Claremont Center for the Mathematical Sciences

Applied Math Seminar: Eulerian Approaches based on the Level Set Method for Visualizing Continuous Dynamical Systems (Shingyu Leung, Department of Mathematics, HKUST)
2019-02-28 16:15 - 17:15 (America/Los_Angeles) | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
One very important concept in understanding a dynamical system is coherent structure. Such structure segments the domain into different regions with similar behavior according to a quantity. When we try to partition space-time into regions according to a Lagrangian quantity advected along with passive tracers, such class of coherent structure is called the Lagrangian coherent structures (LCSs). Among many, a simple definition of an LCS uses the finite-time Lyapunov exponent (FTLE). It measures the rate of separation between adjacent particles over a finite time interval with an infinitesimal perturbation in the initial location. In the talk, we first present various Eulerian-based numerical methods which efficiently compute the flow maps of any continuous dynamical system and, therefore, the corresponding FTLE. Based on these techniques we developed, we will also propose some other useful numerical tools for extracting important structures hidden in the system.

GEMS Workshop: Graph Theory, Part II with Professor Michael Orrison, from Harvey Mudd College
2019-03-02 10:00 - 12:00 | Shanahan 1480, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
TOPIC: Graph Theory, Part II
On the surface, graphs seem to be some of the simplest objects you might encounter in mathematics. After all, they are made up of just two kinds of parts, vertices and edges, and those parts fit together in simple ways. But appearances can be deceiving! In this series of two workshops, we’ll explore several surprising facts about graphs. Along the way, we’ll see how graphs arise in a variety of settings, and how they give rise to many challenging but worthwhile problems.
WHAT IS GEMS:
The Gateway to Exploring Mathematics program (GEMS) is a series of workshops that helps excite the interests and curiosity of young students in mathematics and science
GEMS meets once a month on a Saturday morning from 10:00 AM to 12:00 PM
GEMS is designed to reach 8th, 9th and 10th grade students who have an interest in mathematics and science
Participants interact with excellent and award winning faculty, staff, students and alumni from each of the seven Claremont Colleges
SPRING 2019 DATES:
February 2, 2019
March 2, 2019
April 13, 2019
REGISTRATION:
To register for our next event on March 2, please click on the following link:
https://tinyurl.com/GEMS2019Spring2
ANY QUESTIONS:
Please contact our 2018-2019 GEMS coordinator, Elsa Harris at Elsa.Harris@cgu.edu

Applied Math Seminar: Fluid mechanics at the microscale (Prof. Amy Buchmann, University of San Diego)
2019-03-04 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
I will present mathematical and computational methods used to model interactions between a viscous fluid and elastic structures in biological processes. For example, microfluidic devices carry very small volumes of liquid through channels and may be used to gain insight into many biological applications including drug delivery and development, but mixing and pumping at this scale is difficult. Experimental work suggests that the flagella of bacteria may be used as motors in microfluidic devices, and mathematical modeling can be used to further investigate this idea. Cilia self-organize forming a metachronal wave that propels the surrounding fluid. How this organization occurs is not well understood. Mathematical models can be used to study the role of hydrodynamic interactions in self-organization.

Nonvanishing minors and uncertainty principles for Fourier analysis over finite fields (Daniel Katz, CSUN)
2019-03-05 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Chebotarev’s theorem on roots of unity says that every minor of a discrete Fourier transform matrix of prime order is nonzero. We present a generalization of this result that includes analogues for discrete cosine and discrete sine transform matrices as special cases. This leads to a generalization of the Biro-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions, with some of the simplest examples being even and odd functions. This new uncertainty principle gives a bound that is sharp and, for some classes of functions, stronger than that of Biro-Meshulam-Tao.

Accidental Mathematics (Matt Stamps, Yale-NUs College)
2019-03-06 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Abstract: Growing up, I always loved learning about world-changing scientific breakthroughs that were discovered by accident. Penicillin, artificial sweeteners, X-rays, and synthetic dyes are just a few of the discoveries that were stumbled upon by scientists who had other goals in mind. More recently, I have come to wonder why anecdotes about accidental discoveries in mathematics are not as commonplace. Is it a fundamental difference in they way mathematicians and natural scientists view their work? Are such stories too contrary to the popular perception that success in mathematics is reserved for the genius of a select few? Whatever the reason, I argue that mathematics happens accidentally all the time. In this talk, I will describe two accidental discoveries from my own work involving Penrose tilings, circle packings, chordal graphs, lecture hall partitions, lattice polytopes, and polynomial rings.

