IAP 2022 Classes
For-credit subjects:
Check out the course catalog at http://student.mit.edu/catalog/m18a.html. You can use the Subject Search functionality to limit the search to IAP listings or find Math's IAP offerings here: http://student.mit.edu/catalog/search.cgi?search=18&when=J. Our main offerings in Mathematics are:
18.02A Calculus
- Prof John Bush and staff
- Jan. 3 - 28
- MTWRF12
- TR10-11.30 or TR2-3.30 +final
12 units (only 6 will count toward IAP credit limit)
This is the second half of 18.02A and can be taken only by students who took the first half in the fall term; it covers the remaining material in 18.02.
18.031 System Functions and the Laplace Transform
- Dr Keaton Burns
- Jan. 3-21
- MWF 10-12
(with an extra meeting Tues Jan 18, in place of the MLK Day holiday on the 17th) - This class will be conducted in hybrid mode, with students encouraged to attend in person in room 2-131, but a zoom option will be made available.
3 units (P/D/F graded)
Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering.
18.095 Mathematics Lecture Series
- MWF1-2.30
- R10.30-12 or R1-2.30
- This class will meet in person on campus. Lectures will be held in 2-190, and many should also be recorded. Recitations will meet in 2-132.
6 units (P/D/F graded)
Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session.
Lecture Schedule
| Monday, January 3 | Bill Minicozzi | The Liouville theorem and harmonic functions of polynomial growth | The classical Liouville theorem, named after Liouville but first proved by Cauchy in 1844, states that a harmonic function that is positive on all of R^2 must be constant. This theorem is absolutely fundamental and plays a role in many fields. It can even be used to give a short proof of the fundamental theorem of algebra. I will discuss how this result depends on the geometry of Euclidean space and explain some far-reaching generalizations to different geometries. |
| Wednesday, January 5 | Haynes Miller | Knots and Numbers | Algebraic topology attempts to extract discrete invariants from flexible geometric configurations. One of the most easily visualized type of topological object is a knot -- a closed loop in Euclidean space. Appearance of simplicity notwithstanding, knot theory has been an active area of study for the past 150 years, and especially in the past three decades. In this lecture we'll study fragments of knots known as tangles, and discover a rational number that serves as a complete invariant for a subclass of tangles called rational tangles. The verification will be by square-dance. |
| Friday, January 7 | Wei Zhang | How small can an algebraic integer be? | One can measure the complexity of an integer by the number of digits (or, the number of bits needed to store it in a computer). This leads to a fundamental concept in Diophantine geometry: the height of an integer, or more generally, the height of an algebraic integer (an algebraic integer is by definition a solution to a monic polynomial with integer coefficients). We will address a natural question, so-called “Lehmer's problem”: how small can the height of an algebraic integer be? In particular I hope to present a recent advance, due to Vesselin Dimitrov, towards a variant of Lehmer's problem (replacing “height” by “house” of an algebraic integer), using an interesting criterion of algebraicity of a power series. If time permits, I'll try to indicate some related problems in modern Diophantine Geometry and Algebraic Dynamics. |
| Monday, January 10 | Gigliola Staffilani | The study of wave interactions: where beautiful mathematical ideas come together | Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic interactions gave also the tools to prove a famous conjecture in number theory, or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples. |
| Wednesday, January 12 | Scott Sheffield | What is a random surface? | These constructions have deep roots in mathematics, drawing from classical graph theory (Tutte, Mullin), complex analysis (Gauss, Liouville, Riemann, Loewner), representation theory (Lie, Virasoro, Verma, Kac) and many areas of physics (string theory, Coulomb gas theory, quantum field theory, statistical mechanics, discrete quantum gravity). We present here an informal, colloquium-level overview of the subject. We aim to answer, as cleanly as possible, the fundamental question. What is a random surface? |
| Friday, January 14 | Jorn Dunkel | Minimal mathematical models of living matter | Recent advances in the live-imaging of multicellular systems pose a wide range of interesting mathematical problems, from the compression of video microscopy data to the modeling of gene expression, tissue dynamics and growth during embryonic development. After a brief review of recent experiments, we will introduce and analyze minimal ODE, SDE and PDE models to describe individual and collective cell behaviors. |
| Wednesday, January 19 | David Roe | Factoring Huge Integers | You learned many years ago that any integer N can be factored uniquely into primes. But the algorithm taught in elementary school -- iterate through primes and check whether N is divisible by each one -- quickly becomes impractical when N gets large. Computational number theorists have devised faster methods over the last several decades that make it possible to factor larger integers on a computer, but the problem is still very difficult: the 260 digit RSA-challenge factorization has stood for 30 years. I will give a broad overview of the methods in use today, together with a more detailed description of three: the Miller-Rabin primality test, the quadratic sieve factoring algorithm and the elliptic curve factorization method. |
| Friday, January 21 | Lisa Sauermann | The Odd-Town Theorem | We will discuss the so-called "Odd-Town Theorem", a theorem in extremal combinatorics (or, more specifically, in extremal set theory). Perhaps surprisingly, the proof of this combinatorics theorem relies on linear algebra over the finite field F_2. We will introduce F_2 in the lecture, and discuss the relevant concepts from linear algebra. Using these linear algebra concepts, we will then prove the "Odd-Town Theorem". |
| Monday, January 24 | Valeri Frumkin | Fluidic shaping of optical components | Fabrication of optical components, such as lenses and mirrors, has not changed considerably in the past 300 years, and it relies on mechanical processing such as grinding, machining, and polishing. These fabrication processes are complex and require specialized equipment that prohibits rapid prototyping of optics, and puts a very high price tag on large lenses and freeform designs. In this talk I will present a novel approach that leverages the basic physics of interfacial phenomena for rapidly fabricating a variety of lenses and freeform optical components without the need for any mechanical processing. We will see how such components can be obtained in liquid form, by minimizing the free energy functional of the system, allowing to design various freeform optical topographies. Lastly, I will discuss our collaboration with NASA on the use of this technology of in-space fabrication of optics and for the creation of large space telescopes that overcomes launch constraints. |
| Wednesday, January 26 | John Bush | Surface tension | Surface tension is a property of fluid interfaces that leads to myriad subtle and striking effects in nature and technology. We describe a number of surface-tension-dominated systems and how to rationalize their behavior via mathematical modeling. Particular attention is given to the role of surface tension in biological systems and in hydrodynamic quantum analogs. |
18.S096 Special Subject in Mathematics: Matrix Calculus for Machine Learning and Beyond
- Profs Alan Edelman and Steven Johnson
- Jan 10-28
- MWF 11am-1pm
- This class will be conducted entirely virtually.
