# Department of Mathematics

## Unni Namboodiri Lectures in Geometry and Topology

Unni Namboodiri (1956–1981) was a brilliant student of mathematics at the University of Chicago. He died in an automobile accident in December 1981, just a few days after presenting and defending his doctoral thesis Equivariant vector fields on spheres, written under the direction of J. Peter May. The Unni Namboodiri Lecture Series was established in his memory by his family.

### 2018 LecturesHee Oh (Yale): Dynamics, Fractals and Kleinian groups

#### Lecture 1: Geometric prime number theorems and fractals

Monday, April 16, 2018

The prime number theorem states that the number of primes of size at most $$T$$ grows like $$T/\log T$$, proved by Hadamard and de la Vallee Poussin in 1896. For Gaussian primes, that is, prime ideals in $$\mathbb{Z}[i]$$, not only does the number of Gaussian primes of norm at most $$T$$ grow like $$T/\log T$$ but also the angular components of Gaussian primes are equidistributed in all directions, as proved by Hecke in 1920.

Geometric analogues of these profound facts have been of great interest over the years. We will discuss effective versions of these theorems for hyperbolic 3-manifolds and for rational maps. (The class of quadratic polynomials to which our theorem applies forms an open dense subset of the Mandelbroat set conjectually.)

Both Kleinian groups and rational maps define dynamical systems on the Riemann sphere and they are expected to behave analogously in view of Sullivan’s dictionary. We will also explain how our theorems fit in this dictionary. This talk is based on joint work with Winter, and in part with Margulis and Mohammadi.

#### Lecture 2: Geodesic planes in hyperbolic 3-manifolds

Tuesday, April 17, 2018

By Mostow rigidity, there are only countably many hyperbolic 3-manifolds of finite volume. In such a manifold, a geodesic plane is either closed or dense; this dichotomy was proved by Ratner and Shah independently around 30 years ago.

There was little progress on this question for a hyperbolic 3-manifold of infinite volume until recently. The main obstacle in infinite volume case is scarcity of recurrence of unipotent flows, without which we cannot use unipotent dynamics.

We will discuss recent joint work with McMullen and Mohammadi, which establishes a closed or dense dichotomy for a geodesic plane in the interior of the convex core of $$M$$ for any convex cocompact acylindrical hyperbolic 3-manifold $$M$$. We will explain how we find a compact subset of the frame bundle of $$M$$ of which every element has thick recurrence under unipotent flows. The key ingredient of this construction is to show that circular slices of a Sierpinski curve of positive modulus inherit positivity of the modulus.

#### Lecture 3: Apollonian circle packings

Wednesday, April 18, 2018

An Apollonian circle packing is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices of four mutually tangent circles, via an old theorem of Apollonius of Perga (262-190 BC). They give rise to one of first examples of a fractal in the plane.

In the first part, we will discuss counting and equidistribution results for circles in Apollonian packings in fractal geometric terms and  explain how the dynamics of flows on infinite volume hyperbolic manifolds are related (different parts based on joint works with Kontorovich, Shah and Lee).

A beautiful theorem of Descartes in 1643 implies that if the initial four circles have integral curvatures, then all the circles in the packing have integral curvatures, as observed by Soddy, a Nobel laureate in Chemistry. This remarkable integrality feature gives rise to several natural Diophantine questions about integral Apollonian packings such as "how many circles have prime curvatures?" and "what kind of integers arise as curvatures?". In the second part, we will discuss progress on these questions which are based on expanders and the affine sieve.

### 2017 LecturesMike Hill (UCLA): Evenness in algebraic topology

#### Thematic Abstract

Various notions of "evenness" play a fundamental role in algebraic topology. Spaces with only even cells tend to have more readily computed and conceptual cohomology, and cohomology theories which take non-zero values only on even spheres are the heart of the "chromatic" approach to stable homotopy which links algebraic topology and algebraic geometry. In the most geometric cases, we have both, meaning that geometric objects related to vector bundles and bordism are shockingly well-behaved, with the cells and homotopy groups themselves encoding rich information.

Equivariantly, we have many different notions of spheres and cells built out of representations, and these give other kinds of notions for "even". In this series of talks, I'll discuss a version of even for equivariant spectra which behaves very much like the classical cases. This allows us to understand the spaces that make up the equivariant bordism theory used in the proof of the Kervaire Invariant One problem, to give a conceptual description of the Steenrod algebra for various equivariant cohomology theories, and to explain certain duality phenomena observed in spectral algebraic geometry.

#### Midwest Topology Seminar

This year's Namboodiri Lectures are occurring in conjunction with the Midwest Topology Seminar, which is taking place at UChicago on Saturday, May 13 and Sunday, May 14. See more information here.

