Georgia Institute of Technology College of Sciences

We will try to address a few universality questions for the behavior of large random matrices over finite fields, and then present a minimal progress on one of these questions.

The Euler-Alignment system arises as a macroscopic representation of the Cucker-Smale model, which describes the flocking phenomenon in animal swarms. The nonlinear and nonlocal nature of the system bring challenges in studying global regularity and long time behaviors. In this talk, I will discuss the global wellposedness of the Euler-Alignment system with three types of nonlocal alignment interactions: bounded, strongly singular, and weakly singular interactions. Different choices of interactions will lead to different global behaviors. I will also discuss interesting connections to some fluid dynamics systems, including the fractional Burgers equation, and the aggregation equation.

The Four Color Theorem states that every planar graph is 4-colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}-subdivision is k-colorable. However, Catlin disproved Hajos' conjecture for k >= 6. It is not hard to prove that the conjecture is true for k <= 3. Hajos' conjecture remains open for k = 4 and k = 5. We consider a minimal counterexample to Hajos' conjecture for k = 4: a graph G, such that G contains no K_5-subdivision, G is not 4-colorable, and |V (G)| is minimum. We use Hajos graph to denote such counterexample. One important step to understand graphs containing K_5-subdivisions is to solve the following problem: let H represent the tree on six vertices, two of which are adjacent and of degree 3. Let G be a graph and u1, u2, a1, a2, a3, a4 be distinct vertices of G. When does G contain a topological H (i.e. an H-subdivision) in which u1, u2 are of degree 3 and a1, a2, a3, a4 are of degree 1? We characterize graphs with no topological H. This characterization is used by He, Wang, and Yu to show that graph containing no K_5-subdivision is planar or has a 4-cut, establishing conjecture of Kelmans and Seymour. Besides the topological H problem, we also obtained some further structural information of Hajos graphs.

Let X be a degree d curve in the projective space P^r.

A general hyperplane H intersects X at d distinct points; varying H defines a monodromy action on X∩H. The resulting permutation group G is the sectional monodromy group of X. When the ground field has characteristic zero the group G is known to be the full symmetric group.

By work of Harris, if G contains the alternating group, then X satisfies a strengthened Castelnuovo's inequality (relating the degree and the genus of X).

The talk is concerned with sectional monodromy groups in positive characteristic. I will describe all non-strange non-degenerate curves in projective spaces of dimension r>2 for which G is not symmetric or alternating. For a particular family of plane curves, I will compute the sectional monodromy groups and thus answer an old question on Galois groups of generic trinomials.