Non-existence of epimorphisms between certain genus two handlebody-knot groups (Ryo Nikkuni, Tokyo Woman's Christian University)
2019-03-07 12:00 - 13:30
For two genus $g$ handlebody-knots $H_{1}$ and $H_{2}$, we denote $H_{1} geq H_{2}$ if there exists an epimorphism from the fundamental group of the handlebody-knot complement of $H_{1}$ onto the one of $H_{2}$. In the case of $g = 1$, this order is a partial order on the set of prime knots and has been determined up to $11$ crossings by Kitano-Suzuki and Horie-Kitano-Matsumoto-Suzuki. In this talk, we consider the case of $g = 2$ and exhibit a lot of ordered pairs of irreducible genus $2$ handlebody-knots in the Ishii-Kishimoto-Moriuchi-Suzuki table up to $6$ crossings, each of which does not admit this order. This is a joint work with Y. Ozawa and M. Suzuki.

Applied Math Talk: Cluster analysis on covariance stationary ergodic processes and locally asymptotically self-similar processes (Nan Rao, CGU)
2019-03-11 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
We study the problems of clustering covariance stationary ergodic processes and locally asymptotically self-similar stochastic processes, when the true number of clusters is priorly known. A new covariance-based dissimilarity measure is introduced, from which efficient consistent clustering algorithms are obtained. As examples of application, clustering fractional Brownian motions and clustering multifractional Brownian motions are respectively performed to illustrate the asymptotic consistency of the proposed algorithms.

Indiana Pols Forced to Eat Humble Pi: The Curious History of an Irrational Number (Edray Goins, Pomona)
2019-03-12 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
In 1897, Indiana physician Edwin J. Goodwin believed he had discovered a way to square the circle, and proposed a bill to Indiana Representative Taylor I. Record which would secure Indiana’s the claim to fame for his discovery. About the time the debate about the bill concluded, Purdue University professor Clarence A. Waldo serendipitously came across the claimed discovery, and pointed out its mathematical impossibility to the lawmakers. It had only be shown just 15 years before, by the German mathematician Ferdinand von Lindemann, that it was impossible to square the circle because $pi$ is an irrational number. This fodder became ignominiously known as the “Indiana Pi Bill” as Goodwin’s result would force $pi = 3.2$.

In this talk, we review this humorous history of the irrationality of $pi$. We introduce a method to compute its digits, present Lindemann’s proof of its irrationality (following a simplification by Miklos Laczkovich), discuss the relationship with the Hermite-Lindemann-Weierstrass theorem, and explain how Edwin J. Goodwin came to his erroneous conclusion in the first place.

Some Unexpected Mathematics Arising From Research at NIST ( Hunt, NIST)
2019-03-13 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
A lot of the mathematics done at NIST supports the research on and measurement of advanced materials and technology. In this rather applied context. surprising mathematics makes an appearance. We present a few examples.

Refinements of metrics (Wai Yan Pong, CSUDH)
2019-03-26 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
I will talk about a few graph-theoretic metrics then introduce the concept of refinements on a class of functions that include all metrics. As a case study, we will construct various refinements on the shortest-path distance. Consequently, we obtain a few “better” versions of the Erdos number. In the course of our investigation, we realized various construction of metrics can be unified under a rather natural concept that we called monotonic monoid norm. This is a joint work with Kayla Lock and Alex Wittmond.

Reasoning about Liability of Intelligent Agents ( Naumov, CMC)
2019-03-27 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Abstract: As intelligent agents assume larger role in our daily lives, reasoning by humans about liability of such agents as well as reasoning by the intelligent agents themselves about liability becomes more important. The existing laws, written with humans in mind, will eventually need to be re-interpreted in terms of their applicability in a hybrid environment that consists of humans and intelligent agents. In some cases, new laws will need to be written to redefine liability in the context involving intelligent agents. As a first step in this direction, we need to have a formal definition of liability that can be applied to intelligent agents and logical rules that the machines can use to reason about their own and human liability. In this talk I will discuss several of my recent works on formal logical systems for reasoning about liability.

