November 18

Speaker: Clover May, Norwegian University of Science and Technology
Title: A Structure Theorem for RO(G)graded Equivariant Homology
Abstract: For spaces with an action by a group G, one can compute an equivariant analogue of singular homology (or cohomology) called RO(G)graded Bredon (co)homology. Computations in this setting are often challenging and not well understood, even for G = C2, the cyclic group of order 2. In this talk, I will start with an introduction to RO(G)graded homology and describe a structure theorem for RO(C2)graded homology with (the equivariant analogue of) Z/2coefficients. The structure theorem describes the building blocks for the homology of C2spaces and makes computations significantly easier. It shows the homology of a finite C2 space depends only on the homology of two types of spheres, representation spheres and antipodal spheres. I will give some applications and also talk about recent work with Dugger and Hazel generalizing the structure theorem.

November 11

Speaker: Foling Zou, University of Michigan
Title: The Steenrod Algebra in Equivariant Algebraic Topology
Abstract: In this talk, an extension from the elliptical to nonelliptical Studentt distribution is proposed to improve modelling capability. This extension is based on the meanvariance structure of the t and skewt distributions. Although the probability density distributions of these nonelliptical distributions do not have a close form, the MCMC and EM algorithms have proven to be very efficient to handle these distributions. Financial data and insurance loss data are used for illustration purposes.

November 4

Speaker: Prasit Bhattacharya, New Mexico State University
Title: Higher Homotopy Associativity or Ainfinity Structures
Abstract: In 1958, when Adams proved that S^1, S^3 and S^7 are the only spheres with multiplicative structures, it was unclear if the multiplication on S^7 can be associative up to a homotopy. In 1963, Stasheff answered this question simply by constructing a sequence of polytopes in Euclidean space, but how? This talk will be a survey of Stasheff’s work from a modern point of view. We will discuss Stasheff polytopes, define an Ainfinity operad, and of course, prove that S^7 cannot admit homotopy associative multiplications.

October 28

Speaker: Tom Bachmann, University of Oslo, Germany
Title: Real Étale Motivic Homotopy Theory Revisited
Abstract: The real étale topology on schemes was invented by CosteRoy and has been intensely studied (mainly by semialgebraic geometers) since. Motivic homotopy theory was invented by MorelVoevodsky and has seen some spectacular applications. In a 2018 paper, I showed that there is a close connection between motivic homotopy theory and semialgebraic geometry (via real the étale topology) in the stable setting, that is, for motivic *spectra*. In this talk I will explain an unstable version of this connection, that is, for motivic *spaces*. For example, I will show that if S is a scheme and F is a presheaf of spaces on Sm_S which is A^1invariant and satisfies real étale descent, then F is real étale locally constant.

October 21

Speaker: Jerry Lodder, New Mexico State University
Title: A Cohomology from Tensors: Leibniz Cohomology
Abstract: Recall that the work done in moving a particle through a force (vector) field is independent of the path if the vector field is a gradient field. We use this to review the definition of deRham cohomology in terms of closed and exact forms. These forms are skew symmetric, meaning that the sign changes when the order of dx and dy are switched. We examine a cohomology for differentiable manifolds based on tensors, meaning that there is no sign change when switching dx and dy. A connection on a differentiable manifold is a cochain in this new cohomology, and its coboundary is related to eigenfunctions on the manifold with eigenvalue given in terms of scalar curvature.

October 14

Speaker: Subhadip Dey
Title: Discrete Subgroups of PU(2,1) with Large Critical Exponents
Abstract: Let Γ be a discrete subgroup of PU(2,1), the isometry group of the complexhyperbolic space CH² of (complex) dimension two. Under a suitable normalization of the Riemannian metric of CH², the critical exponent δ(Γ) of Γ, when Γ is a lattice, is 4. In this talk, we discuss an example of a sequence (Γₙ) of discrete subgroups of PU(2,1) such that, for all n \in N, δ(Γ_n) < 4, but δ(Γ_n) → 4, as n → ∞. This example shows that Corlette’s gap theorem on critical exponents of discrete isometry groups of the quaternionichyperbolic spaces does not hold for CH². This talk is based on joint work with Beibei Liu.

September 30

Speaker: Ross Staffeldt, New Mexico State University
Title: The Complex Projective Line and the Fundamental Group of the Circle (Part 2).
Abstract: How are these things related? The concept of "line bundle" provides the link. After reviewing a few basic facts about projective spaces, we will explore the concept through examples. In particular, we will show how to attach elements of the fundamental group of the circle to the normal bundle of the projective line in the projective plane, the canonical bundle over the line, and, possibly, the tangent bundle of the line.

September 23

Speaker: Ted Stanford, New Mexico State University
Title: Free Taylor Polynomials in Lowdimensional Topology
Abstract: Taylor polynomials, sums of powers of (xa), are familiar objects incalculus. There exists an interesting noncommutative generalization ofthis technique to free groups, which turns out to be useful for analyzingvarious groups which arise frequently in lowdimensional topology, such as knotgroups, link groups, and braid groups. I will assume basic familiaritywith groups and rings.

September 9

Speaker: Ross Staffeldt, New Mexico State University
Title: The Complex Projective Line and the Fundamental Group of the Circle
Abstract: How are these things related? The concept of "line bundle" provides the link. After reviewing a few basic facts about projective spaces, we will explore the concept through examples. In particular, we will show how to attach elements of the fundamental group of the circle to the normal bundle of the projective line in the projective plane, the canonical bundle over the line, and, possibly, the tangent bundle of the line.
