Spring 2020

ASTRONOMY AND ASTROPHYSICS: This course will trace our knowledge of the Universe from astronomy's ancient roots in naked eye observations of the sky to the twenty first century studies of extrasolar planetary systems, black holes, and cosmology. Initial topics will include: Newton's laws of motion and gravitation, orbits and space travel, and the properties of planets' surfaces, interiors, and atmospheres. The course will then combine atomic and nuclear physics with stellar and galactic astronomy to describe stars, supernovae, black holes, the interstellar medium, galaxies, the creation of the elements, and the evolution of the universe.

MODERN COSMOLOGY: Cosmology is the branch of physics that studies the Universe on its largest scales and endeavors to understand its origin, evolution, and fate. In this course, we will review the key ingredients and the main observations that contributed to our current understanding of the Universe. We will discover that modern cosmology not only provides an explanation of how structures formed during cosmic history, and evolve on large scales, but that it also answers questions about how nature is organized at a fundamental level. Topics to be explored include: the special and general theories of relativity, geometry, and expansion of the Universe, the Big Bang model, the cosmic microwave background, the large-scale structure of the Universe, dark matter and dark energy.

RELATIVITY AND QUANTUM PHYSICS: Relativity and quantum physics underpin much of our modern understanding of the universe. The first part of the course will present Einstein's special relativity, including topics such as Galilean relativity, Einstein's postulates, time dilation, length contraction, failure of simultaneity at a distance, Lorentz transformations, space-time, four-vectors, the relativistic Doppler effect, Compton scattering, the Einstein and de Broglie relations, and mass-energy equivalence. A brief interlude to general relativity covers the equivalence principle and gravitational redshift. The second part begins with a historical introduction to quantum physics, before moving on to topics such as wave interference, the double-slit experiment, complementarity, the Heisenberg uncertainty principle, the Bohr-Einstein debates, Bohr's atomic model, magnetic monopoles, particle in a box, and zero-point energy. Advanced topics include the two-state quantum system, quantum tunneling, and the Schrodinger equation. Students should have completed pre-calculus.

PARTICLE PHYSICS - EXPLORING MATTER AND FORCES: For more than a century, physicists have probed the inner workings of the atom in order to understand the fundamental constituents of matter and the forces that act between them. This course will present an overview of the Standard Model of particle physics, together with possible new physics at the high energy frontier. Topics will include: high-energy particle accelerators and detectors, quarks and leptons, matter and antimatter, unification of forces, neutrinos, the Higgs boson and the LHC, supersymmetry, and string theory. There will also be a brief discussion of special relativity, quantum mechanics, and the role of symmetries in physics. Recent observations, including the discovery of the Higgs particle, neutrino oscillations, and evidence for dark matter in the universe, will also be explored.

EXPERIMENTS IN PHYSICS: This course will have a combination of laboratory and theoretical work on the properties of electrons and photons, electromagnetism, the interference and diffraction of waves, the structure and dynamics of atoms, the radioactive decay of nuclei and the properties of elementary particles. The laboratory experiments will introduce students to key features of the fundamental particles and forces in nature, and will include a visit to one or more research laboratories on the Columbia campus.

CLASSICAL AND QUANTUM COMPUTING DEVICES: This course will introduce students to various techniques used to create micro-/nano-structures, with an emphasis on devices for classical and quantum information processing. Starting with the pioneering ideas presented by Richard Feynman in his paper "Plenty of room at the bottom", students will learn how those visionary proposals have developed into a discipline undergoing an exponential growth and extremely rapid innovation, particularly CMOS (complementary metal-oxide semiconductor) technology. The course will be highly interactive, including a visit to see examples of various metrology/microscopy tools (such as an atomic force microscope) in quantum materials labs on the Columbia campus. Students will have the opportunity to participate in the fabrication of single atom-thick materials as well as write basic programs to run on IBM's quantum circuit interface. The course will conclude with an introduction to quantum mechanics and the physics of solids, as it relates to quantum information science and technology, while maintaining the focus on the experimental and practical aspects of the discipline.

ORGANIC CHEMISTRY: This course combines lectures, laboratory experiments, and demonstrations to provide an introduction to the principles and exciting frontiers of organic chemistry. Students will be introduced to the synthesis of organic compounds and the reaction mechanisms. Lecture topics will include: chemical bonds, structural theory and reactivity, design and synthesis of organic molecules, and spectroscopic techniques (UV-Vis, IR, NMR) for structure determination. Experiments will introduce common techniques employed in organic chemistry and will include: extraction, recrystallization, thin layer and column chromatography, reflux, and distillation. Note that students must be present for one of the first two classes for mandatory safety training.

