Howard University - Mathematics Colloquium - Spring 2024

• 26 January 2024
  
  Speaker: Rodrigo Treviño, University of Maryland
  Title: The Lorentz Gas and Quasicrystals
  Click to read the Abstract.
  The Lorentz gas model dates to the early 20th century, when it was proposed as a model for the free electron in a crystal. The study of the Lorentz gas has driven many of the major developments in ergodic theory, starting with the works of Sinai in the late 60's and early 70's, and by now the theory of Lorentz gases for solids with periodic structure is very sophisticated and advanced. In the early 80's Shechtman discovered quasicrystals, which are solids whose atomic structure is not modeled by any periodic structure such as a lattice, the discovery of which led to a Nobel prize in 2011. Very little is known about the Lorentz gas for solids with aperiodic structure such as quasicrystals. In this talk I will talk about recent developments in the theory of Lorentz gases for solids with aperiodic structure, including joint work with A. Zelerowicz.
• 2 February 2024
  No Colloquium: Mental Health Day
• 9 February 2024
  
  Speaker: Timothy Myers, Howard University
  Title: Constructive Existence Results for a Nonlinear Elliptic PDE with a Nonlinear Boundary Condition
  Click to read the Abstract.
  Let $\Omega$ represent a bounded domain in $n$-dimensional Euclidean space. We will consider the nonlinear elliptic PDE with a nonlinear boundary condition $$ \begin{cases} -\Delta u+u&=f(x,u)\;\;\;\text{ in }\Omega\\ \phantom{-\Delta u}\displaystyle\frac{\partial u}{\partial \eta}&=g(x,u)\;\;\;\text{ on } \partial\Omega\\ \end{cases} $$ where $f$ and $g$ are real-valued Caratheodory functions respectively on $\Omega\times\Bbb R$ and $\partial\Omega\times\Bbb R$ that simultaneously satisfy a special monotonicity property or growth condition.

  For each of the monotone and non-monotone cases we will construct a weak minimal solution $u_*$ and a weak maximal solution $u^*$ between a given pair of weak sub- and supersolutions $\underline{u}$ and $\overline{u}$. Furthermore, this construction will show that if $u$ is any weak solution bounded by $\underline{u}$ and $\overline{u}$, then in fact u is actually bounded by $u_*$ and $u^*$. This work advances the case where $f = 0$. Also, in the non-monotone case we allow the growth of the non-linearities $f$ and $g$ to go to critical exponents of the appropriate dual Lebesgue spaces; whereas earlier results evidently required strictly larger exponents.

• 16 February 2024
  
  Speaker: Roberto De Leo, Howard University
  Title: The graph of a Dynamical System
  Click to read the Abstract.
  Ever since Poincare' times, understanding the qualitative behavior of a system is one of the main goals of Dynamical Systems' theory. The points with the simplest dynamics are those belonging to periodic orbits. The concept of 'periodic' was then generalized to 'recurrent' (a point $x$ is recurrent if its orbit gets arbitrarily close to $x$ as $t\to\infty$), 'non-wandering' (a point $x$ in non-wandering if, arbitrarily close to $x$, there are orbits that get back close to $x$ for $t\to\infty$) and more sophisticated types. Since Smale showed that the non-wandering set decomposes into basic 'pieces', each piece being the analogue of a 'periodic orbit', the decomposition of the non-wandering set became an important ingredient of the qualitative study of a dynamical system. This decomposition, though, does not say anything about the behavior of points outside of the non-wandering set. In this talk I will present a complementary ingredient recently introduced by Jim Yorke and myself, namely how to encode the full qualitative behavior of a dynamical system into a graph. The nodes of the graph are the pieces of the decomposition above; its edges describe the dynamics of all other points. The exposition will be mainly based on examples and pictures.
• 23 February 2024
  
  Speaker: Carlos Rocha, Instituto Superior Técnico, Lisboa (Portugal)
  Title: Graphs of Sturm global attractors
  Click to read the Abstract.
  Global attractors of semiflows generated by scalar semilinear parabolic partial differential equations are called Sturm global attractors. Here we survey the geometric and combinatorial characterization results for Sturm global attractors. This is based on joint work with B. Fiedler.
• 1 March 2024
  
  Speaker: Canceled,
  Title:
  Click to read the Abstract.
• 8 March 2024
  No Colloquium: Spring recess
• 15 March 2024
  
  Speaker: Dave Anderson, Ohio State University
  Title: Degeneracy loci in geometry and combinatorics
  Click to read the Abstract.
  Given a matrix of homogeneous polynomials, there is "degeneracy locus" of points where specified submatrices drop rank. These loci are ubiquitous, and formulas for their degrees go back to Cayley and Salmon in the mid-1800s. The search for more general and refined degree formulas led to a rich interaction between geometry and combinatorics in the late 20th century, and that interplay continues today. I will describe some of this history, along with recent and new formulas relating the geometry of degeneracy loci with the combinatorics of Schubert polynomials.
• 22 March 2024
  
