We will survey the class of linear groups, and identify three methods of constructing linear groups: algebraic, geometric, and arithmetic. We will propose that every subgroup comes from one of these types of constructions. There are many interesting consequences all across group theory and geometry if this is indeed the case. Focusing on the case of 2-by-2 matrix groups, the algebraic and arithmetic subgroups are very well-understood. We will fill in some understanding of geometric subgroups in this setting, by showing that for any prime p, there is a closed surface group in \({\rm PSL}_2(\mathbb{Z}[1/p])\). These surface groups will have genus proportional to \(p\), and we speculate that the construction is optimal.

Classical extremal results in graph theory (such as Turán's theorem) concern the maximal size of of a graph of given order and without certain subgraphs. Bollobás, Erdős, and Szemerédi in 1975 studied extremal problems in multipartite graphs. One of their problems (in its complementary form) was determining the maximal degree of a multipartite graph without an independent transversal. This problem has received considerable attention and was settle in 2006 (Szabó--Tardos and Haxell--Szabó). Other questions asked by Bollobás, Erdős, and Szemerédi remain open, such as determining:

(1) the maximum degree in a multipartite graph without a partial independent transversal, and;
(2) the minimum degree that forces an octahedral graph in balanced tripartite graphs.

In this talk I will survey recent progress on these and other related problems.

In this talk, I will discuss some new results on intersection graphs of pseudo-segments in the plane and their applications in graph drawing.  These results are joint work with Jacob Fox and Janos Pach.
 

Micrometer-sized particles made from responsive polymer networks (that is, responsive microgel colloids) are of high potential for the design of functional soft materials due to their adaptive compressibility and stimuli-triggered volume transition. In this talk, I will discuss models and theoretical approaches, such as Langevin simulations and classical (dynamic) density functional theory (DFT), to describe the structural and dynamical behavior of dispersions of these responsive colloids in and out of equilibrium. Moreover, I will argue that chemical fueling and the inclusion of chemomechanical feedback loops may lead to excitable and oscillatory dynamics of the active colloids, establishing first steps to a well-controlled design of artificial cells and their emergent behavior.

Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

Aiming to understand the solvability of semilinear elliptic boundary values problems in bounded domains, we will review the best known techniques for establishing the existence of critical points of functionals whose critical points are solutions to such problems. The mountain pass lemma, the Nehari manifold,
the Morse index, and bifurcation analysis will be discussed to conclude the existence of seven solutions for an asymptotically linear elliptic problem.