Applied Math Talk: Repurposing FDA-approved drugs as host-oriented therapies against infectious diseases (Prof. Mikhail Martchenko, KGI)
2019-04-01 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
The traditional method of treating most human diseases is to direct a therapy against targets in the host patient, whereas conventional therapies against infectious diseases are directed against the pathogen. Unfortunately, the efficacy of pathogen-oriented therapies and their ability to combat emerging threats such as genetically engineered and non-traditional pathogens and toxins have been limited by the occurrence of mutations that render pathogen targets resistant to countermeasures. Our work shows that host proteins that are exploited by pathogens (Host Proteins Exploited by Pathogens; HPEPs) contribute to the severity of exposure to pathogenic agents. We find that pathogens recruit HPEPs to bind to, enter, reproduce in, exit from, and kill host cells. Thus, HPEPs are potential targets for therapies. This presentation will discuss examples of our drug discovery efforts to identify host-oriented therapies.

Fibonacci and Lucas analogues of binomial coefficients and what they count (Curtis Bennett, CSULB)
2019-04-02 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
A Fibonomial is what is obtained when you replace each term of the binomial coefficients $ {n choose k}$ by the corresponding Fibonacci number. For example, the Fibonomial
$${ 6brace 3 } = frac{F_6 cdot F_5 cdot dots cdot F_1}{(F_3cdot F_2 cdot F_1)(F_3cdot F_2 cdot F_1)} = frac{8cdot5cdot3cdot2cdot1cdot1}{(2cdot1cdot1)(2cdot1cdot1)} = 60$$
since the first six Fibonacci numbers are 1, 1, 2, 2, 5, and 8. Curiously the Fibonomials are always integers, raising the combinatorial question: what do they count? In this talk we introduce and provide a little history of the Fibonomials. We then provide a simple object the Fibonomials enumerate. We will use this new object to prove various Fibonomial analogues of standard identities on binomial coefficients and discuss further generalizations including the Lucanomials.

On the interplay of functional analysis and operator theory (Puig de Dios, UCR)
2019-04-03 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Abstract: We overview some basic and striking facts concerning the theory of hypercyclic operators (considered to be born in 1982):

1. Hypercyclicity is a purely infinite-dimensional phenomenon: no finite dimensional space supports any hypercyclic operator;

2. It is not easy at all to determine whether a linear operator is hypercyclic. However, the set of hypercyclic operators is dense for the Strong Operator Topology in the algebra of linear and bounded operators;

3. Hypercyclicity is far from being an exotic phenomenon: any infinite-dimensional separable Frechet space supports a hypercyclic operator.

Models of Biological Tissue Electrostatics and Molecular Transport (Jim Sterling, KGI)
2019-04-08 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
In this presentation, some fundamentals of electrostatics in biology will be discussed with focus on the fact that most biological macromolecules including nucleic acids, carbohydrates, and proteins are negatively-charged. Electroneutrality requires cations to move toward the macromolecules where they both screen and bind to the negatively-charged groups. An important class of mathematical models of species-flux and electrostatics are known as the Poisson-Nernst-Planck, or PNP equations. These are partial differential equations describing some important biophysical consequences.

Matrix multiplication: the hunt for $omega$ (Mark Huber, CMC)
2019-04-09 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
For centuries finding the determinant of a matrix was considered to be something that took $Theta(n^3)$ steps. Only in 1969 did Strassen discover that there was a faster method. In this talk I’ll discuss his finding, how the Master Theorem for divide-and-conquer plays into it, and how it was shown that finding determinants, inverting matrices, and Gaussian elimination are the same time complexity as to matrix multiplication.

A General Bayesian Discrete Time Survival Model (King, CPP)
2019-04-10 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Abstract: “We present a general Bayesian statistical model for discrete time, discrete state space stochastic processes. Applications include the modeling of recurrent and episodic disease processes, such as episodes of illicit drug use, as well as social processes such as educational enrollment and employment. We also present Markov chain Monte Carlo inference algorithms for our model, along with a freely available software package called “brea” which implements these methods in the R programming language.”

Geometry of quotient varieties and the algebra of conformal blocks (Han-Bom Moon Fordham University)
2019-04-11 12:15 - 13:30 | Roberts North 104, CMC, 320 E. 9th St., Claremont, CA, 91711, United States
An important question in classical representation theory is when the tensor product of two irreducible representations has another representation as a factor. In this talk, I will introduce a quantum generalization of this question and explain how we may relate this question to geometry of quotients of certain complex manifolds. This is joint work with Sang-Bum Yoo.

