## 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.

### 2016–2017 Lectures

Mike 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.

### 2015–2016 Lectures

Yves 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.