Alp Uzman ([email protected])

Fractal Geometry and Dynamics

In this course we'll investigate fractals and dynamics. Fractals are objects that appear similar at different scales, while dynamics is the mathematical study of time evolutions of closed systems. Often a fractal is the "residue" of a certain dynamical system let to evolve for a long time; for instance, a Barnsley fern is the attractor of a certain iterated function system, and the Mandelbrot set is determined by whether or not the trajectory of a seed runs off to infinity under a certain dynamical system. This deep connection makes studying these two subjects in tandem fruitful.  Registration is by instructor permission.

Graeme Milton ([email protected])

Exploring Field Pattern Materials

Composite materials sometimes have properties unlike those of their constituents and in this context they are known as metamaterials.  A famous example is the Roman Lysurgis glass cup in the British museum which may appear either green or red depending on whether the light source is inside or outside the glass.  Regular arrays of spheres of silica give opals their color.  Wave propagation in static but inhomogeneous media has been well studied.  Now there is great interest in exploring wave propagation in dynamic media, i.e. "space-time materials," where the material properties vary with both time and space.  This class will explore space-time materials for which waves propagate in ordered patterns in space-time.  These are known as field pattern materials.  Previous research has only scratched the surface and there is much to investigate here.  More examples of field pattern materials are needed.  This is just ray tracing, but one needs to position the interfaces so the rays form patterns.  What relations are possible between the periodicity of the material and the periodicity of the field pattern?  (The field pattern can break symmetry just as symmetry is broken in "time crystals.")  If there is no field pattern, but rather just a complicate cascade of wave scattering, can one find something analogous to an ergodic hypothesis to make predictions?  After reviewing the literature, the class will start with easy problems then move on to more challenging problems.  It is anticipated that all students making significant contributions will be coauthors of a publication resulting from the work.  Writing up the paper will be part of the class.  For further background see the SIAM news article, "New Horizons in the Study of Waves in Space-Time Microstructures" and references therein. 

Kurt Vinhage ([email protected])

Dynamics, Coding and Probability

Transformations and flows are natural mathematical models for systems which evolve with time. In this course, we'll investigate flows and transformations defined by smooth functions, differential equations and shifts on symbolic spaces, and the surprising connection between these categories. Interested students should have a good understanding of linear algebra, real analysis on Euclidean space, ordinary differential equations and basic metric space topology. Familiarity with measure theory will be beneficial, but not required.

Sean Lawley ([email protected])

Applied Stochastics

Stochastic models are used extensively across science, engineering, public policy, and many other fields, including areas which do not seem "random."  A few examples include physiology, pharmacology, epidemiology, finance, search processes, and collective decision-making.  In this course, we will learn mathematical fundamentals of stochastic modeling, stochastic simulation methods, and survey a variety of important real-world applications.  Students should expect to work hard solving problems, doing computations, reading, and presenting in class.  The course grade will be a combination of homework, class participation, and a project.

Tommaso de Fernex ([email protected])

Introduction to Algebraic and Tropical Curves

The course introduces basic concepts in algebraic and tropical geometry. Algebraic geometry studies zero sets of polynomials; in tropical algebra, the sum of two numbers is their minimum and the product is their sum. Rather than giving a rigorous, abstract treatment of the theory, we will discuss many examples focusing on the case of plane curves and explore some of the connections between algebra, geometry, topology, and other fields.

Selected Numerical Algorithms and Their Analysis

Computational Mathematics is an essential part of modern applied sciences.  The course will provide an introduction to research in the area of Numerical Analysis and Scientific Computing through lectures, students' presentations and projects.  Examples of topics include selected topics in numerical linear algebra; introduction to selected numerical methods for partial differential equations with applications to problems from Biology, Fluid Dynamics and Materials Science.

Topics in Geometric Group Theory

Symmetry, Manifolds and Flags

What do you remember from your undergraduate math classes? How do they connect to topics in graduate-level mathematics? In this course, we will use the notions of symmetry and manifolds to reexamine some of the big ideas from calculus, linear algebra, analysis, group theory and topology, and look at some advanced topics, such as representations of finite groups, Lie groups and Lie algebras and a bit of the algebraic geometry of flag manifolds.

