[abstract]My talk seeks to demonstrate two
similar proof methods which exhibit a clever shift in perspective to show
existence or non-existence of solutions to difficult elementary problems.
I will begin by explaining the probabilistic method, and present a simple
two line proof to demonstrate its usefulness, I will then demonstrate its
application to a problem of showing existence of sum free subsets of finite
sets of integers of a certain size.
Afterwards I will present a proof technique which mirrors the probabilistic
method, where instead of imposing a probability space, we impose a vector
space, and argue via linear dependence. My talk will conclude by presenting
a solution (using this technique) to a geometric problem highlighted by
Hilbert as “from the discussion of which an advancement of science may be
expected.” In his 1900 lecture presenting a collection of open
mathematics problems.
Riley Wheadon: Modelling hormonal regulation of plant root growth.
[abstract]Root growth in plants is caused by
cell division and elongation. But what regulates these processes?
Experimental results have shown that the hormones auxin and brassinosteroid
are responsible. The distribution of these hormones divides the growing
root into three regions: the differentiation zone, elongation zone, and
division zone. Join me in this talk as I explore the existing literature
on root growth and present a rudimentary model of this fascinating phenomenon.
May 23, 2024:
Christian Campbell: Minimal Surfaces and Complex Geometry
[abstract]Minimal surfaces, or surfaces which
locally minimize area, arise naturally in nature. For example, the surfaces
formed by soap films are minimal surfaces, as a consequence of the laws of
surface tension. In this talk, we will define minimal surfaces (and more
generally minimal submanifolds), give some examples, and discuss some
historical problems in this area. Next, we will explore the deep connection
between minimal surfaces and complex geometry, and sketch a proof that
complex submanifolds of
are always
minimal. Finally, we will conclude with current research questions for
this summer.
Nicholas Rees: The Density Increment Strategy
[abstract]In additive combinatorics, we seek
to find combinatorial estimates of structures related to addition,
e.g. finding the number of three-term arithmetic progressions in an
arbitrary subset of the integers. A useful heuristic here is the dichotomy
between structure and randomness, most famously applied in the proof of
the Green-Tao theorem. This talk will focus on the density increment
strategy, a classic method that formalizes our heuristic into proofs.
After an overview of the strategy, we will see an explicit implementation
in a proof sketch. I will try to stay qualitative and only assume
knowledge of linear algebra.
May 30, 2024:
Jessica Chen:
[abstract]Organoids are lab-grown aggregates
derived from human stem cells that hold promises for making advancements
in drug development and cellular therapies. Previous research shows the
activity level of gene SOX2 affects organoid cell differentiation and is
influenced by chemical stimuli like BMP4.
Thus, using ODES to model gene activity levels can give us insights on
how the shape, size and cellular composition of organoids can be guided by
these chemicals. In this talk, I will go over the ways in which cells are
able to differentiate and sort depending on their environment, and how we
can model these complex systems, both mathematically and computationally.
Shona Sinclair: Modelling asymmetric rotation in cell division
[abstract]Left-right asymmetry (also called
chirality or handedness) is a critical feature in the organization of cells,
tissues, and organs in animals. We seek to understand how chirality arises
from the cellular level by studying C. elegans, an animal whose cells
rotate asymmetrically as they divide.
I will present a model for a dividing cell rotating on a flat
surface, using mechanics at a low Reynolds number. By understanding the
forces that rotating cell experiences and exerts on its surroundings, we
hope to understand how chirality emerges in the C. elegans embryo.`
June 6, 2024:
Sushrut Tadwalker: Unrequitable debts: A brief look at "the birth of Algebraic number
theory"
[abstract]In my talk I will explain the
ideas employed by mathematicians in
the 19th century to try and solve Fermat's last theorem (specifically unique
factorization), and how this resulted in the creation of the field of study
we now know as algebraic number theory. I will do this by broadly talking
about the various approaches they had, the constructions involved, the
problems with said approaches, and the resolutions they settled on. I will
provide examples as often as possible to concretize some of the more
abstract concepts and their behaviours. Virtually no pre-requisites are
required, but basic knowledge in abstract algebra, such as the definition of
a ring, will be helpful.
