### UNDERGRADUATE COLLOQUIUM

The Mathematics Undergraduate Colloquium is held each Wednesday from 12:55 - 1:45 during the regular academic year in LCB 215. Each week a different speaker will present information on a specific subject area in mathematics. Anyone can come by to listen, socialize, get to know members of the department, and hear some interesting information on the many areas of mathematics.

Fall 2022 Schedule

**Introduction to the course**

**Title: Self-similar fractals and dimension**

*Abstract*: You may have heard that the line has dimension 1 and the Cartesian plane has dimension 2. Can things have dimension 1/2? How about other dimensions? This talk will introduce a couple of "self-similar fractals" and a notion of dimension for them which does not have to be a whole number.

**Title: The gambler's ruin problem and an interesting asymptotic extension**

*Abstract*: In 1656, Pascal posited to Fermat something akin to the following: "Players A and
B start with '**a**' and '**b**' points, respectively. Flips of a fair coin determine how points are transferred
between them. The game ends when one player acquires all **a+b** points (and the other player thus has none). Given **a **and **b**, how likely is Player A to win?" In this talk I will answer Pascal's question by
visualizing the game as a 1D random walk with absorbing boundary conditions. I will
then discuss an extension of the problem involving the limit of many random walkers,
which has numerous applications to physics, chemistry, and biology.

**STOCHASTIC ADAPTIVE CHEMOTHERAPY CONTROL OF COMPETITIVE RELEASE IN TUMORS**

*Abstract*: Adaptive chemotherapy seeks to manage chemoresistance by delaying the competitive
release of a resistant sub-population, and to manage cancer by maintaining a tolerable
tumor size rather than seeking a cure. Models typically follow interactions between
infinite populations of sensitive (S) and resistant (R) cell types to derive a chemotherapy
dosing strategy C(t) that maintains the balance of the competing sub-populations.
Our models generalize to include healthy (H) cells, and finite population sizes. With
finite population size, stochastic fluctuations lead to escape of resistant cell populations
that are predicted to be controlled in the deterministic case. We test adaptive schedules
from the deterministic models on a finite-cell (N = 10,000 – 50,000) stochastic frequency-dependent
Moran process model. We quantify the stochastic fluctuations and variance (using principal
component coordinates) associated with the evolutionary cycle for multiple rounds of
adaptive chemotherapy, and show that the accumulated stochastic error over multiple
rounds follows power-law scaling. This accumulates variability and can lead to stochastic
escape which occurs more quickly with a smaller total number of cells. Moreover, we
compare these adaptive schedules to standard approaches, such as low-dose metronomic
(LDM) and maximum tolerated dose (MTD) schedules, finding that adaptive therapy provides
more durable control than MTD even when we include the effects of finite population
size. Although low-dimensional, this simplified model elucidates how well applying
adaptive chemotherapy schedules for multiple rounds performs in a stochastic environment.
Increasing stochastic error over rounds can erode the effectiveness of adaptive therapy.

**Applying for the NSF GRFP**

*Abstract*: This is a panel about applying for the National Science Foundation (NSF) Graduate
Research Fellowship Program (GRFP). This is a program that graduating seniors as
well as first and second year graduate students, may apply for. It provides funding
for 3 years of financial support to US Citizens, US Nationals and permanent residents
who intend or are pursuing a Masters or Ph.D. in various fields including Mathematics.
This panel will provide useful information for anyone who is thinking about graduate
school in Mathematics (whether or not they are eligible to apply for the NSF GRFP
this year).

**Title: What should you do if you miss a dose of medication?**

*Abstract*: Medication adherence is a major problem for patients with chronic diseases that
require long term pharmacotherapy. What should patients do if they miss a dose of
medication? How can physicians design drug regimens to mitigate nonadherence? Why
are some medications effective despite lapses in adherence? In this talk, I will describe
recent efforts to address these questions using mathematical modeling.

