## Undergraduate Mathematics Research Seminar

### Information

**When:**Tuesdays, 12:30PM

**Where:**301 Vincent Hall

**What is the Undergraduate Mathematics Research Seminar?**

A seminar at the University of Minnesota for undergradutes doing mathematics research, undergradutes interested in doing mathematics research, and undergraduates just interested in mathematics! If you'd like to be on the mailing list or may want to give a talk, contact Harini Chandramouli (chand409@umn.edu) for details.

**Other Groups To Check Out**

Association for Women in Math (WiM)

SIAM Student Chapter

Math Club

**Math Department Opportunities**

Undergraduate Research: Here is a list of graduate students interested in supervising undergraduate research projects. They have voluntarily given their names and information so feel free to contact them. I recommend making appointments to meet with at least 3 graduate students and, after meeting, choosing the one that you think would be the best fit. You do not need to prepare anything for the initial meeting, instead just have a brief conversation and ask them to tell you about what they do.

Directed Reading Program: A Directed Reading project works as follows: students get paired with graduate student mentors with whom they will choose a mathematical topic of mutual interest and decide upon some focused mathematical readings. Over the course of the semester, the mentor and mentee will regularly meet (~one hour per week) to unpack and explore the readings, to clarify confusions, and to master the notoriously difficult but invaluable skill of effective mathematical discourse. The project culminates in a short presentation before an audience of the other DRP participants. Learn more here.

### Schedule

Date | Speaker | Title & Abstract |
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CANCELLED.Due to the extreme weather conditions, seminar on this day will be cancelled and postponed to next week. |
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Numerical Methods: SVD with its Applications Based on the basic ideas about matrix decomposition including eigen value decomposition (Schur Decomposition) and QR Decomposition, we focus on singular value decomposition (SVD) this time. The classical method to compute SVD turns out to not be stable according to our analysis. Instead, we use a numerical method involving Krylov subspaces and Arnoldi iteration to compute SVD. Then we will introduce some common applications with SVD including pseudo-inverse, least squaresp roblem, and data dimension reduced strategy by principle component analysis (PCA). |
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Zero-Free Regions of the Riemann Zeta FunctionIn 1859, Riemann showed that the zeta function can be seen as a complex valued function that is well defined almost everywhere and has a surprisingly close connection to prime numbers. Basically, knowing how many primes there are up to a certain number comes down to understanding where the zeros of the zeta function are located. The problem is that no one has been able to prove where they are. The famous Riemann Hypothesis says that they are located in the complex plane line $\text{Re}(s) = \frac{1}{2}$. Even though this hasn't been proven or disproven, some mathematicians have been able to find regions where zeta is not zero, giving us some "reassurance" on the hypothesis. This talk is an introduction to the Riemann Hypothesis and the methods used to find zero-free regions of the zeta function. |
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QuadratureI will prove the area of a Lune using only "compass and straight edge" as the Greeks described it. More specifically, from the book "Journey Through Genius," I will be summarizing the Great Theorem: The Quadrature of a Lune. |
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Counting Perfect Matchings of Snake GraphsSnake Graphs are used to determine arc lengths in polygon triangulations by taking the sum of each weighted perfect matching obtained from the snake graph. We will have a sagemath demonstration to better understand this relationship. However, snake graphs can also be viewed as abstract objects with their own unique rules and properties. All snake graphs can be decomposed into segments that consist of the two basic types of snake graphs, straight and zig-zag. We will show that an $n$-tile straight snake graph will have $Fn+1$ perfect matchings. We then will prove that a zig-zag snake graph consisting of $n$-tiles will have $(n+1)$ perfect matchings. |
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Objects Derived from Triangulated Polygons$T$-paths between two vertices of a triangulated polygon can be used to give a bijection between the terms in the expression of the length of the connecting arc and the set of admissible sequences between these vertices. These $T$-paths can also be used to construct the perfect matchings of the related snake graph. The way we derive a snake graph from triangulation is time-consuming. We will introduce a faster way to draw snake graphs by observing the shape of any arbitrary arc and its intersection arcs from the triangulation. |
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An Optimal Policy for an Assignment of Police Patrols in the UMPD PrecinctsHow should University of Minnesota Police Department (UMPD) allocate their police officers and give assignments? What kind of assignments may result in less crime in the entire region? Public safety around campus is one of the prominent issues and concerns for students, parents, and faculties around the world. In this article, we studied the best ways to assign police officers to precincts of University of Minnesota Police Department (UMPD) to minimize the total number of crimes. We obtained the crime data for the past two years for regions under the UMPD. Distributions of crimes were analyzed, and the crime levels of those regions were defined. With the assumption that an increased presence of police officers leads to a decrease in the number of crimes in the region, we developed a model applying the concepts of Markov Chain Process and Dynamic Programming, where the states were defined as the crime levels and the actions were defined as the assignments of the police officers to regions. We found one best allocation of officers for all the states to minimize the expected total number of crimes of the following shifts. In addition, our model was able to consider both the short-term effects and the long-term effects of the assignments. Most of the results were within expectation, but there were some interesting results for a few states which we specifically addressed in this paper. Our model can be applied to other police precinct areas to find optimal policies for police assignments. |