Spring 2023

• Joint talk with Department Colloquium 

• • Date: Friday, January13

  • Time: 3:30-4:30 PM 

  • Location: MP3314 

  • Speaker: Dr. Yangbo Ye (University of Iowa)

  • Title: Distribution of neighboring values of the Liouville and Möbius functions

  • Abstract: The security of most public-key cryptography systems is based on the belief that prime factorization of a large integer n cannot be done in polynomial time of log n (in P). This belief, however, has no theoretical or scientific evidence. An increasing number of number theorists, including the speaker, believe that the opposite is actually true. To look for evidence of a possible factorization algorithm in P, one considers a seemingly simpler question: are there algorithms for the Liouville function λ(n) and/or the Möbius function μ(n) in P? These functions are defined multiplicatively and contain much less information than a full factorization of n, but the most efficient methods to compute them are still based on factorization of n.

While finding divisors of an integer by a multiplicative method is not in P, finding the greatest common divisor of two integers is in P by the Euclidean algorithm. Note that the Euclidean algorithm is an additive algorithm. This suggests that efficient methods to compute λ(n) and μ(n) might be additive algorithms. The goal of this talk is thus to find additive relations among values of λ(n) and among values of μ(n).

• • Date: Friday, March 10

  • Time: 3:30-4:30 PM 

  • Location: MP3314 and ZOOM

  • Speaker: Dr. Shijun Zheng (Georgia Southern University)

  • Title: Energy-critical Solitons in a magnetic field 

  • Abstract: The existence problem of magnetic soliton states has been a fascinating topic 

in geometric optics, plasma, condensed matter as well as superfluids. In quantum physics, a vortex type soliton solution indicates a certain topological defect, where quantized vorticity is a signature phenomenon of fluid flow. 

In this talk, I shall review recent progress on the threshold dynamics for rotating BEC and magnetic NLSE. Numerical construction will be given for vortex solitons with rotating phase which satisfies Euler equation for classical fluids.  We shall observe the symmetric breaking in the presence of an anisotropic potential along with their dynamical trajectories in the sense of statistical "clouds".

• • Date: Friday, March 24

  • Time: 3:30-4:30 PM 

  • Location: MP3314 and ZOOM

  • Speaker: Dr. Shibo Liu (Florida Institute of Technology)

  • Title: Morse theory and its applications to nonlinear differential equations

  • Abstract: In the first part of this talk we quickly review some basic concepts in Morse theory (critical groups and More inequalities), then in the setting of saddle point reduction, we investigate the relation between the critical groups of the original functional and the reduced functional. The abstract results are used to get multiple solutions for elliptic BVPs over bounded domain.

In the second part, using Morse theory we discuss the existence of nonzero solutions for stationary nonlinear Schrodinger type equations with indefinite Schrodinger operator. Finally, by applying a variant of the Clark’s theorem we study a quasilinear Schrodinger equation whose nonlinear term can grow super critically. Infinitely many solutions with negative energy are obtained.