#### Lecture 1

Friday, May 12, 2017, 4:00pm–5:00pm in Ryerson 251

Abstract: Complex projective space plays a fundamental role in algebraic topology as a space which simultaneously represents line bundles and the second cohomology group with integral coefficients. This space has an additional extremely useful feature: it has cells only in even dimensions (and homotopy only in even dimensions). This makes many algebraic topology constructions exceptionally easy. The Grassmanians of $$n$$-planes in $$\mathbb{C}^{\infty}$$ all also have cells only in even dimensions, meaning that the they share many of the properties of complex projective space. More surprisingly, Wilson in his thesis showed that the even spaces in the complex bordism spectrum have only even cells and only even homotopy groups. This talk will explore these classical results and their consequences before exploiting the natural $$C_2$$-equivariance of the spaces to describe a similar Real version.

#### Lecture 2

Monday, May 15, 2017, 4:00pm–5:00pm in Eckhart 202

Abstract: Extending to larger groups: the norm, $$G$$-equivariant Wilson spaces, and the equivariant Steenrod algebra. Central to the Hill-Hopkins-Ravenel proof of the Kervaire invariant one problem was a well-behaved multiplicative induction functor, the norm. The norm of $$MU_\mathbb{R}$$ from $$C_2$$ to $$C_{2^n}$$ was the basic object of study in that proof, and as a Thom spectrum, this encodes significant geometric information. Moreover, the Hill-Hopkins-Ravenel approach gave a way to understand various spectra built out of the norm of $$MU_\mathbb{R}$$, including the one giving the homology. This talk will describe this set-up together with how one can use this to vastly generalize the earlier results about the Real Wilson spaces and how one can compute the $$C_{2^n}$$-equivariant Steenrod algebra with constant coefficients.

#### Lecture 3: Towards $$RO(G)$$-graded algebraic geometry: explorations of duality for Galois covers via equivariant homotopy

Tuesday, May 16, 2017, 4:30pm–5:30pm in Eckhart 206

Abstract: The unifying theme from the first two talks is the lifting of ordinary, non-equivariant maps from spheres to equivariant maps from representation spheres. This procedure also arises in a surprising way in spectral algebraic geometry, where we can use these techniques to understand a spectral version of Serre duality for certain derived moduli problems. This talk will focus primarily on several examples related to the theory of topological modular forms with level structure, where the equivariance coming from the level, coupled with refinements of homotopy from $$\mathbb{Z}$$-graded maps to $$RO(G)$$-graded maps, gives a conceptual and computationally useful approach to duality.

### 2016 LecturesYves Benoist (Université Paris-Sud) On spectral gaps in simple Lie groups

#### Lecture 1: Dense subgroups in $$\mathrm{SL}(2,\mathbb{R})$$, with N. de Saxce

Monday, May 16, 2016, 4:00pm–5:00pm in Ryerson 251

Abstract: I will focus on proper Borel measurable dense subgroups of $$\mathrm{SL}(2,\mathbb{R})$$ and explain that their Hausdorff dimension is zero. I will also explain why the convolution of sufficiently many compactly supported continuous functions is differentiable. We will see how these two questions are related. The main tool will be a local spectral gap property.

#### Lecture 2: Random walk on $$\mathrm{SL}(2,\mathbb{R})$$, with JF. Quint

Tuesday, May 17, 2016, 4:30pm–5:30pm in Eckhart 202

Abstract: I will focus on random walks on $$\mathrm{SL}(2,\mathbb{R})$$ and explain the law of large numbers for the coefficients. I will also focus on the regularity of the stationary measure on the projective line. We will see how these two questions are related. The main tool will be a spectral gap property.

#### Lecture 3: Tempered homogeneous spaces, with T. Kobayashi

Thursday, May 19, 2016, 4:30pm–5:30pm in Eckhart 202

Abstract: I will focus on unitary representations of $$\mathrm{SL}(n,\mathbb{R})$$ and give a criterion for the temperedness of the regular representations on its algebraic homogeneous spaces. I will also explain the relation between temperedness and induction. We will see how these two questions are related. The main tool will be a uniform spectral gap property.

Past Namboodiri Lecturers include: J. Frank Adams, William Thurston, Shing-Tung Yau, Daniel Quillen, Robert MacPherson, Simon K. Donaldson, Edward Witten, Mikhael Gromov, Gregory A. Margulis, Graeme Segal, Andrew J. Casson, Julius Shaneson, Nigel Hitchin, Dusa McDuff, Clifford Taubes, Alain Connes, Ib Madsen, Michael Hopkins, Yakov Eliashberg, David Gabai, Curt McMullen, Gang Tian, Michael Weiss, John Baez, Étienne Ghys, Nigel Higson, Mladen Bestvina, Douglas C. Ravenel, Danny Calegari, and Larry Guth.