Department of Mathematics
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Department Colloquia

The UChicago math department hosts colloquia to introduce its members, and anyone else who wishes to attend, to recent developments and important ideas in mathematics.

Hugh Woodin (Harvard): Beyond the age of independence by forcing?

Wednesday, May 31, 2017, 3:00pm–4:00pm, Eckhart 206

Abstract: Gödel’s consistency proof for the Axiom of Choice and the Continuum Hypothesis involves his discovery of the Constructible Universe of Sets. The axiom "\(V = L\)" is the axiom which asserts that every set is constructible. This axiom settles the Continuum Hypothesis and more importantly, Cohen’s method of forcing cannot be used in the context of the axiom "\(V = L\)".

However the axiom \(V = L\) is false since it limits the fundamental nature of infinity. In particular the axiom refutes (most) strong axioms of infinity.

A key question emerges. Is there an "ultimate" version of Gödel’s constructible universe L yielding an axiom "\(V = \text{ultimate } L\)" which retains the power of the axiom "\(V = L\)" for resolving questions like that of the Continuum Hypothesis, which is also immune against Cohen’s method of forcing, and yet which does not refute strong axioms of infinity?

Until recently there seemed to be a number of convincing arguments as to why no such ultimate \(L\) can possibly exist. But the situation is now changed.

Emmy Murphy (Northwestern): Planar graphs, Legendrian surfaces, and contact homology

Wednesday, May 10, 2017, 3:00pm–4:00pm, Eckhart 206

Abstract: To any cubic planar graph, we can associate a surface inside \(\mathbb{R}^5\), which is nicely compatible with a certain canonical geometry, called a contact structure. From here we can apply techniques from contact geometry to study graph theory; for example it was shown by Treumann-Zaslow that the space of constructable sheaves with singular support on the surface recovers the chromatic polynomial of the graph. Another strategy is to look at pseudo-holomorphic curves with boundary conditions on the surface, which defines an algebraic package known as Legendrian contact homology.

We'll explain what this gadget looks like in purely graph theoretical terms. In particular, the augmentation variety of this contact homology recovers the chromatic polynomial of the graph, though it does this in a highly non-trivial way and therefore yields a novel combinatorial definition of a graph coloring. But it also contains more subtle information about the graph, concerning how trajectories in the plane are required to interact with the graph. From here we can draw connections with mathematical physics, such as the spectral networks of Gaiotto-Moore-Neitzke and mirror symmetry in the style of Aganagic-Vafa.

Ciprian Manolescu (UCLA): The Triangulation Conjecture

Wednesday, April 26, 2017, 3:00pm–4:00pm, Eckhart 206

Abstract: The triangulation conjecture stated that any \(n\)-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group).

James Lee (University of Washington): Discrete conformal metrics and spectral geometry on distributional limits

Wednesday, March 29, 2017, 3:00pm–4:00pm, Eckhart 206

Abstract: Random triangulations have been studied as a discrete model for 2D quantum gravity since the 1980s. While there are many conjectures about the emergent geometry of such models, mathematical progress has been somewhat slower. Toward this end, Benjamini and Schramm (2001) defined the distributional limit of a sequence of finite graphs; the limit object is a unimodular random graph in the sense of Aldous and Lyons. Benjamini and Schramm showed that every distributional limit of finite planar graphs with uniformly bounded degrees is almost surely recurrent (the random walk returns to its starting point infinitely often almost surely).

Their approach uses the Koebe-Andreev-Thurston circle packing theorem to uniformize the geometry of the limit graph. In contrast, we consider intrinsic deformations of the path metric by a (random) weighting of the vertices. This leads to the notion of the conformal growth exponent of a unimodular random graph, which is the best degree of volume growth of balls that can be achieved by such a weighting of "unit area." The conformal growth exponent carries information about the underlying geometry; in particular, it bounds the almost sure spectral dimension.

We show that distributional limits of finite graphs that can be sphere-packed in \(\mathbb{R}^d\) have conformal growth exponent at most \(d\), and thus the connection to spectral dimension yields \(d\)-dimensional lower bounds on the heat kernel. When the conformal growth exponent is bounded by 2, one obtains more precise information, including a conjectured generalization of the Benjamini-Schramm Recurrence Theorem to larger families of graphs.

These methods extend to models with unbounded degrees, giving new proofs of almost sure recurrence for the extensively studied uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ). The latter results were established only recently by Gurel-Gurevich and Nachmias (2013). Our approach yields quantitative lower bounds on the heat kernel, spectral measure, and speed of the random walk.

Melanie Wood (UW Madison): Random groups from generators and relations

Wednesday, January 11, 3:00pm–4:00pm, Eckhart 206

Abstract: We consider a model of random groups that starts with a free group on \(n\) generators and takes the quotient by \(n\) random relations. We discuss this model in the case of abelian groups (starting with a free abelian group), and its relationship to the Cohen-Lenstra heuristics, which predict the distribution of class groups of number fields. We will explain a universality theorem, an analog of the central limit theorem for random groups, that says the resulting distribution of random groups is largely insensitive to the distribution from which the relations are chosen. Finally, we discuss joint work with Yuan Liu on the non-abelian random groups built in this way, including the existence of a limit of the random groups as \(n\) goes to infinity.

Nick Rozenblyum (UChicago): Shifted symplectic structures, quantization, and applications

Wednesday, October 26, 3:00pm–4:00pm, Eckhart 206

Abstract: Many moduli problems of interest, such as moduli spaces of local systems, come equipped with a natural symplectic structure. Moreover, quantization of these symplectic structures is closely related to counting problems in geometry and topology, such as the Casson invariant and its generalizations, as well as Feynman integration in physics. The theory of shifted symplectic structures, a vast generalization of algebraic symplectic geometry, provides a natural framework for constructing and studying such symplectic structures and their quantizations. I will give a brief overview of this theory and describe applications in geometry and physics.

Doug Ravenel (Rochester): What is a \(G\)-spectrum?

Wednesday, May 25, 2016, 3:00pm–4:00pm, Eckhart 206

Abstract: Spectra in the sense of stable homotopy theory have been a major object of study in algebraic topology for half a century. During that time the basic definition has undergone some major revisions, including a major breakthrough in 1993 due to Peter May and three coauthors. Remarkably, these shifting foundations have not affected any of the computations made using earlier definitions. In the talk I will describe how the use of category theory has led to major simplifications.