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This page contains information on the 2020 Virtual Recruitment event for the UM Math/AIM graduate programs. The event will take place on Friday, March 13 and Saturday, March 14, 2020. Below is a schedule, along with titles and abstracts of the talks.
The event will take place through a video conferencing website called BlueJeans. You can access the event either using a web browser or by downloading the app. You should have received an email invitation with a link to access the meeting. (If not, please email email@example.com)
When you enter the meeting, your audio will be muted, but video will be unmuted. You can unmute your audio at any time. There is also a chat window on the right. The meeting is set up to be open from 5:15pm Thursday March 12 through 8pm Saturday March 14. You may enter and leave at any time, and then re-enter when you wish. Once you receive the invitation, please take a moment to enter the meeting and check that your audio/visual equipment works. You may do this on Thursday evening or Friday morning before the official start at 10am Friday.
All times below are Michigan time (Eastern Daylight Savings)
Friday, March 13
10am -- 12pm: Discussion of the Math/AIM PhD programs, Q&A session with Prof. Silas Alben (AIM Director) and Prof. Kartik Prasanna (Admissions Director);
Life in Ann Arbor (Google presentation)
12 pm -- 1pm AIM student forum (Q&A with current AIM PhD students Christiana Mavroyiakoumou, Ryan Kohl, Yili Zhang, Preetham Mohan, April Nellis)
1pm -- 2pm Talk by Prof. Jennifer Wilson
2pm -- 3:30pm Math student forum (Q&A with current Math PhD students Elizabeth Collins-Wildman, Jason Liang, Ruian Chen)
4pm -- 5pm Talk by Prof. Peter Miller
5pm -- 7pm Social hour: prospective students can use this time to chat with each other. No students or faculty from Michigan will be present.
Saturday, March 14
10am -- 10:30am Talk by AIM PhD student Christiana Mavroyiakoumou
10:40am -- 11:10 am Talk by AIM PhD student Will Clark
11:20am -- 11:50 am Talk by AIM PhD student Hai Zhu
12pm -- 12:50 pm Talk by Math PhD student Bradley Zykoski
1pm -- 1:50pm Talk by Math Postdoctoral Assistant Professor Karol Koziol
2pm -- 4pm Social hour: prospective students can use this time to chat with each other. No students of faculty from Michigan will be present
Titles and Abstracts (in the order of the talks)
(In)stability in configuration spaces
In this talk, we will explore certain patterns arising in families of topological spaces called configuration spaces, and their associated braid groups. These patterns are a first example of a phenomenon called "representation stability", a subject of growing research interest.
Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy We describe how to use a recently-proposed robust inverse scattering transform for arbitrary spectral singularities to study fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large-order. We establish the existence of a limiting profile of the rogue wave when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables. This profile, which we call the rogue wave of infinite order‚ also satisfies ordinary differential equations with respect to space and time, and the spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We obtain the far-field asymptotic behavior of this near-field solution and also compute it numerically. The same solution is also found to describe near-field asymptotics in an analogous large-order limit of solitons on a zero-background and a certain type of extreme focusing in geometrical optics. These properties lead us regard the rogue wave of infinite order as a new special function. At the end of the talk we will highlight some other projects that were carried out jointly with PhD students at Michigan.
Large-amplitude membrane flutter in inviscid flow
We study the dynamics of thin membranes---extensible sheets with negligible bending stiffness---initially aligned with a uniform inviscid background flow. This is a benchmark fluid-structure interaction that has previously been studied mainly in the small-deflection limit, where the flat state may be unstable. Related work includes the shape-morphing of airfoils and bat wings. We study the initial instability and large-amplitude dynamics with respect to three key parameters: membrane mass density, stretching rigidity, and pretension. When both membrane ends are fixed, the membranes become unstable by a divergence instability and converge to steady deflected shapes. With the leading edge fixed and trailing edge free, divergence and/or flutter occurs, and a variety of periodic and aperiodic oscillations are found. With both edges free, the membrane may also translate transverse to the flow, with steady, periodic, or aperiodic trajectories.
Nonintegrable Constraints in Dynamics and Mechanics
Classical mechanical systems are generally described by the least action principle; a consequence is the impossibility of asymptotic stability. On the contrary, systems with nonholonomic constraints (systems with nonintegrable velocity constraints) do not obey the least action principle which can lead to more exotic behavior. This talk will introduce some basic concepts of nonholonomic mechanics with various examples.
Boundary integral methods for Stokes flow in complex geometries
Developing an efficient numerical solver for simulating fluid phenomena dominated by viscous effects in complex domains is a challenging problem. A boundary integral method is presented which achieves spectral accuracy of handling the hydrodynamic interactions, including boundaries with sharp corners or in close proximity. We will also showcase simulation on cilia driven mixing and transport in complex geometries and optimization of Stokesian peristaltic pump for particle transport.
Riemann surfaces and dynamics
The trajectory of a billiard ball, on a table of arbitrary shape, is a nice example of a dynamical system. When the table is a polygon whose angles are rational multiples of pi, we may construct an associated Riemann surface together with a holomorphic 1-form. This allows us to enter the world of complex algebraic geometry, which is a rich landscape of intricate tools and powerful theorems. I will discuss the passage from billiards to Riemann surfaces, and give an example of the application of algebraic geometry to basic questions in billiards dynamics.
Galois groups and Galois representations
The absolute Galois group of Q is an object of fundamental importance in number theory, encoding the symmetries of all algebraic extensions of the rational numbers. Despite its significance, there are still many open questions about this group. (For example: is every finite group a quotient of the absolute Galois group of Q?) One way of understanding this creature is by linearizing the situation; that is, we want to examine how this group acts on vector spaces. I'll give a brief introduction to this world of Galois representations, and how they lead us into the Langlands program.