Colloquium

Biquasiles and Dual Graph Diagrams

01/25/2017 - 4:15pm
01/25/2017 - 5:15pm
Speaker: 
Sam Nelson
Abstract: 

Dual graph diagrams are an alternate way to present oriented knots and links in $R^3$. In this talk we will see how to turn dual graph Reidemeister moves into an algebraic structure known as biquasiles and use this structure to define new integer-valued counting invariants of oriented knots and links.

 

Where: 
Shanahan B460, Harvey Mudd College

Saving California Newts Two Models at a Time

03/22/2017 - 4:15pm
03/22/2017 - 5:15pm
Speaker: 
Courtney Davis (Pepperdine University)
Abstract: 
Santa Monica Mountain (SMM) streams are home to the California newt (Taricha torosa), a species of special concern in California.  Our historically severe drought as well as stream invasion by nonnative crayfish (Procambarus clarkii) that prey upon newt eggs have decimated local newt reproduction.  This has led to localized newt extinctions in some SMM streams.  Restorative measures are currently underway in some SMM streams to remove crayfish through trapping in order to prevent or slow the decline of native species.
 
In collaboration with biologists and undergraduate mathematics students, we have created discrete mathematical models to study the population dynamics of the California newt under drought and crayfish predation.  In this talk, I will describe how we construct two nonlinear systems of discrete equations that include demographic parameters such as survival rates for newt life stages and egg production, which depend upon habitat availability and rainfall. We estimate these demographic parameters using 15 years of stream survey data collected from SMM streams.  Our models capture the observed decline of the studied newt population and replicate crayfish trapping data.  Our drought model makes predictions about how the length and severity of drought can affect the likelihood of persistence and the time to critical endangerment of a newt population.  With our crayfish trapping model, we evaluate the persistence or the time to extinction for newt populations under crayfish trapping regimes when varying the quantity of trapping resources, frequency of trapping implementation, and susceptibility of the crayfish population to trapping. Predictions made with both models inform restorative efforts and crayfish management.

 

Where: 
Shanahan B460, Harvey Mudd College

Planar Algebras: Calculating with Pictures

04/19/2017 - 4:15pm
04/19/2017 - 5:15pm
Speaker: 
James Tener (UCSB)
Abstract: 

Planar algebras were first introduced in the late 90's by Vaughan Jones as an axiomatization of the standard invariant of a subfactor. Jones' idea was that the structure of standard invariants had a description in terms of planar diagrams, and that one could compute things about the subfactor by manipulating the pictures. Since then, planar algebras have been used extensively as a framework for performing rigorous calculations by manipulating diagrams. In this talk I will give an example-driven introduction to planar algebras and diagrammatic calculation, and demonstrate some of the features of working with pictures. If time permits, I will also discuss the state of the on-going project to classify `small' planar algebras, as well as the role played by planar algebras in constructive quantum field theory.

 

 

Where: 
Beckman B460, Harvey Mudd College

Multi-scale Modeling in Biology and Medicine

05/03/2017 - 4:15pm
05/03/2017 - 5:15pm
Speaker: 
Mark Alber (Notre Dame /UC Riverside)
Abstract: 

Population of bacteria P. aeruginosa, main infection in hospitals, will be shown to propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Biologically-justified cell-based multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depend upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. P. aeruginosa efficiently colonizes surfaces by controlling the physical forces responsible for expansion of thin liquid films and by propagating towards the tendril tips. Therefore, P. aeruginosa can efficiently colonizes surfaces by controlling the physical forces responsible for expansion of thin liquid films and by propagating towards the tendril tips. The model predictions of wave speed and swarm expansion rate as well as cell alignment in tendrils were confirmed experimentally. The model was also used for studying mechanism of drug resistance [1].

In the second half of the talk, a three-dimensional multi-scale modeling approach will be described for studying fluid–viscoelastic cell interaction during blood clot formation, with cells modeled by subcellular elements (SCE) coupled with fluid flow sub model [2]. Using this method, motion of a viscoelastic platelet in a shear blood flow was simulated and compared with experiments on tracking platelets in a blood chamber. It will be shown that complex platelet-flipping dynamics under linear shear flows can be accurately recovered with the SCE model.

References

[1] Morgen E. Anyan, Aboutaleb Amiri, Cameron W. Harvey, Giordano Tierra, Nydia Morales-Soto, Callan M. Driscoll, Mark S. Alber, Joshua D. Shrout [2014], Type IV Pili Interactions Promote Intercellular Association and Moderate Swarming of Pseudomonas aeruginosa, Proc. Natl. Acad. Sci. USA vol. 111, no. 50, 18013-18018.

[2] Wu Z, Xu Z, Kim O, Alber M. [2014], Three-dimensional multi-scale model of deformable platelets adhesion to vessel wall in blood flow. Philosophical Transactions of the Royal Society A 372: 20130380.