GEMS Workshop: "Graphs, matrices, and recurrences" with Professor Lucas Bang, from Harvey Mudd College
2019-04-13 10:00 - 12:00 | Shanahan 1480, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
TOPIC: Graphs, matrices, and recurrences
Abstract: In mathematics, we are often surprised to find that problems that look very different are actually the same problem in a different guise! In this seminar, we will build on the previous discussions about graph theory and describe how other areas of math are closely related to graphs. Specifically, we will learn how graph problems can be translated to and from matrix problems. We will also see how both graphs and matrices are related to recursive integer sequences (like the famous Fibonacci numbers).
WHAT IS GEMS:
The Gateway to Exploring Mathematics program (GEMS) is a series of workshops that helps excite the interests and curiosity of young students in mathematics and science
GEMS meets once a month on a Saturday morning from 10:00 AM to 12:00 PM
GEMS is designed to reach 8th, 9th and 10th grade students who have an interest in mathematics and science
Participants interact with excellent and award winning faculty, staff, students and alumni from each of the seven Claremont Colleges
SPRING 2019 DATES:
February 2, 2019
March 2, 2019
April 13, 2019
REGISTRATION:
To register for our workshop on April 13, please click on the following link:
https://tinyurl.com/GEMS2019Spring3
ANY QUESTIONS:
Please contact our 2018-2019 GEMS coordinator, Elsa Harris at Elsa.Harris@cgu.edu

Applied Math Talk: Solving Complex Public Health Problems—Cancer, Obesity and Aging (Jessica Dehart, CGU)
2019-04-15 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
Abstract: Remember smoking? What’s the new public health problem? In the US, we are currently entangled within three converging and intertwined complex problems: Cancer, Obesity, Aging. There are over 16 million cancer survivors living in the US as we speak. Over 50% of our society is overweight and/obese. Our society is aging and the age distribution is much older than a few years back. Cancer, obesity and aging share several risk factors, biological mechanisms and patterns. Given the multidimensionality and complexity of these issues, only a transdisciplinary approach will have the best chances of success in sustaining a health society. This talk will discuss the problems and potential transdisciplinary approaches—including math—to finding successful solutions.

Chow rings of heavy/light Hassett spaces via tropical geometry (Dagan Karp, HMC)
2019-04-16 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
In this talk, I will try to give a fun introduction to tropical geometry and Hassett spaces, and show how tropical geometry can be used to compute the Chow rings of Hassett spaces combinatorially. This is joint work with Siddarth Kannan and Shiyue Li.

Unravelling Biochemistry Mysteries: Knot Theory Applied to Biochemistry (Price, University of San Diego)
2019-04-17 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Abstract: Mathematical modeling is an effective resource for biologists since it provides ways to simplify, study and understand the complex systems common in biology and biochemistry. Many mathematical tools can be applied to biological problems, some traditional and some more novel, all innovative. This presentation will review the mathematical tools that are used to model and study biological issues of DNA-protein interactions.

Enhancements of the quandle coloring invariant for knots (Karina Cho, Harvey Mudd College)
2019-04-18 12:00 - 13:30 | Roberts North 104, CMC, 320 E. 9th St., Claremont, CA, 91711, United States
Quandles are algebraic structures that play nicely with knots. The multiplicative structure of finite quandles gives us a way to “color” knot diagrams, and the number of such colorings for a given knot and quandle is called the quandle coloring invariant. We strengthen this invariant by examining the relationships between the colorings, which are given by endomorphisms. This can be visualized using a directed graph that we call the quandle coloring quiver. We will show that the quandle coloring quiver is a strict enhancement of the quandle coloring invariant and discuss further enhancements of this invariant that arise from quandle cohomology. This work is a senior thesis project under the advising of Sam Nelson.

Applied Math Talk: Nonlocal problems for linear evolution equations (Prof. Smith David Andrew, Yale-NUS College, Singapore)
2019-04-22 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
Linear evolution equations, such as the heat equation, are commonly studied on finite spatial domains via initial-boundary value problems. In place of the boundary conditions, we consider “multipoint conditions”, where one specifies some linear combination of the solution and its derivative evaluated at internal points of the spatial domain, and “nonlocal” specification of the integral over space of the solution against some continuous weight.

