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MATH 4993-100 (12849) and 4994-100 (14050), Spring 2019

Undergraduate Mathematics Seminar

Syllabus

MATH 4993-100 (12849)

MATH 4994 (14050)

Title:

Undergraduate Mathematics Seminar I

Undergraduate Mathematics Seminar II

Credits:

1

2

Requisites:

MATH 3300 Calculus III and (MATH 3200 Applied Linear
Algebra or MATH 3210 Linear Algebra) and (MATH 3050
Discrete Math or CS 3000) and (Jr or Sr). (You may see 6
hours MATH 4200-4799 also listed as a prerequisite, but
that is no longer true.)

MATH 4993

Tier III:

Taking 4993 and then 4994 is a Tier III equivalent.

Major requirements:

Taking 4993 and then 4994 counts as one 4xxx course
toward Mathematics major requirements.

Catalog Description:

The student participates in a weekly seminar on topics in
mathematics that are beyond the material covered in our
regular courses. During the first semester the student will
develop a proposal for a topic of interest to be presented in
the second semester.

The study topic will be presented in
the weekly public seminar and a final written report
will be submitted to the instructor.

Desired Learning Outcomes:

The student will become familiar with various advanced
topics in mathematics or applications of mathematics.

The student will develop an individual plan for further
study and a seminar in a chosen topic.

The student will develop proficiency at presenting
a mathematical topic in a seminar format.

The student will develop proficiency at presenting
a mathematical topic in an expository article format.

Description:

Undergraduate students will participate and give presentations in a
weekly seminar. Topics will vary depending on the interests of the
students. All of the topics covered will be advanced material not
covered in regularly offered courses. Participation will

build upon and thus reinforce and contextualize more
basic material,

broaden the students mathematical horizons,

spark interest in further study in specific areas of
mathematics and allied fields.

In addition to mathematical goals, the activities are designed to
improve oral and written communication skills.

Instructor:

Martin J. Mohlenkamp,
mohlenka@ohio.edu,
(740)593-1259, 315-B Morton Hall.
Office hours:
Monday, Wednesday, and Friday 2:00-2:55pm,
or by appointment.
Please communicate with me using your @ohio.edu email address. I will not send any private information to other addresses.

Thursdays 5:15-6:10pm. We meet in 322 Morton Hall when there is a speaker and 314 Morton Hall (the computer lab) when there is not.

(a second hour, arranged)

Attendance:

Attendance is part of your grade. Up to
3 missed classes may be made up by attending Mathematics
department research seminars or colloquia and filling out a Presentation Rating form.
Basic participation, such as providing feedback on the work of
other students, is considered part of attending.

Main Products:

You will develop a topic on which you can give an hour
public seminar in part II.
See the topic guide and
topic development guide/template
(pdf, tex),
which uses a graphic OHIOCLR.pdf.

You will develop and deliver a public seminar talk on
your topic and write a report on the topic. See the
slides template
(pdf, tex),
which uses a graphic OHIOCLR.pdf
and the report guide/template (pdf, tex).

Final Exam:

Our scheduled final exam is Tuesday April 30 at
4:40pm. There will not be an exam, but your final
topic development or report is due at that time.

Grade:

Your grade is based on attendance 30%,
first draft of the topic proposal 5%,
final version of the topic proposal 15%,
first draft of the topic contents 15%,
and final version of the topic contents 35%.

Your grade is based on attendance 15%,
oral presentation 35%,
first draft of the report 15%,
and final version of the report 35%.

An average of 90% guarantees
you at least an A-, 80% a B-, 70% a C-, and 60% a D-.

Academic Dishonesty:

Cooperation and the use of outsides sources
is encouraged, but you must acknowledge in writing what
help you received and from whom or where. Use of sources
without proper citation is considered plagerism. Serious
or second violations will result in failure in the class
and be reported to
the Office
of Community Standards and Student Responsibility,
which may impose additional sanctions. You may appeal any
sanctions through the grade appeal process.

Special Needs:

If you have specific physical, psychiatric, or learning
disabilities and require accommodations, please let me
know as soon as possible so that your learning needs may
be appropriately met. You should also register
with Student
Accessibility Services to obtain written documentation
and to learn about the resources they have available.

Responsible Employee Reporting Obligation:

If I learn of any instances of sexual
harassment, sexual violence, and/or other forms of
prohibited discrimination, I am required to report
them. If you wish to share such information in confidence,
then use one of the confidential resources listed by
the Office
of Equity and Civil Rights Compliance.

Writing Environment:

We will use the cloud computing environment CoCalc for writing proposals, talks, and reports, using \(\LaTeX\).
Sign up for a free account, using your @ohio.edu email address.

Title: Guide to Mathematics Job and Internship Searches

Abstract: Want an internship or a job? It will
never be easy. This talk will present job search
strategies and resources for students in the math
department from the beginning stages of exploration
through the internship and job search process. The
guide and resources will help you find, apply to,
interview for, and get the job that is right for
you.

Individual consultations on slides

3

Jan 31

Initial discussion of topic ideas. Practice with \(\LaTeX\).

Practice talk using slides

4

Feb 7

4993 draft 0 of Topic Proposal due; you will get feedback on it but no grade.
Guest Speaker:
Christopher Renner.

