Posted September 6, 2023

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm - 2:30 pm Lockett 233
Nilangshu Bhattacharyya, Louisiana State University

Characteristic Classes - Lecture 3

Posted August 30, 2023

Last modified September 1, 2023

Xiaoqi Huang, Louisiana State University

Curvature and growth rates of log-quasimodes on compact manifolds

We will discuss the relation between curvature and L^q norm estimates of spectral projection operators on compact manifolds. We will present a new way that one can hear the shape of a connected compact manifold of constant sectional curvatures, if the shape refers to curvature, and the radios used are the L^q norm of quasimodes. This is based on ongoing work with Christopher Sogge.

Posted August 30, 2023

Last modified September 18, 2023

Xiaoqi Huang, Louisiana State University

Curvature and growth rates of log-quasimodes on compact manifolds

We will discuss the relation between curvature and L^q norm estimates of spectral projection operators on compact manifolds. We will present a new way that one can hear the shape of a connected compact manifold of constant sectional curvatures, if the shape refers to curvature, and the radios used are the L^q norm of quasimodes. This is based on ongoing work with Christopher Sogge.

Posted September 6, 2023

Last modified September 22, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Ana Bălibanu, Louisiana State University

Moment maps and multiplicative reduction

Symplectic reduction is a process that eliminates the symmetries of a Poisson manifold equipped with a Hamiltonian group action. Many algebraic varieties which are of interest to representation theory arise as reductions of symplectic spaces associated to algebraic groups. We introduce several new reduction procedures, some of which are multiplicative analogues of ”classical” examples of symplectic reduction. This is joint work with Maxence Mayrand.

Posted September 18, 2023

Colloquium Questions or comments?

3:30 pm - 4:20 pm Lockett 232
Wilhelm Schlag, Yale University

Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory

In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.

Posted August 18, 2023

Last modified September 11, 2023

Control and Optimization Seminar Questions or comments?

10:30 am - 11:20 am Zoom (Click “Questions or Comments?” to request a Zoom link)
Cristina Pignotti, Università degli Studi dell'Aquila

Consensus Results for Hegselmann-Krause Type Models with Time Delay

We study Hegselmann-Krause (HK) opinion formation models in the presence of time delay effects. The influence coefficients among the agents are nonnegative, as usual, but they can also degenerate. This includes, e.g., the case of on-off influence, namely the agents do not communicate over some time intervals. We give sufficient conditions ensuring that consensus is achieved for all initial configurations. Moreover, we analyze the continuity type equation obtained as the mean-field limit of the particle model when the number of agents goes to infinity. Finally, we analyze a control problem for a delayed HK model with leadership and design a simple control strategy steering all agents to any fixed target opinion.

Posted September 25, 2023

Combinatorics Seminar Questions or comments?

2:30 pm - 3:30 pm Zoom (Please email zhiyuw at lsu.edu for Zoom link)
Xiaonan Liu, Vanderbilt University

Counting Hamiltonian cycles in planar triangulations

Whitney showed that every planar triangulation without separating triangles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $\log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and, in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles. Joint work with Zhiyu Wang and Xingxing Yu.

Posted September 24, 2023

Algebra and Number Theory Seminar Questions or comments?

3:20 pm - 4:10 pm Lockett 233 or click here to attend on Zoom
Cheng Chen, University of Minnesota

TBA

Posted September 6, 2023

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm - 2:30 pm Lockett 233
Colton Sandvick, Louisiana State University

Characteristic Classes - Lecture 4

Posted August 23, 2023

Last modified September 25, 2023

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Yuan Yao, Sorbonne University

Morse-Bott theory and embedded contact homology

I will give an overview of embedded contact homology as a Floer theory that counts pseudo-holomorphic curves asymptotic to Reeb orbits. Then I shall explain how to compute this homology theory in the Morse-Bott setting, via an enumeration of J-holomorphic cascades