**Gradient Dynamics in a Valley: Super Slow Motion of Interfaces**

*Abstract:*Paul Fife and Charles Conley studied a dynamical system modeling the spread of a genetic
trait through a population over many generations. In doing so, they developed an abstract theorem, using a topological
tool, with applications going far beyond the biological model. Consider a curve (manifold) M of stationary solutions to a dynamical
system and suppose that each point of M is exponentially stable in directions perpendicular to M. They showed that small
changes in the dynamical system retained some semblance of M, namely, traveling waves in the perturbed dynamical systems. I will
have more to say about invariant manifolds in Thursday’s talk. Now for something completely different. Not really, but I can’t resist
Monty Python lines sometimes (ask your parents, sorry, grandparents about MP). Diverging slightly from the above, in this talk, I
will go on to talk about places like Alta, where there are mountains and valleys and how something like gravity tended to propel me into the
valleys from where I would walk out by going along the valley to what I hoped would be a route to the chairlift, so I could do
it again. The dynamics I described is a gradient dynamical system, where I would go down in the steepest way possible until getting
into a valley, where motion became rather slow. I will describe a class of dynamical systems, being the gradient of an “energy” J,
that has a peculiar property for points near M, a set of points forming a curve (manifold): If the distance of an initial point u0 from M is
less than some number g, and if its energy J(u0) is low, then that distance is less than b, for some number less than g! The solution then
stays within distance b from M. One such dynamical system is a vector version of the system studied by Fife and Conley and also having
small diffusion. I’ll indicate how the statement on boldface applies and produces incredibly slow motion of interfaces in the spatial
profile of a solution. This is joint work with Giorgio Fusco (U. L’Aquila) and Georgia Karali (U. Crete).

**Panel: Population Health Sciences PhD, Emphasis in Biostatistics**

We train the next generation of biostatisticians with the methodological and collaborative skills needed to design and analyze studies aimed at addressing important population health problems. Our graduates are well positioned for meaningful careers in academia, industry, or government.

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**Panel: Healthcare Data Science 2023 Internship Program**

Do you enjoy solving real life problems with math and programming? Does innovation and automation of AI and Machine Learning in healthcare excite you? Are you motivated by helping people live the healthiest lives possible?

Learn more about internship and full-time employment opportunities with Intermountain Healthcare in Date Science.

**Cryptography, Freedom, DemocracyHow Basic Science Affects Everyone**

To most people, research in basic science seems irrelevant, and

consequently, citizens, legislators, government funding agencies, and corporations are disinclined to support it. Nevertheless, basic science can have deep impacts on our lives. This talk examines two developments in basic science in the Twentieth Century. The first of them, Albert Einstein's work in 1905, changed the field of physics, and the course of history. The second, the invention of public-key cryptography in 1975, has important consequences for secure communications. Many of mankind's discoveries have potential for both good and bad. The talk concludes with a discussion of some recent uses of technology that pose the very serious risk of our complete loss of privacy, freedom, and democracy.**Title: Power Grid Operation and Planning: Convergence of Engineering, Economics, and
Machine Learning**

*Abstract: *Power grid fuels our everyday life and has successfully contributed to the economic
surplus of all countries across the globe. Today, the power grid is undergoing a massive
change with the integration of renewable energy resources, and faces a growing number
of natural disasters and cyber attacks that threaten the reliable delivery of power
to the communities. This talk will discuss how new mathematical optimization and machine
learning models are changing the way power grids are operated to adapt to the challenges
the grid faces.

**Topic: Yoneda Lemma**

*Abstract: *I'll give a quick introduction to Category theory and state Yoneda Lemma. Then If
time permits, I'll give a couple of quick application.

**Title: Area without Numbers**

** ***Abstract: *Ancient Greek civilization didn't have the greatest set of numbers. Therefore the
way they approached geometry was a little different from what we're used to. This
talk will explore the notion of quadrature -- producing a square with an area equal
to that of a given figure.

**Title: Area without Numbers**

** ***Abstract: *Ancient Greek civilization didn't have the greatest set of numbers. Therefore the
way they approached geometry was a little different from what we're used to. This
talk will explore the notion of quadrature -- producing a square with an area equal
to that of a given figure.

**Title: What is Representation Theory? **

** ***Abstract: *Representation theory is a fundamental field of mathematics and has been shown to
be useful in nearly every other area. The main idea behind this is to transform a
hard problem about symmetry to a simpler problem about linear algebra. In this talk
we will look at why such a thing is possible and how understanding the representation
theory of a group can tell us about its structure in the case of finite groups.

## Math 3000 (Receive Credit for Attending)

The Undergraduate Colloquium is open to anyone to attend; however, if students would
like to receive credit, you may register for **Math 3000**.

This is a 1 credit hour CR/NC course. To receive credit:

- You may not miss more than 2 of the colloquia
- You will need to write a short paper on one of the topics presented during the semester.

Course Coordinators

Kevin Wortman | Lisa Penfold |

Course Instructor | Administrative Coordinator |

ugrad_services@math.utah.edu |