Prerequistes:

Math 2270 (Linear Algebra) and 3210-20 (Foundations of Analysis), Instructor Consent

Mathematics of Cancer

Cancer is the failure of regulation of cell replication in multicellular organisms. Understanding and treating cancer thus requires linking processes within individual cells with their environment, including non-cancerous cells, physical barriers, resources, and the immune system. It is through a sick, twisted version of evolution that the barriers inhibiting growth and spread are overcome. Treatment seeks to halt or reverse this process without causing undue damage to the host. 

We will work together to build mathematical models of several key cancer processes, none of which are fully understood either biologically or mathematically, and then focus the second half of the semester on projects.

Polynomials

Polynomials are central to the study mathematics. For example, we learn in Calculus that Taylor polynomials are used to approximate exponential, logarithmic, and trigonometric functions. Behind these simple mathematical objects is a rich theory which unites many branches of mathematics. In this course we will explore analytic, algebraic, geometric, and number theoretic properties of polynomials in hopes of connecting the dots between seemingly different branches of mathematics.

Financial Machine Learning 

In modern day finance, with intrinsic nonlinearities in the models and vast amount of data sets available, machine learning (ML) is destined to transform the financial world as we know it, ranging from customer services and security measures. In this course we will discuss two particular data analysis subjects closely related to traditional quantitative financial analysis: portfolio selection and algorithmic trading. We will begin with a quick survey of a wide variety of data structures available and the challenges presented, and the basic notions of machine learning tools. The nature of finance makes it particularly difficult for standard machine learning tools to apply and yield successful results consistently. The rate of failure in financial ML is rather high and we would like to explain the reasons and provide clues to recognize the shortcomings. One area we would like to address is the assessment of values of strategies, and another is the detection of structural breaks. Regarding models, we will discuss the basic ideas in cross-validation and backtesting. For asset allocation, we will discuss approaches beyond the traditional quadratic optimizers that can compute a portfolio on ill-degenerated covariance matrix that is quite practical in reality.

Data analysis in Neuroscience

The advancement of neuroscience techniques has allowed collection of vast and increasingly complicated sets of data. What can we learn from them? What are useful ways to analyze and present them? In this course we will explore methods of data analysis that are employed in modern neuroscience. We will learn about relevant techniques through a combination of reading, presentations, discussions and solving problem sets. We will then apply these techniques to actual data collected in neurophysiological and behavioral experiments. Students should expect to work hard solving problems, doing computations, reading and presenting in class. The course grade will be a combination of homework, class participation and a project.

Representation Theory of Finite Groups

Roughly speaking, group theory is the mathematical study of symmetry. Shapes in the plane may have rotational or reflectional symmetry; a collection of n objects can be permuted in n! different ways; a Rubik’s cube can be configured in roughly 43 quintillion different ways. Symmetries can be composed and in general the order of composition matters. The resulting mathematical objects - groups - have a rich structure. Understanding this structure can be quite challenging. In this course we’ll start with some down-to-earth examples of groups to build some intuition and the ability to do computations. Then we’ll dive into the basics of Representation Theory - a field that studies how to represent abstract groups as a collection of square matrices. We’ll see that the trace of these matrices is something very worthy of study - the character of the representation. Students should expect to work hard solving problems, doing computations, reading and presenting proofs in class. The course grade will be a combination of homework, class participation and an open-ended project.

 Graph Theory

This course, which is intended for advanced mathematics majors and computer science majors, is about graph theory. A graph is a simple mathematical structure that stores information about how a set of objects is connected. The definition of graph is so natural that although graphs arise frequently in mathematics and computer science, they rarely get the attention they deserve. That is about to change. We will construct graphs, prove theorems about graphs, and study algorithms that solve graph-theoretic problems relating to enumeration, subgraphs, graph coloring, finding routes, network flows, graph decompositions, and more. See Wikipedia’s page on graph theory for an introduction to these topics. Students will be encouraged to do graph computations in a programming language of their choice. Based on students’ interests, we will consider more advanced applications of graphs to linear algebra, group theory, topology, algebraic geometry, logic, complexity theory, theorem proving, data mining, and studying large networks.

