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Title: Laplacian Growth: selection and singularities Speaker: Artem Abanov, Texas A&M University Time: 3:00pm‐4:00pm Place: PHY 120
Laplacian growth is the simplest nontrivial growth process. It describes the growth in many different situations, from the growth of bacterial colony to the propagation of lightning to the growth of the domain of inviscid liquid inside the viscose one. It also has many connections to the integrable models, condensed matter, and matrix models. In this talk I will present a simple introduction to the field. I will show the integrability of the problem and outline some of the open questions.
Title: Chinese Rings, coverings, and iterated things Speaker: Zoran Sunik, Texas A&M University Time: 3:00pm‐4:00pm Place: PHY 120
We model the Chinese Rings Puzzle and the Hanoi Towers Problem by self-similar groups of rooted tree automorphisms. We use these examples as an excuse to talk about iterated monodromy groups and self-coverings of topological spaces.
Title: On the Study of Singular Nonlinear Traveling Wave Equations: Dynamical System Approach Speaker: Jibin Li, Kunming University of Science and Technology Time: 4:00pm‐5:00pm Place: PHY 120
Nonlinear wave phenomena are of great importance in the physical world and have been for a long time a challenging topic of research for both pure and applied mathematicians. There are numerous nonlinear evolution equations for which we need to analyze the properties of the solutions for time evolution of the system. As the first step, we should understand the dynamics of their traveling wave solutions.
The aim of this talk is to give a more systematic account for the bifurcation theory method of dynamical systems to find traveling wave solutions with an emphasis on singular waves and understand their dynamics for some classes of the well-posedness of nonlinear evolution equations.
Title: Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system Speaker: Maoan Han, Shanghai Normal University Time: 3:00pm‐4:00pm Place: PHY 120
In this work, we study the analytical property of the first Melnikov function for general Hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of the expansion we give some conditions for the perturbed system to have \(4\), \(5\) or \(6\) limit cycles in a neighborhood of loop. As an application of our main results, we consider some polynomial Lienard systems and find \(4\), \(5\) or \(6\) limit cycles.
Title: Multiple Solutions With Precise Sign Information for Nonlinear Elliptic Equations With Combined Nonlinearities Speaker: Nikolaos S. Papageorgiou, National Technical University of Athens Athens, GREECE Time: 3:00pm‐4:00pm Place: PHY 108
We consider nonlinear elliptic equations driven by the \(p\)-Laplacian differential operator and with a reaction term which involves the combined effects of concave and convex terms or concave and \(p\)-linear terms. Using variational methods based on critical point theory and truncation techniques, we prove multiplicity theorems and provide information about the sign of the solutions (constant sign or nodal solutions). In the semilinear case, i.e., \(p=2\), using Morse theory, we produce additional solutions.
Title: \(n!\) conjecture and the family of ideals Speaker: Kyungyong Lee, Purdue University Time: 3:00pm‐4:00pm Place: PHY 120
The famous \(n!\) conjecture can be stated in an elementary language. In fact it asserts that the dimension of the vector space spanned by all derivatives of a certain bivariate analogue of the \(n\times n\) Vandermonde determinant is equal to \(n!\). This seemingly elementary conjecture was solved by M. Haiman, with a highly nontrivial machinery. The proof is closely related to the geometry of a certain moduli space of ideals, which is isomorphic to a family of ideal projectors on the \(2\)–dimensional plane. These families arise naturally in many branches of mathematics, and many interesting questions remain open. I'll discuss how some of the results in the plane case can or cannot be generalized to the higher dimensional case.