  Speaker: Michael Scioletti, West Point
  Title: Understanding the importance of heuristics in computational multivariable constrained nonlinear optimization
  Click to read the Abstract.
  Often students ignore checking of the underlying assumptions or conditions in optimization, relying strictly on computer algebra systems such as Maple, Mathematica, and/or solvers in Excel or other off the commercial software to deliver the solution. In an example from our nonlinear optimization classes, we see a rather simple problem that mimics typical convexity rules that allow for most if not all, analytical constrained optimization methods that we normally teach; however, it fails sufficiency conditions and students miss it. Instead, they report non-optimal solutions. We provide the example, discuss the issues, show enumeration examples in Excel and code in Python for obtaining the optimal solution, and recommend a common heuristic procedure and provide code in MATLAB and Python to solve the problem with a basic introduction to help them understand associated parameter settings. Needless to say these type problems might appear more often in business, industry or government. We feel solution methods should be taught to deal with these type problems. At the very least, we are teaching students to be thorough and try multiple solution techniques, which I like to call quality reps (albeit in the classroom)!
• 29 March 2024
  
  Speaker: Hans Engler, Georgetown University
  Title: The Lorenz System of 1996
  Click to read the Abstract.
  The meteorologist and applied mathematician Edward Lorenz is famous for discovering chaotic behavior in dynamical systems in 1963. In 1996, Lorenz introduced a dynamical system that describes very simple ``weather'' on a cartoon planet: a scalar quantity evolves on a circular array of N sites, undergoing radiative forcing, dissipation, and nonlinear advection. Lorenz proposed this system as a test bed for numerical weather prediction. Since then, it has found much use as a test case in data assimilation. Related systems have been studied by other authors earlier.

  Mathematically, this is an nonlinear N-dimensional dynamical system that is invariant under rotating the sites. There is a single parameter, namely forcing strength. For small forcing strength, there is no ``weather'': the only possible stable solutions are constant in space and time. As the strength of the forcing increases, periodic wave patterns appear that move around the circle of sites. These periodic patterns are not unique - the same forcing strength may be associated with stable patterns that are qualitatively different, depending on the system's initial state. For even larger forcing, the motion becomes chaotic. Regular wave patterns are replaced by moving irregular wave trains that are short-lived, similar to changing weather systems that move across a landscape.

  The talk will introduce the main properties of this and of related systems. The appearance of periodic solutions can be explained with bifurcation theory for all these versions. By using the discrete Fourier transform, the stability of bifurcating periodic solutions and the coexistence of multiple such solutions for the same radiative forcing can be understood with techniques from elementary complex analysis. This is joint work with John Kerin, a former Georgetown University student.
• 5 April 2024
  
  Speaker: Pierre Berger, Institut de Mathematiques de Jussieu-Paris Rive Gauche, Paris (France)
  Title: Emergence of wild dynamics
  Click to read the Abstract.
  In dynamical systems, a main question is the description of the asymptotic and statistical behaviors of most of the orbits for a typical system. The quasi-periodic and chaos theories provide many examples of ergodic systems: a.e. orbit is equi-distributed following the same law. We will present examples of dynamics which are wild in the sense that their statistical behavior is extremely complicated; in particular they display infinitely many simultaneous statistical behaviors. We will discuss the typicality of such systems and how to understand them. The presented results will be in many settings: dissipative dynamics, symplectic dynamics, complex dynamics or steady Euler flow.
• 12 April 2024
  
  Speaker: Andrea Giacobbe, University of Catania (Italy)
  Title: A geometric approach to bifurcations of linear systems
  Click to read the Abstract.
  A parameter-dependent dynamical system is always subject to bifurcations that effect its qualitative evolution in general, and the stability of the equilibria in particular. The results of Routh, and the so called Routh-Hurwitz conditions are the most commonly used techniques to investigate the linear stability of an equilibrium, and to define the domain in parameter space in which a given equilibrium is stable. In this presentation we revisit these results concentrating on marginal ipersurfaces: at which bifurcations takes place.

  The marginal ipersurfaces are algebraic and semi-algebraic varieties in invariant space, that can be pulled-back to the space of parameters of the dynamical system. They possess a stratified structure and a labelling due to the type and the codimension of bifurcations.

  Due to Grobman-Hartman theorem, generic equilibria can be associated to four natural numbers (whose sum is the dimension of the phase space), a bifurcation generates a change in such numbers, and is connected to one of the following marginal events:

  • an eigenvalues vanishes;
  • two complex-conjugate eigenvalues touch the purely imaginary axes;
  • a real eigenvalue has higher multiplicity.

  An analysis of the characteristic polynomial allows to characterise these marginal events, to define ipersurfaces that separate the parameter space in connected domains in which the four natural numbers are constant, and to characterise the ipersurfaces depending on the type of change that takes place crossing them. The four natural numbers can be computed analytically, using residues and a sequence of integers called Sturm's sequence.

  The analysis can be extended to the Hamiltonian case, and to Floquet theory.

• 19 April 2024
  
  Speaker: Jim Wiseman, Agnes Scott College
  Title: TBA
  Click to read the Abstract.
• 26 April 2024
  
  Speaker: TBA,
  Title: TBA
  Click to read the Abstract.