Speaker: Jashan Bal
Title: Reproducing Kernel Hilbert Spaces and the Lebesgue Decomposition
Abstract: We introduce Reproducing Kernel Hilbert space theory. These are Hilbert spaces of functions from a set X to the complex numbers with the added property that, evaluation functionals are bounded. This additional property give rise to many interesting results. We then construct the Lebesgue Decomposition of finite, positive, Boreal measures on the unit circle in terms of Reproducing Kernel Hilbert Spaces.
Speaker: Mahsa Nasrollahishirazi
Title: Erdős-Ko-Rado theorem for perfect matchings
Abstract: One of the important results in extremal set theory is the Erdős-Ko-Rado (EKR) theorem which gives a tight upper bound on the size of intersecting sets. The focus of this talk is on extensions of the EKR theorem to perfect matchings. Two perfect matchings are said to be t-intersecting if they have at least t edges in common. In 2017, Godsil and Meagher algebraically proved the EKR theorem for 1-intersecting perfect matchings on the complete graph with 2k vertices. Later in 2017, Lindzey presented an asymptotic refinement of the EKR theorem on perfect matchings. In this talk, we extend their results to 2-intersecting and also to set-wise 2-intersecting perfect matchings. These results are not asymptotic.
Speaker: Junyu Lu
Title: Left Orderability and 3-Manifold Groups
Abstract: If the elements of a group can be given a strict total ordering which is invariant under multiplication on the left, the group is said to be left-orderable. The theory of orderable groups has deep connections with topology and the dynamics of group actions on the circle and real line. I will give a short introduction to left-orderable groups and the L-space conjecture.
Speaker: Romain Branchereau
Title: Modular forms: from number theory to geometry
Abstract: Modular forms are complex functions that play a central role in number theory because they carry a lot of arithmetic informations. I will try to explain what these functions are and how they can sometimes be related to the geometry of certain spaces.
Speaker: Fouad Naderi
Title: An introduction to non-commutative function theory
Abstract: Our view in classical multi-variable function theory towards functions is prejudice, i.e., in formulas like f(x,y,z)=sin(x+y) +xy+x2yz we have given too much priority to real or complex one-dimensional variables x, y and z in (x,y,z). However, Non-commutative function theory (NCFT) attempts to relax this condition by allowing for matrix variables (of the same size), and turning our functions to the so-called non-commutative ones, and it then seeks what happens for the calculus of such functions. Thus, NCFT is a generalization of multi-variable complex analysis to the setting of several non-commuting variables. Though NCFT can be seen as some sort of functional calculus or non-linear operator theory, it is not quite like them since in the NCFT we do not bother ourselves with the spectrum of matrices/operators at all. Furthermore, NCFT should say something about differential calculus, Taylor series, and integral calculus of these new non-commutative functions. In this talk, I will try to touch upon the most important ideas of NCFT without going into technical details and motivate the audience.