On Thursday, January 19th, we’ll be holding our Mathematics Colloquiumfest! This will feature multiple colloquia being presented in parallel sessions both in the morning and the afternoon; see below for the full schedule. For junior and senior math majors, this is a great opportunity for you to get multiple colloquium credits in one day (as many as 6)! There will also be a pizza lunch in the Frank Morgan Math/Stat Library from 11:45am to 1pm, open to anyone attending any of the talks.
Here’s a detailed schedule, including speakers, titles, abstracts, rooms (all in Wachenheim), and times:
9:30a.m. Room 114 Petros Markopoulos
The Axiom of Choice and Equivalent Statements
Mathematician Jerry Bona said: “The Axiom of Choice is obviously true, the well-ordering principle
obviously false, and who can tell about Zorn’s lemma?” but they are all equivalent statements. In this
talk, we will first discuss the statements and some of their consequences and present a proof of their
equivalence.
10:15a.m. Room 017 Will McCormick
A Hemispherical Basis for Efficient Rendering
Reflectance Transformation Imagery (RTI) is a form of texture mapping with the possibility of
interacting to change an image’s light direction. This powerful tool has improved the studies of ancient
writings, inscriptions, and more. This talk discusses the two algorithms developed for RTI:
Hemispherical Harmonics Mapping (HSH) and its predecessor, Polynomial Texture Mapping (PTM).
Throughout this discussion, we will uncover why HSH is often preferred in modern-day practice over
PTM.
10:15a.m. Room 114 Brandt Mandia
Predicting MLB win-loss records by Run Scored vs Runs Allowed
Bill James empirically discovered a good predictor of a team’s winning percentage is RS^2 / (RS^2 +
RA^2), where RS is the average number of runs scored per game, and RA the average number allowed.
We discuss a derivation of this result using techniques from probability and statistics, under the
assumption that runs scored and allowed are independent random variables drawn from a three
parameter Weibull. If time permits we discuss how to estimate these values from seasonal data.
11a.m. Room 017 Alex Cardonick
From Chomp to Chess: Examining Winning Strategies for Weakly and Partially Solved Games
For 1500 years, chess has dominated popular culture, and its complexity has made it the subject of
countless hours of study in attempts to find a winning strategy. However, the game remains only
partially solved. Using the game Chomp as an investigative tool, we will define the class of “weakly
solved” games in order to better understand varying degrees of game solvability. We will then examine
solvable subgames of Chomp to show how games without an explicit solution can be solved under
certain conditions.
11a.m. Room 114 Kevin Ryan
Modeling Epidemics Using an SIR Model with Built-In Notions of Digital and Manual Contact Tracing
In this talk, I will be discussing the Susceptible-Infected-Recovered (SIR) Model which uses differential
equations to make predictions about the infection rates and impacts of the spread of disease. After going
over the basic model, I will be examining how adding mathematical notions of both manual and digital
contact tracing to this model affects the spread of diseases and infectivity based on the paper, “Epidemic
models with digital and manual contact tracing” by Dongni Zhang and Tom Britton. Further, I will also
discuss the ramifications of the findings of this paper in regard to the Covid-19 pandemic.
11:45a.m.-1p.m. Pizza lunch in the Frank Morgan Library
1:15p.m. Room 017 Sarah Shi
Bézout’s Theorem and the Intersections of Two Plane Curves
How many times will two plane curves intersect? How many roots will two polynomials share? Bézout’s
Theorem provides an answer to both of these questions. It tells us that a curve of degree n will intersect
a curve of degree m exactly nm times, with some caveats, in the complex projective plane. Equivalently,
a homogeneous polynomial of degree n will share exactly nm roots, counted with multiplicity, with a
homogeneous polynomial of degree m with which it shares no common factors. We will prove Bézout’s
Theorem using the concept of resultants — the determinant of a matrix created using the coefficients of
two polynomials.
2p.m. Room 116 Jonathan Rogers
Divergent Series
We know what it means for a series to converge, and what it means for it to diverge. But it’s also
possible to extend the definition of converging series in a way to make series that classically diverge,
like 1-1+1-1…, convergent to a value. We will go over a few ways to do so, as well as give criteria on
how we can manipulate series to show that these extended definitions for converging series make logical
sense.
2p.m. Room 017 Christina Zhou
An Acute Case of Cloning: Constructing Self-Similar Polygonal Tilings
Penrose tiling are famous examples of non-local, aperiodic tilings constructed from two primitive
shapes: the “kite” and “dart”. We will consider similar tilings that involve only one primitive shape,
namely some polygon. Beginning with “Golden Bee” tilings, we will generalize the construction of self-
similar polygonal tilings and show how they tile the plane in beautiful ways.
2:45p.m. Room 017 Balint Szollosi
Why Don’t Some Knots Come Loose?
What prevents a knot from becoming untied? Following a brief overview of some basic knot-theoretic
concepts, we introduce the edge number, with which we can begin modeling knots via graphs. This way,
we can identify so-called deadlocks in the knot that prevent it from becoming untied.
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