Seminars and Colloquia by Series

Lyapunov exponent of random dynamical systems on the circle

Series
CDSNS Colloquium
Time
Friday, March 12, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add'l notes for link)
Speaker
Dominique MalicetUniversity Paris-Est Marne la vallée

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

We consider a sequence of compositions of orientation preserving diffeomorphisms of the circle chosen randomly with a fixed distribution law. There is naturally associated a Lyapunov exponent, which measures the rate of exponential contractions of the sequence. It is known that under some assumptions, if this Lyapunov exponent is null then all the diffeomorphisms are simultaneously conjugated to rotations. If the Lyapunov exponent is not null but close to 0, we study how well we can approach rotations by a simultaneous conjugation. In particular, our results can apply to random products of matrices 2x2, giving quantitative versions of the classical Furstenberg theorem.

Linear multistep methods for learning dynamics

Series
School of Mathematics Colloquium
Time
Thursday, March 11, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Qiang DuColumbia University

Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. The latter has been around in various settings such as the model reduction of multiscale processes. It has received particular attention recently in the data-driven modeling via deep/machine learning. Indeed, solving both forward and inverse problems forms the loop of informative and intelligent scientific computing. A natural question is whether a good numerical integrator for discretizing prescribed dynamics is also good for discovering unknown dynamics. This lecture presents a study in the context of Linear multistep methods (LMMs).

Bekolle-Bonami estimates on some pseudoconvex domains

Series
Analysis Seminar
Time
Wednesday, March 10, 2021 - 02:00 for 1 hour (actually 50 minutes)
Location
Speaker
Nathan WagnerWashington University, St Louis

The Bergman projection is a fundamental operator in complex analysis. It is well-known that in the case of the unit ball, the Bergman projection is bounded on weighted L^p if and only if the weight belongs to the Bekolle-Bonami, or B_p, class. These weights are defined using a Muckenhoupt-type condition. Rahm, Tchoundja, and Wick were able to compute the dependence of the operator norm of the projection in terms of the B_p characteristic of the weight using modern tools of dyadic harmonic analysis. Moreover, their upper bound is essentially sharp. We establish that their results can be extended to a much wider class of domains in several complex variables. A key ingredient in the proof is that favorable estimates on the Bergman kernel have been obtained in these cases. This is joint work with Zhenghui Huo and Brett Wick. 

Degree conditions for Hamilton cycles

Series
Graph Theory Seminar
Time
Tuesday, March 9, 2021 - 23:30 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Richard LangHeidelberg University

Please Note: Note the unusual time!

A classic theorem of Dirac (1952) states that a graph in which every vertex is connected to half of the other vertices contains a Hamilton cycle. Over the years this result has been generalized in many interesting ways. In this talk, I will give an overview of these efforts and then explore some of the more recent developments.

Group Synchronization via Cycle-Edge Message Passing

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 8, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Gilad LermanUniversity of Minnesota

The problem of group synchronization asks to recover states of objects associated with group elements given possibly corrupted relative state measurements (or group ratios) between pairs of objects. This problem arises in important data-related tasks, such as structure from motion, simultaneous localization and mapping, Cryo-EM, community detection and sensor network localization. Two common groups in these problems are the rotation and symmetric groups. We propose a general framework for group synchronization with compact groups. The main part of the talk discusses a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios. Under our mathematical model of adversarial corruption, it can be used to infer the corrupted group ratios and thus to solve the synchronization problem. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish competitive theoretical results under a uniform corruption model. Finally, we discuss the MPLS (Message Passing Least Squares) or Minneapolis framework for solving real scenarios with high levels of corruption and noise and with nontrivial scenarios of corruption. We demonstrate state-of-the-art results for two different problems that occur in structure from motion and involve the rotation and permutation groups.

Rotor-routing reachability is easy, chip-firing reachability is hard

Series
Combinatorics Seminar
Time
Friday, March 5, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/751242993/PASSWORD (To receive the password, please email Lutz Warnke
Speaker
Lilla TóthmérészEötvös Loránd University

Chip-firing and rotor-routing are two well-studied examples of Abelian networks. We study the complexity of their respective reachability problems. We show that the rotor-routing reachability problem is decidable in polynomial time, and we give a simple characterization of when a chip-and-rotor configuration is reachable from another one. For chip-firing, it has been known that the reachability problem is in P if we have a class of graphs whose period length is polynomial (for example, Eulerian digraphs). Here we show that in the general case, chip-firing reachability is hard in the sense that if the chip-firing reachability problem were in P for general digraphs, then the polynomial hierarchy would collapse to NP.

Talk based on https://arxiv.org/abs/2102.11970

Dynamics of movement in complex environments

Series
Mathematical Biology Seminar
Time
Friday, March 5, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Sarah OlsonWorcester Polytechnic Institute

In this talk, we will highlight two different types of movement in viscosity dominated environments: sperm navigation and centrosome clustering in dividing cells.  Sperm often interact with chemicals and other proteins in the fluid, changing force generation and emergent swimming trajectories. Recently developed computational methods and asymptotic analysis allow for insight into swimming efficiency and hydrodynamic interactions of swimmers in different fluid environments. We will also show how parameter estimation techniques can be utilized to infer fluid and/or swimmer properties. For the case of centrosome movement, we explore how cancer cells can cluster additional centrosomes and proceed through either a bipolar or multipolar division. The models focus on understanding centrosome movement during cell division, which is the result of complex interactions between stochastic microtubule dynamics and motor proteins in the viscous cytoplasm of the cell.

Meeting Link: https://gatech.bluejeans.com/348270750

Synchronization in Markov random networks

Series
CDSNS Colloquium
Time
Friday, March 5, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see add'l notes for link)
Speaker
Shirou WangU Alberta

Please Note: Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09

Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenario.  By studying the asymptotics of the Markov perturbed stationary distributions, probabilistic characterizations of the alternating pattern between synchronization and desynchronization behaviors is given.  More precisely, it is shown that if a random network without intrinsic noise perturbation is synchronized, then after intrinsic noise perturbation high-probability synchronization and low-probability desynchronization can occur intermittently and alternatively in time, and moreover, both the probability of (de)synchronization and the proportion of time spent in (de)synchrony can be explicitly estimated.

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