Events

We consider the problem of finding harmonic maps to the circle with a prescribed singular set in an arbitrary Riemannian manifold and characterise their uniqueness in terms of the "one-dimensional topology" of the ambient space. We then show how these maps can be used to define new notions of (n-2)-volume, leading to a promising approximation scheme for classical codimension 2 minimal surfaces.

Intersection theory on A_g and on Mumford's partial compactification

Abstract: Machine learning and data science competitions, wherein contestants submit predictions about held-out data points, are an increasingly common way to gather information and identify experts. One of the most prominent platforms is Kaggle, which has run competitions with prizes up to 3 million USD. The traditional mechanism for selecting the winner is simple: score each prediction on each held-out data point, and the contestant with the highest total score wins. Perhaps surprisingly, this reasonable and popular mechanism can incentivize contestants to submit wildly inaccurate predictions. The talk will begin with intuition for the incentive issues and what sort of strategic behavior one would expect---and when. One takeaway is that, despite conventional wisdom, large held-out data sets do not always alleviate these incentive issues, and small ones do not necessarily suffer from them, as we confirm with formal results. We will then discuss a new mechanism which is approximately truthful, in the sense that rational contestants will submit predictions which are close to their best guess. If time permits, we will see how the same mechanism solves an open question for online learning from strategic experts. Bio: Rafael (Raf) Frongillo is an Associate Professor of Computer Science at the University of Colorado Boulder. His research lies at the interface between theoretical machine learning and economics, primarily focusing on information elicitation mechanisms, which incentivize humans or algorithms to predict accurately. Before Boulder, Raf was a postdoc at the Center for Research on Computation and Society at Harvard University and at Microsoft Research New York. He received his PhD in Computer Science at UC Berkeley, advised by Christos Papadimitriou and supported by the NDSEG Fellowship.

For which link diagrams can we tell, simply with the 2-dimensional diagram information, whether the associated link is prime? Menasco showed that we can for alternating diagrams, and Cromwell, for positive braids. The latter further conjectured that the same would hold for any diagram for which Seifert's algorithm yields a minimal genus surface. This minimality property, if we also required that the diagram be a braid closure, is equivalent to the braid being homogeneous. Since these links are fibered, we can use open book techniques. I will present a criterion for how primeness of fibered links behaves under an operation called Murasugi sums, which we consider to be of independent interest; and explain how we can use it to prove Cromwell's conjecture for braid closures, i.e. for homogeneous braids. This is joint work with Peter Feller and Lukas Lewark.

Let d>1 be an integer and let k be an algebraically closed field with char(k)=0 or char(k)>d. Given a polynomial f in k[x], its critical values C_f are the branching points of the induced map f: P^1 —> P^1. When suitably normalized, polynomials of degree d with given critical values can be parametrized in terms of their critical points, the roots of the derivative, yielding a covering T_d: A^{d-1} —> A^{d-1}. We briefly discuss the geometry of these coverings, giving a generalization of a classical result of Arnold. Motivated by this and by some applications, we consider two questions of arithmetical flavour: given a subfield L of k, for which subsets V of k is there a degree d polynomial f in L[x] with C_f=V, and are two such polynomials always related by a linear change of variable? We answer these questions in the case of quartics over any number field and discuss a link to the arithmetic of elliptic curves, as well as (time permitting) the case of higher degrees.