Title: Numerical Methods for Solving Integral Equations

Abstract: Integral equations arise in many
applications in engineering, physics, and computer
science. Most integral equations cannot be solved
analytically, so the solution must be approximated
using numerical methods. An introduction to the types
of integral equations as well as multiple numerical
techniques will be given. Numerical methods are
derived from all subjects in math; linear algebra,
Fourier analysis, topology, and statistics to name a
few. One method is not optimal for all types of
integral equations. New methods are being developed
and compared to existing techniques for specific
integral equations in order to improve their use in
applications.

Binary operations with additional properties have
been widely studied: one has semigroups, monoids,
groups, etc. In its most general setting, when one has
a binary operation over which no additional properties
are assumed, the structure is said to be a magma. We
use the notation M(S) (the magma of S) to denote the
set of all binary operations on the set S (all magmas
on the set S.)

Many interesting algebraic structures deal with a
pair of operations, one of which distributes over
the other (i.e. an equation similar to a(b+c) = ab
+ ac holds). Normally, the operations involved
satisfy nice properties of their own in addition to
the distributivity that relates them. For instance,
a ring involves an additive abelian group structure
and a monoid multiplicative structure where
multiplication distributes over addition. Likewise,
a triple (S,o ,*) where (S,o) and
(S,*) are magmas is said to be a left (
resp. right or two-sided) distributive magma if
* left ( resp. right or two-sided) distributes
over o. In the absence of other assumptions
about * and o, many interesting
questions arise.

How long of a sequence of operations can you have such that o_1 distributes over o_2, o_2 distributes over o_3, etc.?

Can an operation distribute over itself?

Can two operations distribute over one another?

These and many other interesting questions will be explored in this presentation, which is part of an ongoing project with Sergio Lopez-Permouth and Asiyeh Rafieipour.

Reference: S. Lopez-Permouth and L. H. Rowen, Distributive hierarchies of binary operations. Advances in rings and modules, 225-242, Contemp. Math., 715, Amer. Math. Soc., Providence, RI, 2018.

Practice talk using chalk

7

Feb 28

Further discussion and work on topics and slides.

Practice talk in final form

8

Mar 7

4993 final version of Topic Proposal due.
Guest Speaker: Michael John Burton

Title:

Information, Entropy, and Target Selection

Abstract:

Political marketing professionals mobilize supporters to cast ballots and they persuade voters to cast ballots the 'right' way. To do so, professionals distinguish high-value targets from the rest of the population - essentially, partitioning the electorate into a target array and a rejection array. Target-identification models have been hand-crafted for many decades. In the Digital Age, this task is increasingly given to machine-learning classifiers, a type of algorithm that often relies on Claude Shannon's mathematical theory of information entropy. The seminar will focus on uses of Shannon's theory in machine-driven target identification. Our discussion will show how elementary combinatorics can reveal direct connections among algorithmic procedures, political marketing, and statistical thermodynamics.

Practice talk in final form

Spring Break

9

Mar 21

4993 draft 0 of Topic Content due; you will get feedback on it but no grade.
4994 draft 0 of Report due; you will get feedback on it but no grade.
Student Speaker: Haihai Song

Title:

NBA Heat Maps and the Heat Equation

Abstract:

A heat map gives a visual way to understand basketball data. The data is analyzed reasonably and effectively so that conclusions can be drawn. The corresponding game strategy and training arrangement can then be formulated and the most valuable player of the season can be judged based on the statistics and analysis of the data. In this presentation I discuss the relationship between a heat map and the heat equation. I show how to use mathematics and the heat equation to build a heat map. I start by showing heat diffusion in one dimension.

Have you ever wanted to see sound? Through the help of many different graphs, systems, and formulas we can! Utilizing ratios, we can find a relation between frequencies in order to find the tones that appeal to us via western music. We can map out sound to show how it can be used to find amazing chords and tones, as well as take a sound and be able to break it down into its different notes! I'd just love to be able to see how music just flows out of my saxophone, oh wait, I can!

Individual consultations on reports

11

April 4

Presentation debriefings.
Work on topic development and reports.

Individual consultations on reports

12

April 11

4993 draft 1 of Topic Content due.
4994 draft 1 of Report due.
Student Speaker: Jingmin Gao

Title:

Prisoner's Dilemma and the Nash Equilibrium

Abstract:

In the prisoner's dilemma, two criminals are captured by the police and interrogated separately. Each has to decide whether to keep silent or squeal. Game theory tells us that each will try to minimize their own jail time by squealing. This is an example of a non-cooperative game where the Nash equilibrium ends up worse for both participants. In the Cournot duopoly model, two companies have to decide how much of a product to produce. When each maximizes their own profit, the amount they produce is given by a Nash equilibrium and the customers will pay less than the monopoly price. These two representative examples help explain common choices and behaviors in our life.

The Ocean Cleanup Project: Vertical Distribution of Plastics

Abstract:

There are over five TRILLION pieces of plastics in the world's oceans. There is a team, The Ocean Cleanup Project, set to do something about this. To help build the most efficient system possible, one of the matters the team studied was the vertical distribution of plastics in the ocean. Studying this brings in calculus, mathematical modeling, and many useful statistics. Every study that is done about the plastics in the ocean gives a little more knowledge and gets us that much closer to having plastic free oceans.

Individual consultations on reports

14

April 25

Presentation debriefings.
Work on topic development and reports.

Individual consultations on reports

15

Tues Apr 30

4993 final version of Topic Content due 4:40pm.
4994 final version of Report due 4:40pm.