Undergraduate Seminars II

Spring 2023

https://www.math.columbia.edu/~bayer/S23/Seminar

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Spring 2023 Mathematics |
MATH UN3952

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507 Mathematics

hindy.drillick@columbia.edu

Hindy Drillick

In probability theory, a Markov chain is a sequence of random states with the condition that the state you go to next is only influenced by your current state, and not by any of your previous states. This property is also known as memorylessness since the process does not retain memory of its past states. A typical example of a Markov chain is a random walk where at each time you take either one step to the right or one step to the left with a 50/50 probability. This is sometimes known as the drunkard’s walk, as the drunkard in this example does not remember where they are coming from (or going to) and therefore just takes a step in a random direction away from their current position.

In this seminar, we will see examples of many different types of Markov chains, and we will learn how to answer questions such as

How often (if ever) does a Markov Chain return to a specific state?

What is the long-term behavior of a given Markov Chain? Does it tend towards a stationary/equilibrium state?

Markov chains are used heavily in finance, biology, physics, and computer science. Two particular applications that we will look at this semester are

- The PageRank algorithm behind Google’s search engine
- Markov Chain Monte Carlo—a class of algorithms that use Markov Chains as a tool to help computers simulate complicated probability distributions

Prerequisites: Students should be familiar with linear algebra as well as basic notions from probability such as random variables, expectations, and independent events. We will review the necessary probability theory at the beginning of the semester.

Primary Reference Markov Chains by J.R. Norris (see Courseworks for a PDF copy of the textbook)

507 Mathematics

mhaseliu@math.columbia.edu

Matthew Hase-Liu

Fundamental Groups (section page)

See “Advertisement for class” for a video presentation describing this seminar.

507 Mathematics

ccl@math.columbia.edu

Cailan Li

We will learn about the algebraic properties of symmetric polynomials and the combinatorics that arises from their study. In particular, we will go over Macdonald polynomials, the final boss of symmetric functions, in great detail.

Symmetric Functions (section page)

507 Mathematics

mmiller@math.columbia.edu

Michael Miller Eismeier

We’ll study the area of dynamical systems. A dynamical system is some collection of objects with a rule for how they evolve over time; if y(t) measures population of a bacteria on a Petri dish, the classic example is the fact that d/dt y(t) = c y(t): the population grows proportionally to the size of the current population, and this leads to the exponential growth of that population y(t) = e^{ct+d}. Many processes from many different fields of science can be studied in the language of dynamical systems, including biology, electrical engineering, and a great deal of physics. The notion of chaos stems from the existence of dynamical systems which behave very differently even after changing the starting position of the system by a very small amount. There is also plenty of purely mathematical interest in understanding the geometric behavior of dynamical systems.

Our goal will be to cover the mathematical foundations of the subject, and work through some of its many applications, as well as get a sense for chaotic dynamical systems and where they come from. Much of dynamical systems is founded on ordinary differential equations (“ODE”), so this seminar is recommended for students who have some experience or interest in differential equations.

507 Mathematics

History of Mathematics (section page)

7:30 pm to 8:45 pm Eastern Time

This is a message regarding Math UN3952 - Undergraduate Seminars II. There will be an organizational meeting this Thursday evening:

Organizational Meeting
Thursday, January 19, 2023
7:30 pm to 8:45 pm Eastern Time

508 Mathematics (Lounge)

We will have an organizational meeting on Thursday, January 19 from 7:30 pm to 8:45 pm. We ask that anyone interested in taking an undergraduate seminar attend this meeting. At this meeting, we will present the seminar topics and discuss how seminars work.

There will be a choice of seminars. Each seminar will meet once a week for two hours.

After this meeting, a survey will be distributed to all registered students, asking for your seminar preferences and your available times. We ask that you complete the survey by Noon Friday, as we need to fix times and reserve rooms.

If you cannot attend this organizational meeting or complete the survey, please contact me after the meeting for a seminar assignment.

Please see the course web page for further information:

https://www.math.columbia.edu/~bayer/S23/Seminar

Thanks,

Dave Bayer