BIOCHEMISTRY: This course will provide a foundation for understanding the chemical basis of biological processes. The course will explore how molecules such as DNA, RNA and proteins are made and how their structure confers their function. Students will learn how biochemists clone out a selected gene from the entire genome of any organism, mass-produce protein from the gene, and purify it in order to study its biochemical properties and determine its structure. Students will be exposed to cutting-edge technologies such as X-ray diffraction, cryo-electron microscopy, and nuclear magnetic resonance used to determine protein structures at atomic resolution. The course will also cover fundamental metabolic pathways involving the break-down of carbohydrates, lipids and fatty acids and the crucial biological machines that carry out these processes. Students will learn how perturbation in molecular processes leads to complex pathologies, and understand how protein structures can be used to design novel therapeutic compounds in the fields of metabolic engineering and synthetic biology. By the end of the course, students will be asked to present their own ideas on a current innovative research concept and its potential applications.

EXPERIMENTS IN GENETICS AND MOLECULAR BIOLOGY: Through a sequence of experiments and analysis of large datasets, students will be introduced to some fundamental principles and basic techniques of genetics and molecular biology. Topics will focus on molecular evolution. Experiments include: DNA purification & amplification, sequencing, bacterial conjugation, and fruit fly mating. Students will also analyze datasets of human genetic variation to assess ancestry and potentially disease-causing mutations. These topics are designed to introduce students to a fundamental question of biology: how does DNA produce phenotypes (traits)?

VIROLOGY: This course will provide an understanding of how viruses work, using both historical and current examples. Students will learn about different types of viruses that infect animals, plants and bacteria, causing diseases from cold sores to hemorrhagic fevers. The course will also cover vaccines, host-pathogen interactions and gene therapy. While highly interactive and including group work, the course is primarily lecture-based.

HUMAN PHYSIOLOGY: This course provides an introduction to the major systems of the human body, including the cardiovascular, respiratory, digestive, endocrine, immune, and nervous systems. Discussions will progress from general system structure to function on a cellular level. An overview of pathology and current research will also be presented.

BIOINFORMATICS: The study of biology is changing rapidly thanks to the advent of DNA sequencing technology. This technique produces so much data that researchers must use tools from computer science, statistics, and physics to make sense of it all, in a new field broadly referred to as bioinformatics. In this course, we will explore diverse topics in bioinformatics ranging from genome wide association studies, to functional cancer genomics, to the human microbiome. Our goal is to showcase how data science can be applied to real-world problems across many areas of biology. Some coding experience will be helpful, but is not required.

NEUROSCIENCE - EXPLORING THE BRAIN: This course will provide a comprehensive overview of what we currently know about the brain and how we study it. We will explore the organization, structure, and function of this fascinating organ which enables us to sense, move, sleep, feel, and think. Going from single molecules to cells, from cells to neural circuits, and from networks to behavior, our journey will feature a description of how we perceive, process, store, and retrieve information, as well as how these processes are altered during disease states such as Alzheimer's, Parkinson's, depression, addiction, schizophrenia, and autism. Topics will include: anatomical and cellular organization of the brain, electrical impulses and signaling in neurons, neurodevelopment, sensory perception, movement, sleep, and higher cognitive functions such as language, emotions, learning, and memory.

UNDERSTANDING EARTH'S CLIMATE SYSTEM AND CLIMATE CHANGE: In this course, students will explore the Earth's climate system. We will learn about the physics of climate, how it affects life on Earth, and how humans are changing it. We will discuss the models and tools used by climate scientists and apply one of these methods on real climate data. Toward the end of the course, we will read from an international climate assessment and consider possible solutions.

TOPOLOGY: This course will give an introduction to topology. Roughly speaking, topology is the study of shape. To a topologist, a square and a circle have the same shape since lengths and angles do not affect shape. We will study properties that can describe and distinguish different shapes (Why does a donut have a different shape than a beach ball?). Using these properties, we will be able to prove things like the fundamental theorem of algebra (every polynomial has a root), Nash's equilibrium theorem, "there is a location on the earth where the wind is not blowing", and more! Other topics include: colorings of maps, the classification of surfaces, homotopy groups, the Ham Sandwich theorem, manifolds, knot theory, and homology groups. We will also see applications of topology to questions in data science, biology, and sociology via topological data analysis. No special mathematical background is required.

NON-EUCLIDEAN GEOMETRY: An introduction to geometry beyond the Euclidean geometry taught in high school and assumed in calculus. We will discuss hyperbolic geometry (mathematically inclined students are frequently interested in the hyperbolic tessellations of M.C. Escher) together with some of its applications, including the classical constructions of non-Euclidean geometries inside of Euclidean geometry, demonstrating the independence of Euclid's axiom about parallel lines from the other axioms. We will then learn about Bezout's theorem by experimenting with intersection points of curves in a plane, noting that if we extend our notion of geometry to the complex projective plane we get more consistent answers, and then further study projective geometry. Time permitting we will discuss additional topics including applications to physics such as general relativity.