 Random Walks with Algebraic Combinatorics

If you have ever found yourself in an unknown city then likely you performed some version of a random walk: not knowing which street to take next you randomly chose among the available options and then repeated. Although it’s a simple mechanism, the statistics of the walk produced in this way turn out to be ubiquitous across mathematics. If the geometry of the city has an underlying “group structure” then from a mathematical point of view the random walk process is particularly interesting. Commonly the statistical properties of the random walker are studied by simple counting arguments, and if one exploits the underlying group structure to do so then the tools of algebraic combinatorics become available. The course will start with the basics of simple random walk on integer lattices, with an emphasis on studying it through combinatorial ideas. Many simple and cute, but powerful, methods of counting will be used. We will then move into the study of random walks on graphs and groups and along the way encounter many interesting objects such as Young tableaux, the RSK algorithm, and the matrix­tree theorem. We will also briefly discuss the deep connections to probability, statistics, differential equations, geometry, and number theory.

Network Inverse Problems

Networks can be used to model many physical phenomena such as electricity conduction and vibrations of an elastic body. We focus on the inverse problem, i.e. the question: Can one recover properties of the network from measurements made at a few nodes? A classic example is to recover the position and weight of beads in a vibrating string from measuring how the string responds to being plucked at one end. This class explores connections between physics, graph theory, partial differential equations, linear algebra and stochastic processes. Applications include medical imaging and geophysical prospecting.

 Algebra, Geometry and a little bit of String, Theory

It turns out that String Theory in Physics is built on some very concrete algebra and geometry. In this course we will begin with the algebra and geometry that students bring to the course and inch our way toward the geometry of the other six dimensions that String Theory postulates in order to allow for an 'understandable' universe.

Introduction to tropical geometry

The purpose of this course is to give an introduction to algebraic curves and tropical curves. The origins of algebraic geometry lie in the study of zero sets of systems of polynomials. These objects are algebraic varieties, and they include familiar examples such as plane curves and surfaces in three-dimensional space. In tropical algebra, the sum of two numbers is their minimum and the product of two number is their sum. It makes perfect sense to define polynomials and rational functions over the tropical semiring. The functions they define are piecewise-linear. Also, algebraic varieties can be defined in the tropical setting. They are now subsets of ℝnRn that are composed of convex polyhedra. Thus, tropical algebraic geometry is a piecewise-linear version of algebraic geometry.

An inverse problem: finding boundary fields which produce breakdown

Most materials break down if the fields are high enough, this breakdown may be mechanical fracture or plastic yielding (for elasticity) or electrical shorting (in dielectric media) and in general one wants to prevent this. It is obviously important to know what boundary conditions necessarily lead to dangerously high internal fields. For a homogeneous body this is straightforward as one could solve for the internal fields, but what if the body is inhomogeneous, say containing two materials (or one material with holes) in an unknown geometry? Here we will explore this question and the ultimate goal of the course will be to produce a scientific paper on the problem, with the class contributing to the research and coauthoring the paper which will then be submitted to a scientific journal. Elementary analysis and numerical computation will be required, though it is expected that the class will have different strengths in different areas. Prerequisites are basic PDE theory and linear algebra.

Selected Numerical Algorithms and Their Analysis

Computational Mathematics is an essential part of modern applied sciences. The course will provide an introduction to the research in the area of Numerical Analysis and Scientific Computing through lectures, students' presentations and projects. Example of Topics: Selected topics in numerical linear algebra, introduction to numerical methods for partial differential equations involving interfaces and irregular domains, meshfree approximation methods. The discussion on each topic will be self-contained.

Inquiry Into Mathematics

This course gives a problem-based introduction to the methods of mathematical research with a focus on topics within discrete mathematics, algebra, and geometry. Through a combination of lectures, problem-sessions and projects we will take a look at various examples of how mathematical theory can build from asking simple questions and generalizing. Topics will vary from week to week. A small sampling includes: special numbers, the map colour problem, incompleteness, projective geometry.

Polynomials

This course will be a mixture of algebraic geometry, number theory and some topics of a contemporary nature. It is about polynomials in one and several variables: their algebraic properties, the geometry of their real, complex and tropical solution sets, and the number theory of their rational and integer solution sets. Polynomials have fascinated mathematicians for thousands of years, and yet most people probably can't say a single intelligent thing about them. You will have lots to say after taking this course.

The Mathematics of Disease

Introduction to Tropical Geometry

To A_D_E and Beyond; Dynkin Diagram and Classifications

Symmetry & Goups

Metamaterials and cloaking theory

Graph Theory

Random Walk: Modeling, Theory, and Applications

Knot Theory

Metric Spaces, The Contraction Mapping Principle, Fractals & Other Applications

Fractals

Topics in Mathematical Finance