3 units
We all know that calculus courses such as 18.01 and 18.02 are univariate and vector calculus, respectively. Modern applications such as machine learning require the next big step, matrix calculus.
This class covers a coherent approach to matrix calculus showing techniques that allow you to think of a matrix holistically (not just as an array of scalars), compute derivatives of important matrix factorizations, and really understand forward and reverse modes of differentiation. We will discuss adjoint methods, custom Jacobian matrix vector products, and how modern automatic differentiation is more computer science than mathematics in that it is neither symbolic nor finite differences.
Prereq: Linear Algebra such as 18.06 and multivariate calculus such as 18.02
18.S097 Special Subject in Mathematics: Introduction to Metric Spaces
- Paige Dote, paigeb@mit.edu, Faculty Advisor: Prof Larry Guth
- Jan 3-21
- TR 1-2:30pm
- This class will meet in person on campus in room 2-131. We hope to record most meetings for those who must occasionally miss a class.
3 units (P/D/F-graded)
Covers metrics, open and closed sets, functional spaces, continuous functions (in the topological sense), completeness and compactness. Covering pp. 229-266 in Lebl’s Basic Analysis I: Introduction to Real Analysis, vol. 1 (available as a free PDF download at https://www.jirka.org/ra/).
Prerequisites/Audience: 18.100A/P is the recommended prerequisite for this class. (18.100B/Q will have covered the material in this class.) Intended to bridge the gap between 18.100A and 18.100B for students with a basic understanding of material covered in 18.100A, ideally making further classes such as 18.101, 18.102, 18.103, 18.901 more accessible.
Non-credit activities and classes:
Math Lecture Series (non-credit version)
The same ten lectures listed above for 18.095 are also open to the public and you may attend as many or as few as you wish. Check back often to see new postings, including lecturers, titles and abstracts.
(Students wishing to receive course credit must register for 18.095, attend all ten lectures plus weekly recitations, and complete problem sets.)
Introduction to LaTeX
Enrollment: Unlimited, but sign-up required to have Canvas access (sign up at https://forms.gle/PRF4BhrmqWNmA95G9)
This will be an introduction to LaTeX, the programming language used for writing math (and other equation-dense) papers. Students learn basic codes, packages, and formatting they will need for writing papers in Course 18 CI-Ms, including 18.100P/Q, 18.200, 18.821. This self-paced, asynchronous workshop is geared for Course 18 majors, but might also be of interest to other equation-dense fields like Course 8 or Course 6. Asynchronous session uploads each Tuesday: January 4, 11, 18, 25.
Contact: Malcah Effron, (617) 324-2302, meffron@MIT.EDU
Integration Bee
Like solving integrals? Like watching other people solve integrals? Come try your problem solving skills against some of the best at the 41st Annual MIT Integration Bee! Compete for the chance to win prizes and the prestigious title of Grand Integrator! To qualify for the bee, you must be a student and come to 4-163 for a 20-minute written test any time between 4pm and 6pm on Tuesday, January 18th. The top sixteen will be invited to the main event, which will be held on Thursday, January 20th at 6:00pm in 32-123. Come try out, and invite your friends to watch the finale! Whether you pass the qualifier or not, come watch your classmates show off their mad integrating skills at the main event will be held on Thursday, January 20th at 6:00pm in 32-123. Bring your friends!
Qualifying Rounds (Open to participants only)
- Tuesday, January 18th
- 4:00pm-6:00pm
- 4-163
Finals (Open to the public)
- Thursday, January 20th
- 6:00pm-10:00pm
- Room 32-123
Music Recital
- Thursday, January 27th
- 3:00pm-5:00pm
- Killian Hall
The MIT math department music recital will be returning once again this IAP, taking place in Killian Hall on 1/27/2022 from 3pm-5pm. The recital is a yearly tradition where we gather to listen to music performed by members of the math department. All styles of music are encouraged. Classical (Indian and western), jazz, video game, Latin-American, and Scandinavian folk music, as well as original compositions have all previously been featured.
Those interested in performing in the recital should contact Dominic Skinner at dskinner@mit.edu