Please join us for a colloquium presentation this Friday (09/16) at 3:30pm in Keller 302.
The speaker will be our new colleague Chuang Xu. Please find the title and abstract below.
Title: Mean field limit of heterogeneous networks of higher order
Abstract: Networks are widely used in modelling science phenomena
(chemical reaction networks, brain networks, epidemic networks,
ecological networks, etc.). Almost all networks in nature are
heterogeneous. Such heterogeneity together with the huge size of the
network makes it challenging to study the dynamics of these networks
analytically or numerically. It is well known that network of higher
order interactions (H.O.I.) can invoke new dynamics. In this talk, I
will briefly introduce a recent work on the mean field limit of
heterogeneous networks of H.O.I.. Hypergraphs are generally used to
capture the underlying graphical structure of these networks. I will
first review the literature and the concepts of hypergraphs and graph
limits, and explain the main difficulty. Then I will introduce our work
on the mean field limit of heterogeneous networks, by introducing limits
of hypergraphs. Finally, I will mention some applications in physics,
epidemiology and ecology. This is a joint work with Christian Kuehn.
Join us for a colloquium presentation this Friday (09/23) at 3:30pm in Keller 302.
The speaker will be our colleague Yash Lodha.
—-Title: Adventures in the land
Abstract: The groups of homeomorphisms of the real line and the circle exhibit a remarkable subgroup structure, and consequently, a plethora of topological, algebraic and combinatorial phenomena. In this talk I shall provide some historical motivation to study such groups, and describe some concrete examples that illustrate the theory. If time permits, I will present some recent advances in the area.
formalization of multivariable calculus in Mizar. We are now formalizing
multivariable calculus with the aim of formalizing differential
manifolds, partial differential equations, and numerical analysis. In
the formalization of multivariable calculus, the handling of
higher-order partial derivatives tends to be more troublesome than in
the one variable case, but this can be alleviated somewhat by
introducing the Fréchet derivative in Banach space. I would like to give
an overview of the Mizar project and discuss topics ranging from the
definition of higher-order partial derivatives in normed spaces to
formalization of the implicit and inverse function theorems.</p>
Daniel Erman (University of Wisconsin)
Ultraproducts, Hilbert’s Syzygy Theorem, and Stillman’s Conjecture
When and where:
3-3:50pm, December 7, in Keller 403
Hilbert’s Syzygy Theorem is a classic finiteness result about
a construction in algebra known as a free resolution.
Stillman once proposed an analogue of Hilbert’s result, which involved potentially considering polynomials in infinitely many variables.
Stillman’s Conjecture was recently solved, and perhaps the simplest proof is based upon a novel use of an ultraproduct.
I’ll give an expository overview of the history of Stillman’s Conjecture (very little algebraic background will be assumed), and then explain how and why ultra products came to play such a key role.