as Stokes' theorem on manifolds.

    This project will develop the concept of a manifold and that of a differential form. We will learn how to integrate these forms on manifolds and eventually prove the most general version of Stokes' theorem. Book: The Symmetries of Things , by J. Conway, H. Burgiel, and C. Prerequisites: Only an enthusiasm for mathematics is required for this project.

    This project can very much be tailored to the mathematical background of the mentee.

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    Albrecht Pfister, Johannes Gutenberg Universität Mainz, Germany. Subjects: Number Theory, Recreational Mathematics, Geometry and Topology, Mathematics. Chapter 2 - Multiplicative Quadratic Forms. Quadratic Forms With Applications To Algebraic. Geometry And Topology. Albrecht Pfister. Book review Introduction to quadratic forms over fields by T. Y. Lam.

    Symmetries and symmetric patterns surround us in all aspects of our lives. We would like to be able to describe symmetrical patterns mathematically, and the first task will be to enumerate them. The more advanced students will begin with spherical and frieze patterns, and will discover how group theory governs the possible symmetries that can occur. A more advanced student can go even further, venturing into hyperbolic space and flat universes. A course in real analysis e. We will start with some topology: introduction to topological spaces, homotopy and homology, and depending on the student this can be enough for the whole reading project.

    On the other hand, we can also move on to differentiable manifolds and Lie groups. The chosen book is especially approachable, because these are introduced in a more explicit, hands-on, way. The book then moves on to the application of all of these to the study of Gauge Fields, which lie at the heart of modern physics, especially particle physics, and which became a stage for deep collaborations between geometers and theoretical physicists.

    Links for the lazy

    Consider a function with multiple inputs and one output. For instance a function that takes in a point on the xy plane and outputs a single real number. In Calculus 3, presumably, we learned how to find minima of this equation: assuming our function is "nice" we check the partial derivatives.

    Imagine now we have an infinite dimensional space as an input still only a single real number as output. The theory of the calculus of variations gives us tools to sensibly think about finding minima of these functions in some special cases. Troutman introduces the reader to this theory via its many applications e. In this DRP, we follow Troutman through Chapter 6 - which he estimates is an introduction to the basics. Topics include: examples of problems that may be addressed, appropriate notions of derivatives in this setting, a discussion of the restrictions we require of functions in this setting, and analysis of some necessary and sufficient conditions for finding minima.

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