Justin Wan: Integer Sequences
[abstract]In this talk, I will present
a way to create a generalisation of
binomial coefficients/Pascal's Triangle from any given sequence of positive
integers. When all coefficients generated by a sequence are integers, we
call the sequence "binomid". We will explore binomid sequences through the
lens of divisibility, and prove properties of some sequences that will lead
to being binomid.
June 13, 2024:
Haad Bhutta: A data-driven approach for modeling
epigenetic dynamics in stem-cell differentiation
[abstract]Embryonic stem cells
(ESCs) proceed through four general stages of growth
as they differentiate. ESCs develop into hemangioblasts (HB), then into the
hemogenic endothelium (HE), and then into hematopoietic progenitors (HP).
In this process, DNA is transcribed into mRNA and this mRNA is translated
into specific proteins which play key roles in ESC differentiation.
Epigenetic components called transcription factors and histone
modifications regulate this differentiation process. We seek to formulate
ODE and PDE systems that capture the dynamics of these epigenetic
components in each stage while obeying current biological literature by
employing a data-driven modeling approach.
Young Lin: Analysis of Markov Chain mixing times
[abstract]Markov chains are a very
intuitive type of stochastic model, and Markov chain convergence is
applicable to many areas, including Markov Chain Monte
Carlo methods and simulations. Mixing time quantitively describes the time
required by a Markov chain for the distance to stationarity to be small. In
this talk, I will discuss various examples of analysis of mixing times that
I found interesting and briefly discuss the cutoff phenomenon and our
research question for this summer.
June 20, 2024:
Lucas O'Brien: Optimal transport and the average-distance problem
[abstract]
The Monge-Kantorovich transportation problem is introduced, as well as a
related problem in which we optimize a region over which the transport cost
vanishes. Properties of optimal regions and variants of the problem will be
discussed, as well as real-world applications.
Rain Zimin Yang: Minimal Surfaces and Weierstrass-Enneper Representation
[abstract]
Minimal surfaces, like soap films stretching across wireframes,
are shapes that locally minimize surface area. In this talk, I'll introduce
these fascinating surfaces and we'll explore the Weierstrass-Enneper
representation, a tool that uses complex analysis to describe and
understand minimal surfaces.
June 27, 2024:
Mark Ewert: An inductive proof illustrated geometrically
[abstract]
In this talk I will be going through a proof by induction that arose when
trying to answer the question “If player A flips a fair coin
times and
player B flips a fair coin times,
what is the probability that A flips
more heads that B?” I will start with a short story to give some background
on my attempt at solving this problem and then transition into the proof.
This proof by induction has a very good geometric visualization to go along
with it and the end result cleans up quite elegantly. No prerequisites are
needed for this presentation other than a basic understanding of
combinatorics.
July 4, 2024:
Hanna Khan
[abstract]
In this talk, I will be going through the intuition behind modelling
oncolytic virus dynamics in a paper I studied over the summer. My goal is to
demonstrate how models are developed and built upon to reflect the behavior
of biological processes.
July 18, 2024: two talks.
July 25, 2024:
Sushrut Tadwalkar and Tighe McAsey: Blocks and Blueprints: Simplifying the Science of Self-Assembly
[abstract]
Particle self-assembly is a rich and active area of interdisciplinary
research, uniting all major branches of science. Self-assembly has the
potential to revolutionize medicine and technology, by allowing us to build
structures on the scale of microns, which has historically been an
impossible task. Our talk will present new regimes to vastly improve self
assembly, and outline powerful simulation tools for their verification, and
development. We will bring the subject to life with diagrams, images and
animations. No prerequisites are required apart from the basic terminology
of graph theory and set theory.
Young Lin
[abstract]
In my talk I will present a cute topic in probability theory, namely
the branching process. Branching process models growth and development of
species, and ask questions about their survival. I will try to explain some
of its properties and connections to percolation theory.
