Science Talk for Families — Richard Schwartz

Richard Schwartz
Chancellor’s Professor of Mathematics, Brown University
October 10, 2015
Richard Schwartz, Member (2003–04) in the School of Mathematics and the Chancellor’s Professor of Mathematics at Brown University, will talk about really large numbers, as well as the different sizes of infinity and the different forms of infinite space. He will illustrate the talk with excerpts from his books, Really Big Numbers, Life on the Infinite Farm, and Gallery of the Infinite. All children and families are welcome. The talk is geared for ages 8 and up.

Factors of polynomials of low individual degree

Rafael Oliveira
Princeton University
October 12, 2015
In [kal89], Kaltofen proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen's work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in [KSS14], and the question of bounding the depth of the circuits computing the factors of a polynomial.

3-manifold groups

Ian Agol
University of California, Berkeley; Distinguished Visiting Professor, School of Mathematics
October 12, 2015
I'll review recent progress on properties of 3-manifold groups, especially following from geometric properties. Then I'll discuss some open questions regarding 3-manifold groups, including their profinite completions, torsion in covers, orderability, and $PD(3)$ groups.

Non-constructive combinatorics

Noga Alon
Tel Aviv University; Visiting Professor, School of Mathematics
October 13, 2015
I will describe several old and new applications of topological and algebraic methods in the derivation of combinatorial results. In all of them the proofs provide no efficient solutions for the corresponding algorithmic problems. Finding such solutions is an intriguing challenge.

The four-color theorem and an instanton invariant for spatial graphs I

Peter Kronheimer
Harvard University
October 13, 2015
Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional $\mathbf{Z}/2$ vector space. The main result about the instanton homology is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, its instanton homology is non-zero.

The four-color theorem and an instanton invariant for spatial graphs II

Tomasz Mrowka
Massachusetts Institute of Technology
October 13, 2015
Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional $\mathbf{Z}/2$ vector space. The main result about the instanton homology is a non-vanishing theorem, proved using techniques from 3-dimensional topology: if the graph is bridgeless, its instanton homology is non-zero.