Theory of vertex Ho-Lee-Schur graphs (Sin-Min Lee, SJSU)
2019-04-23 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
A triple of natural numbers (a,b,c) is an S-set if a+b=c. I. Schur used the S-sets to show that for n >3, there exists s(n) such that for prime p > s(n), x^p + y^p = z^p (mod p) has a nontrivial solution. A (p,q)-graph G is said to be vertex Ho-Lee-Schur graph if there exists a bijection f: V(G) –> {1,2,…,p} such that for each C3 subgraph of G with vertices {x,y,z} the triple (f(x),f(y),f(z)) is an S-set. The VHLS deficiency of G is the smallest k such that GU Nk, where Nk is null graph, is a vertex Ho-Lee-Schur graph. We determine VHLS deficiency of some graphs and show that no Kuratowski type characterization of non-vertex Ho-Lee-Schur graphs. Some relation of integer partitions and this theory is explored. We will also introduce some unsolved problems and invite the audience to solve them.

A Conformal Mapping Approach to Shape Optimization Problems. (Kao, CMC)
2019-04-24 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Abstract: In this talk, a conformal mapping approach to shape optimization problems on planar domains will be discussed. In particular, spectral methods based on conformal mappings are proposed to solve Steklov eigenvalues and their related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problems, we use gradient ascent approaches to find optimal domains that maximize objective functions involving Steklov eigenvalues.

A (Z⊕Z)-family of knot quandles (Jim Hoste, Pitzer College)
2019-04-25 12:00 - 13:30
Suppose K is an oriented knot in a 3-manifold M with regular neighborhood N (K). For each element γ ∈ π 1 (∂N (K)) we define a quandle Q γ (K; M) which generalizes the concept of the fundamental quandle of a knot. In particular, when γ is the meridian of K, we obtain the fundamental quandle. The collection of all such quandles gives a (Z⊕Z)-family of quandles. If K is a knot in M and γ is a primitive element, then we show that there exists a knot K’ in a 3-manifold M’ such that Q γ (K; M ) ∼= Q μ (K’ ; M’) where μ is the meridian of K’ . Starting with a partially framed link L in the 3-sphere where the framed components give a surgery description of the manifold M and a single unframed component represents K we can derive a similar surgery description of K’ in M’ . Using results of Fenn and Rourke, we may then use this description of K’ to record a presentation of the quandle Q γ (K; M). We describe a number of examples of these quandles for knots
in various manifolds.

Applied Math Seminar: The Kaczmarz Algorithm and its Applications to Data Science (Anna Ma, UCSD)
2019-04-29 16:15 - 17:15 | Emmy Noether Room, Millikan 1021, Pomona College, 610 N. College Ave., Claremont, California, 91711
Data is exploding at a faster rate than computer architectures can handle. For that reason, mathematical techniques to analyze large-scale data need be developed. Stochastic iterative algorithms have gained interest due to their low memory footprint and adaptability for large-scale data. In this talk, we will study the Randomized Kaczmarz algorithm for solving extremely large linear systems of the form Ax=y. In the spirit of large-scale data, this talk will proceed under the assumption that the entire data matrix A cannot be loaded into memory in a single instance. We consider different settings including when a only factorization of A is available, when x is sparse, and a time-varying model. We will also present applications of these Kaczmarz variants to problems in data science.

What Did Ada Do? Digging into the Mathematical Work of Ada Lovelace (Gizem Karaali, Pomona)
2019-04-30 12:15 - 13:10 | Millikan 2099, Pomona College, 610 N. College Ave., Claremont, CA, 91711, United States
Augusta Ada Byron King Lovelace (1815-1852) is today celebrated as the first computer programmer in history. This might be confusing to some because in 1852 there were no machines that looked like what we call computers today. In this talk I attempt to explain what Ada really did, and delineate the mathematics involved. Bernoulli numbers will definitely come into play, but there may also be other fun distractions along the way, possibly including some juicy gossip about Ada’s life.

Is My Subgroup Normal? How Math Communities Differand Why it Matters (Sinclair, Google)
2019-05-01 16:15 - 17:15 | Shanahan B460, Harvey Mudd College, 301 Platt Blvd., Claremont, CA, 91711, United States
Mathematics isnt done in a void: its done by groups of people. Those groups have
different norms and values, which affect both who wants to engage in math and the mathematics itself
being done. When thinking about diversity and inclusion, explicitly examining norms within our
communities can get us a long way. Through a Thomas J Watson Fellowship, I had the opportunity
to experience mathematics competitions communities in Brazil, Argentina, Senegal, Singapore and
England. Come hear about the differences I found within those communities, and how Ive continued
to connect those learnings to technical communities back in the United States.