Where: 
Shanahan B460, Harvey Mudd College

Modeling Diffusion and Capture

04/12/2017 - 4:15pm
04/12/2017 - 5:15pm
Speaker: 
Alan Lindsay (Notre Dame)
Abstract: 

Diffusion is a fundamental transport mechanism whereby spatial paths are determined from probabilistic distributions. In examples such as the pollination of a flower or immune response to infection, the arrival of a single particle can initiate a cascade of events. The movement of these particles is driven by random motions, yet these systems largely function in an ordered and predictable way. This process, and others like it, can be modeled as a problem for the arrival time of a diffusing molecule to hit a small absorbing target.  In this talk I will discuss the mathematical models of this phenomenon and introduce the governing equations which are PDEs of parabolic and elliptic type with a mix of Dirichlet (Absorbing) and Neumann (Reflecting) boundary conditions. A particular feature of cellular problems is that the absorbing set has a large number of very small sites. I will present a new homogenized theory which replaces the heterogeneous configuration of boundary conditions with a uniform Robin type condition. To verify this limit, I will describe a novel spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. The numerical formulation reduces the problem to a linear integral equation supported on each of the absorbing sites. Real biological systems feature thousands of absorbing sites and our numerical method can rapidly resolve this realistic limit to high accuracy.


Where: 
Shanhan B460, Harvey Mudd College

Uniform Bounds of Families of Twists

02/15/2017 - 4:15pm
02/15/2017 - 5:15pm
Speaker: 
Bianca Thompson (HMC)
Abstract: 
The study of discrete dynamical systems boomed in the age of computing. The Mandelbrot set, created by iterating 0 in the function z^2+c  and allowing c to vary, gives us a wealth of questions to explore. We can ask about the number of rational preperiodic points (points whose iterates end in a cycle) for z^2+c. Can this number be uniform as we allow c to vary? It turns out this is a hard question to answer. Instead we will explore places where this question can be answered; twists of rational functions. 
Where: 
Shanahan B460, Harvey Mudd

Constructing extremal lattices in Euclidean spaces

02/01/2017 - 4:15pm
02/01/2017 - 5:15pm
Speaker: 
Lenny Fukshansky (CMC)
Abstract: 

Lattices are discrete periodic structures in Euclidean spaces, which are essential in many areas of mathematics and their applications. In particular, lattices are vital in number theory, discrete geometry, discrete optimization, digital communication, and other areas. Some of the most interesting lattice constructions come from various algebraic contexts, resulting in lattices with remarkable geometric properties: it appears that algebraic structure often informs the geometry. One of such properties is extremality, which ensures certain strong optimization potential, however extremal lattices are not so easy to construct. In this talk we will give a basic overview of lattice theory with a view towards optimization properties, and will exhibit some new systematic constructions of extremal lattices from surprisingly simple algebraic objects.

Where: 
Shanahan B460, Harvey Mudd

Averaging hyperbolic-type metrics

04/05/2017 - 4:15pm
04/05/2017 - 5:15pm
Speaker: 
Zair Ibragimov (CSU Fullerton)
Abstract: 

We begin with a brief discussion of the classical hyperbolic metric on the unit disk and present several of its generalizations, commonly known as hyperbolic-type metrics. In some details we will discuss two particular metrics, Scale-Invariant Cassinian metric and Mobius-Invariant Cassinian metric. We end the talk with a discussion of averaging hyperbolic-type metrics and its potential applications in geometric function theory.

Where: 
Shanahan B460, Harvey Mudd

Totally geodesic submanifolds in Riemannian geometry

03/08/2017 - 4:15pm
03/08/2017 - 5:15pm
Speaker: 
Thomas Murphy (CSU Fullerton)
Abstract: 

One of the main avenues of research in Riemannian geometry has been submanifold geometry, which studies how one manifold "sits" (embeds)  inside another. It is analogous to studying subgroups of a given group. Totally geodesic embeddings are the simplest cases to study, but the problem is fiendishly difficult. I will explain carefully the objects mentioned in the title of talk, outline  the history and importance of the classification problem, and explain some  work in progress with Fred Wilhelm (UCR) concerning their existence in generic settings.

Where: 
Shanahan B460, Harvey Mudd

Repetitivity of patterns in mathematical quasicrystals

03/29/2017 - 4:15pm
03/29/2017 - 5:15pm
Speaker: 
Alan Haynes (Univ. of Houston)
Abstract: 

In this talk we will begin with a brief history of the mathematics of aperiodic tilings of Euclidean space, highlighting their relevance to the theory of physical materials called quasicrystals. Next we will focus on an important collection of point sets, cut and project sets, which provide us with mathematical models for quasicrystals. Cut and project sets have a dynamical description, in terms of return times to certain regions of linear R^d actions on higher dimensional tori. As an example of the utility of this point of view, we will demonstrate how it can be used, in conjunction with input from Diophantine approximation, to classify a subset of `perfectly ordered’ quasicrystals.

Where: 
Shanahan B460, Harvey Mudd
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