#
SCIENCE HONORS PROGRAM

COURSE DESCRIPTIONS

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.