#
SCIENCE HONORS PROGRAM

COURSE DESCRIPTIONS

Fall 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.

**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. While the course is usually highly interactive, in light
of the pandemic, in-person activities will be replaced with virtual
experiences including a visit to see examples of fabrication
facilities as well as 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 a
virtually guided tour and preparation of single atom-thick materials
as well as write basic programs to run on IBM's quantum circuit
interface. The course will conclude with introductory lectures on
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, virtual 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. Recordings of experiments and follow-up discussions
will introduce common techniques employed in organic chemistry and
will include: extraction, recrystallization, thin layer and column
chromatography, reflux, and distillation.

**BIOCHEMISTRY:** Living organisms are complex and highly organized. Yet
the biological basis of life boils down to complex interactions
between molecules and biomolecules. This course will provide a
foundation for understanding the chemical basis of biological
processes. We 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. The course will also cover fundamental metabolic pathways
involving the breakdown of carbohydrates, lipids and fatty acids and
the crucial biological machines that carry out these
processes. Students will be exposed to cutting-edge technologies such
as liquid chromatography, mass spectrometry, and metabolomics used to
profile the metabolome and analyze metabolic fluxes. Students will
also learn how perturbation in molecular processes leads to complex
pathologies, and how protein structure, enzyme kinetics, and metabolic
activity can be leveraged by biochemists to design novel therapeutic
compounds. 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.

**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 cancer and
hemorrhagic fevers. Classes will explore the molecular biology of
viruses, their replication cycles and the unique features that
distinguish them from all other forms of life. 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.

**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.

**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.

**KNOT THEORY:** Take a string, knot it in some complicated way around
itself and tie the ends together. One may ask: What hidden structures
lurk behind these seemingly innocuous one-dimensional objects?
Surprisingly, it turns out that these objects have a lot of hidden
structure, for example: polynomial invariants and their
categorification connect these objects with representation theory;
considering the compliment of the knot gives a three-manifold, and
therefore these objects are connected to algebraic topology; the
polynomial invariants can be viewed as partition functions, and
therefore these objects are connected to statistical mechanics; the
geometric aspects of the compliment of the knot connects these objects
to differential geometry. The beauty of knot theory lies in the fact
that it is an intersection point for many different aspects of
mathematics: representation theory, algebraic topology, probability,
and differential geometry. These seemingly unrelated areas of
mathematics thereby have rich and surprising interconnections with one
another through their common lens of knot theory.

**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. The course will cover real world applications
like PageRank (ranking web pages), Maps, hashing in cryptocurrency
etc.

**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.