DIAGRAMMATIC ALGEBRA: In high school, when you think of algebra, you often think about 'solving for x'. That is, we have a notion of multiplication, a notion of addition, and hence a notion of a polynomial - and now it makes sense to 'solve for x'. The multiplication you think of is algebraically defined (via multiplication of real numbers). But, what if instead of numbers, we could instead think of pictures, and we could replace multiplication of numbers by a 'joining-of-pictures'? Instead of the real numbers, we obtain some new structure ('an algebra'), and we may now try to solve for x in this new algebra. This is the basic idea behind diagrammatic algebra - defining algebraic relations via pictures, and looking for structures. It is an interesting blend of algebra, topology, combinatorics, and geometry. For instance, braid groups can be viewed topologically as the fundamental group of configuration space, and geometrically as the mapping class group of the n-punctured disk; one is interested in algebraic features such as the representation theory of braid groups, and also the topological features such as the cohomology of configuration spaces. In this course, we will use braid groups as a starting point, and address a whole host of other interesting diagrammatic constructions such as Temperley-Lieb algebras, Deligne categories, diagrammatic invariant theory, and zigzag algebras, to name a few.

ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS: Algebraic combinatorics is a very modern field of mathematics. It uses algebraic methods such as representation theory to address various combinatorial questions. In this course we will start with generating functions, discuss Catalan, Fibonacci, Bernoulli numbers, Bernoulli-Euler triangle, enumeration of trees, graphs on surfaces which leads to important results in Gromow-Witten theory such as Harer-Zagier formula. We will discuss various bases in the ring of symmetric functions such as Schur functions and their deformation Macdonald functions, and if we have time, its relation to quantum algebras and knot theory. We will also explore recent work which has revealed the power of algebraic combinatorics in quantum field theory and string theory. Students should have some knowledge of basic calculus.

REPRESENTATION THEORY AND PHYSICS: Our modern understanding of mathematics and physics relies greatly on uncovering and making use of (hidden) symmetries. For example, the laws of physics do not change over time; this is a symmetry called time-translation invariance, and the law of conservation of energy is a direct consequence of this symmetry. The mathematical toolkit used to understand symmetries is called representation theory. In this course we will see the beautiful ideas behind the representation theory of finite groups, used to describe discrete symmetries, and also the representation theory of Lie groups, used to describe continuous symmetries. Applications will be drawn from various fields, e.g. combinatorics, number theory, but primarily from physics. No special math or physics background will be assumed.

GRAPH THEORY BY EXAMPLE: Graph theory is a new and exciting area of discrete mathematics. Simply put, a graph is just a collection of points joined by certain pairs of these points, yet many real-world problems (i.e. traffic flow, school admissions, scheduling) can be formulated as such. Although many problems in graph theory can be easily stated, these problems often have complex solutions with far reaching implications and applications. Problem solving, class discussions, and student examples will be the major proportion of this course. Rigorous proofs will also be presented in the lecture. In addition to exploring the mathematics of graph theory, we will also see how graph theory arises in fields such as computer science, chemistry, game theory, and many others.

COMPUTER PROGRAMMING IN PYTHON: Students will learn the basics of programming using Python. Topics will include: variables, operators, loops, conditionals, input/output, objects, classes, methods, basic graphics, and fundamental principles of computer science. Approximately half of the class time will be spent working on the computer to experiment with the topics covered. Some previous programming experience will be helpful but is not required.

INTRODUCTION TO ALGORITHMS: This course motivates algorithmic thinking. The key learning objectives are the notions of run-time analysis of algorithms, computational complexity, algorithmic paradigms and data structures. Content will primarily be based on high-school algebra and calculus. A tentative list of topics includes: run-time analysis of algorithms, sorting, searching, hashing, computational complexity and complexity classes, graph algorithms, and dynamic programming.

EXPLORATIONS IN DATA SCIENCE: In this course, students will carry out a series of explorations in data science to learn about statistical thinking, principles and data analysis skills used in data science. These explorations will cover topics including: descriptive statistics, sampling and estimation, association, regression analysis, etc. Classes will be organized to have a lecture component and a hands-on exploration component each session. In the lecture session, an introductory curriculum on data science will be given. In the exploration session, students will be led through data analysis exercises using the statistical analysis language R. These exercises are designed to use open data, such as NYC open data that contain interesting information about neighborhoods of New York City. No prior programming experience is required.

Columbia University Science Honors Program.