Howard University - Mathematics Colloquium - Fall 2024

• 13 September 2024
  
  Speaker: Stefano Olla, University of Paris Dauphine (France) and GSSI l'Aquila (Italy)
  Title: Diffusive behavior in extended completely integrable systems (Video, Slides)
  Click to read the Abstract.
  On a diffusive space-time scaling density fluctuations behave very differently in extended completely integrable systems with respect to chaotic systems. I will expose some recent results concerning the one dimensional hard rods infinite dynamics and the box-ball cellular automata (an ultradiscretization of KdV equation). Joint works with Pablo Ferrari, Makiko Sasada, Hayate Suda.
• 16 September 2024
  
  Speaker: Ravi Vakil, Stanford, President-elect of the AMS
  Title: The Mathematics of Doodling (Video, Slides)
  Click to read the Abstract.
  Doodling is a creative and fundamentally human activity, resulting in doodles with intricate and often hidden implicit structure. We will treat doodles as an example for how mathematics done is by starting with some doodles, we will ask ourselves some natural questions and see where they take us. They will lead us to some unexpected places, and to some sophisticated mathematics.
• 20 September 2024
  
  Speaker: ,
  Title:
  Click to read the Abstract.
• 27 September 2024
  
  Speaker: Jim Yorke, University of Maryland and IPST
  Title: What is the graph of a dynamical system?
  Click to read the Abstract.
  Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations.
  We define two types of graphs: what we call a "trajectory graph" and a "Morse graph", which adds subtleties. Each node of a trajectory graph is a limit set of a trajectory, and there is an edge between two nodes if, for every $\varepsilon>0$, there is a trajectory passing within $\varepsilon$ from both nodes. In particular, there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Morse graphs are more varied, and the literature has several definitions of graphs of this type. In this talk we will illustrate some examples of graphs of dynamical systems, including infintie-dimensional ones.
• 4 October 2024
  
  Speaker: Eitan Tadmor, University of Maryland
  Title: Swarm-Based Gradient Descent Methods for Non-Convex Optimization
  Click to read the Abstract.
  We discuss a novel class of swarm-based gradient descent (SBGD) methods for non-convex optimization. The swarm consists of agents, each is identified with position, x, and mass, m. There are two key ingredients in the SBGD dynamics: (i) persistent transition of mass from agents at high to lower ground; and (ii) time stepping protocol which decreases with m.

  The interplay between positions and masses leads to dynamic distinction between "leaders" and "explorers": heavier agents lead the swarm near local minima with small time steps; lighter agents use larger time steps to explore the landscape in search of improved global minimum, by reducing the overall "loss" of the swarm.

  Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer.
• 25 October 2024
  
  Speaker: Christopher Kim, Howard University
  Title: Recent advances in neuroscience and the role of computational models of the brain
  Click to read the Abstract.
  Thanks to recent technological advances in neuroscience, scientists are now able to view and monitor the activity of multiple regions of the brain at cellular resolution. It is possible to reconstruct the connectivity between hundreds of thousands of neurons, identify the types of neurons in the whole brain, and measure and perturb neural activity of individual neurons in multiple brain regions. On the other hand, computational models of neurons or network of neurons have been investigated since the development of Hodgkin-Huxley models to understand the computational principles of neural system. How can we utilize computational models of network of neurons to better make sense of newly emerging large-scale experimental datasets? In this talk, I will briefly survey the technological advances in neuroscience and discuss recent work on how biologically plausible neural network models can be interweaved with experimentally recorded neural activity through machine learning methods to provide insights into neural computations performed by the brain.
• 1 November 2024
  
  Speaker: Mohamed Omar, York University (Canada)
  Title: Search for the Lost Determinant
  Click to read the Abstract.
  For the past 7-8 years I've been working on trying to prove a class of matrices (that arise in a combinatorial setting) have determinant at most 1. Every year or so I come back to the problem with renewed light and almost every time I learn new linear algebra theorems and how they apply. This talk is a walk through the journey in this almost decade-long escapade.
• 15 November 2024
  
  Speaker: Tomas Wanner, George Mason University
  Title: Bifurcation point validation using extended systems
  Click to read the Abstract.
  Parabolic partial differential equations are fundamental models for many applied phenomena. For example, pattern formation in diblock copolymers is described by the evolution of the fourth-order Ohta-Kawasaki equation, while competition in the interaction of populations can be modeled by second-order reaction-diffusion systems such as the Shigesada-Kawasaki-Teramoto model. Essential for a deeper understanding of the long-term dynamics of such problems are the sets of equilibria as described by the associated bifurcation diagrams. In this talk, we present computer-assisted proof techniques which can be used to validate and continue bifurcation points through the use of suitable extended systems. This includes not only fold points, but also pitchfork and transcritical bifurcations which are the result of group actions beyond forcing through involutions. Our results apply to both one- and two-dimensional domains, and they can also be used to treat certain bifurcation points with higher-dimensional kernels.
• 22 November 2024
  
  Speaker: Mark Meyer, Georgetown University
  Title: TBA
  Click to read the Abstract.