Colloquium: Spring 2021

Time & Location:  All talks are on Thursdays in Gibson Hall 126A at 3:30 pm unless otherwise noted.  Refreshments in Gibson 426 after the talk.

Organizer:  Gustavo Didier

 

January 25

Title: Modeling and analysis of complex systems — with a basis in zebrafish patterns
Alexandria Volkening - Northwestern University

Abstract: Many natural and social phenomena involve individual agents coming together to create group dynamics, whether they are cells in a skin pattern, voters in an election, or pedestrians in a crowded room. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that appear on their bodies and fins. Mutant zebrafish, on the other hand, feature different skin patterns, including spots and labyrinth curves. All these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. The longterm motivation for my work is to better link genes, cell behavior, and visible animal characteristics — I seek to identify the specific alterations to cell interactions that lead to mutant patterns. Toward this goal, I develop agent-based models to simulate pattern formation and make experimentally testable predictions. In this talk, I will overview my models and highlight future directions. Because agent-based models are not analytically tractable using traditional techniques, I will also discuss the topological methods that we have developed to quantitatively describe cell-based patterns, as well as the associated nonlocal continuum limits of my models.

Join us: 
Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30

 

January 28

Title: Mathematically Modeling the Mechanisms Behind Intra-Droplet and Droplet Field Patterning in Phase Separated Systems
Dr. Kelsey Gasior - Florida State University

Abstract: 

Join us: 
Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30

 

February 2

Title: Rigidity of plane frameworks with forced symmetry
Daniel Bernstein - Fields Institute

Abstract: Rigidity theory asks and answers questions about how a given mechanical structure can deform. This area extends back into the nineteenth century with work of Cauchy and Maxwell, and continues to be an active area of research with a wide range of applications. I will begin my talk with a broad overview of this area. Then, I will narrow my focus onto symmetry-forced rigidity of plane frameworks to discuss a recent result. I will discuss the algebraic-geometric ideas involved in the proof, and how these same ideas can be used to address certain problems in matrix completion.

Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30

 

February 4

Title: The Batchelor spectrum and mixing in stochastic fluids
Samuel Punshon-Smith | Brown University (Scott McKinley)

Abstract: In 1959 George Batchelor predicted that a passively advected quantity in a fluid (like small temperature fluctuations or some chemical concentration), in a regime where the scalar dissipation is much lower than the fluid viscosity, should reach an equilibrium with an L2 spectral density proportional to 1/|k| over an appropriate inertial range, known as "the Batchelor spectrum". This prediction has since been observed experimentally and in various numerical experiments. However, despite strong evidence in its favor, rigorous derivations are only known in very special circumstances.
 
In this talk, I will consider the problem of a passive scalar undergoing advection-diffusion when the advecting velocity field belongs to a class of stochastic incompressible fluid motions, including models like the 2d incompressible stochastic Navier-Stokes equations (among a host of other stochastic fluid models in both 2d and 3d). I will discuss how a version of Batchelor's prediction is actually a general consequence of uniform-in-diffusivity exponential mixing properties of a fluid. Based on an argument inspired by techniques of Furstenberg for random dynamical systems, I will present a result deducing almost sure chaotic motion of the particle trajectories, known as Lagrangian chaos. I will then discuss how Lagrangian chaos can be used along with spectral theory and quantitative techniques from the ergodic theory of Markov processes to deduce almost sure, uniform-in-diffusivity exponential mixing, a powerful property that is not known to hold in the deterministic setting. This result shows that a version of Batchelor's prediction is indeed fairly robust and holds quite generally for a variety of suitably non-degenerate, stochastic incompressible fluid motions.

Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30

 

February 9

Title: Brownian Dynamics with Constraints
Brennan Sprinkle | New York University

Abstract: At the scale of a few micrometers, objects suspended in a fluid are subject to random kicks from collisions with the solvent molecules. This leads to a random motion of the suspended objects which must be reconciled with any geometric or mechanical constraints, like rigidity or inextensibility. After discussing numerical methods to simulate the Brownian dynamics of rigid bodies, I will primarily focus on simulation methods for inextensible filaments. Filaments at the cellular scale can take the form of beating flagella that propel sperm cells and bacteria; or they can tangle into the vast, interconnected networks that make up the cellular cytoskeleton. I’ll introduce a method where fibers are treated as a chain of beads and use it to interrogate experimental observations on magnetic filaments which can be made to swim using an applied field.  Motivated by this study, I’ll present ongoing work concerning a method more suited to fiber networks in which inextensible fiber motions are parametrized as curves on the unit sphere.

Zoom access: Contact mbrown2@math.tulane.edu
Time: 3:30

Special Colloquium

Title: Edge Ideals of Random Graphs
Arindam Banerjee | (at Ramakrishna Mission Vivekananda Educational and Research Institute)

Abstract: The theory of edge ideal studies finite simple graphs from an algebraic perspective. It attaches an ideal with every finite simple graph and tries to interpret the combinatorics of the graph in terms of various algebraic invariants of that ideal. Philosophically speaking, one may ask what happens to those invariants on an average when one considers all possible graphs. The study of the edge ideals of Erdos-Renyi random graphs gives a nice mathematical framework to properly pose that philosophical question. In this talk we shall develop this framework and use that to discuss average behaviours of some important algebraic invariants. In particular we shall discuss a new result which shows that algebraic invariants Krull Dimension and  Castelnuovo-Mumford regularity (a measure of size and a measure of complexity respectively) satisfy some law of large number when number of vertices of the underlying Erdos-Renyi random graph goes to infinity.

Zoom access: Contact mbrown2@math.tulane.edu
Time: 10:30AM – 11:30AM

 

February 11

Title: Class numbers of quadratic fields
Olivia Beckwith | University of Illinois Urbana-Champaign (Host: Lisa Fauci)

Abstract: Gauss was the first to count classes of binary quadratic forms with a fixed discriminant up to matrix equivalence. The number of equivalence classes, the class number, measures the obstruction to unique factorization into primes for quadratic number fields. Information about class numbers percolates into many branches of number theory, including the theory of L-functions via Dirichlet's class number formula, and elliptic curves in view of the work and conjecture of Birch and Swinnerton-Dyer. This talk will begin with a brief introduction to algebraic number theory and class numbers, as well as some of the important results in the history of their study. Then I will discuss some of my work in this area, which is about the divisibility properties of class numbers.

Location: TBA

Time: 3:30

 

February 18

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Speaker - Institution (Host: TBA)

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February 25

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March 4

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Mark Girolami - Cambridge (Host: Glatt-Holtz)

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March 11

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Speaker - Institution (Host: TBA)

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March 18

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March 25

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April 1

Spring Break

 

April 8

Title: Mixing, transport, and enhanced dissipation
Anna Mazzucato - Penn State (Host: Glatt-Holtz)

Abstract: I will discuss transport of passive scalars by incompressible flows and measures of optimal mixing and stirring.  I will present two examples of opposite effects of mixing: one leads to irregular transport and a dramatic, instantaneous loss of regularity for transport quation, the other is enhanced dissipation, which can lead to global existence in non-linear, dissipative systems. In particular, I will  show how mixing leads to global existence for the 2D Kuramoto-Sivashisky equation, a model for flame propagation.

Location: TBA

Time: TBA

 

April 15

Title: Analysis of free boundary problems in fluid mechanics
Ian Tice - Carnegie Mellon University (Host: Glatt-Holtz)

Abstract: A free boundary problem in fluid mechanics is one in which the fluid domain is not specified a priori and evolves in time with the fluid.  Such problems are ubiquitous in nature and occur at a huge range of scales, from dew drops, to waves on the ocean, to the surface of a star.  In this talk I will review the basic features of free boundary problems and how we incorporate interesting interfacial physics effects.  I will also survey recent work on various viscous free boundary problems.

Location: TBA

Time: 3:30

 

 

April 22

Title: Examples of MM Algorithms
Ken Lange - UCLA (Host: Ji, Xiang)

Abstract: 

Location: TBA

Time: 3:30

 

 

April 29

Title: TBA
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