Fall 2023

 November 10

Speaker:  Andrew Senger, Harvard University

Title:  The mod (p,v_1) K-theory of Z/p^n

Abstract:  While Z/p^n is a relatively simple ring, its algebraic K-theory groups are very complicated, and their computation has been an open problem for many years. Recently, prismatic methods have made the computation of these groups a purely algebraic question, and Antieau--Krause--Nikolaus have leveraged this to (among other results) write a computer program which can compute K_* (Z/p^n) in a range of stems. In this talk, I will describe a different approach to the computation of K_* (Z/p^n). Instead of computing the groups in a range, we completely compute the mod (p,v_1) algebraic K-theory pi_* K (Z/p^n)/(p,v_1), which is substantially simpler.

 November 3

Speaker:  Connor Malin, University of Notre Dame

Title:  Koszul Duality With Applications to Axiomatic Functor Calculus

Abstract:  Koszul duality is a general philosophy asserting that given an algebraic objects A, the indecomposables A/decom(A) should support a ''Koszul dual'' algebraic structure. Separately, there is a notion of functor calculus which seeks to understand the category of functors C -> D. The goal of this talk is to show that any version of functor calculus with a reasonable theory of derivatives is tied to Koszul duality (in a useful way).

 October 20

Speaker:  Noah Mark Riggenbach, Northwestern University

Title:  TR is Locally Even in Characteristic p

Abstract:  Soon after topological restriction homology (TR)was introduced it was quickly noticed, first by Hesselholt and then built uponby Hesselholt-Madsen, Geisser-Hesselholt, and many others, that there is aclose connection between de Rham-Witt forms and TR. In this talk I will explainwork, joint with Micah Darrell, which shows that in characteristic p the recentbreakthrough techniques pioneered by Bhatt, Morrow, and Scholze around theseobjects are in fact the same observations described above. 

 September 15

Speaker:  Yang Hu, NMSU

A Calculus Approach to the Enumeration of Topological Vector Bundles.

It is a classical problem in topology to enumerate complex vector bundles of a fixed rank r over a given complex projective space CP^n (where 1 < r < n), with fixed Chern class data. A particular case is when the Chern data is trivial, which we call the vanishing Chern enumeration. In this talk, we apply the unitary calculus of Weiss to produce the vanishing Chern enumeration in the first two unstable cases (which belong to what we call the “metastable” range, following Mark Mahowald), namely rank (n-1) bundles over CP^n for n > 2, and rank (n-2) bundles over CP^n for n > 3. Time permitting, we will discuss work in progress (joint with Hood Chatham and Morgan Opie) where more metastable vector bundles are detected using EO-theory.

 September 8

Speaker:  Ross Staffeldt, NMSU

Topological Invariants of Certain Join Constructions

In even dimensions there is a hierarchy of geometric structures: smooth manifolds, almost complex manifolds, and complex manifolds.  The cartesian product of almost complex manifolds is another almost complex manifold and the product of complex manifolds is another complex manifold.

In odd dimensions there are analogues of almost complex manifolds called K contact manifolds and analogues of complex manifolds called Sasakian manfolds.  Since the cartesian product of odd dimensional manifolds is even dimensional, the product of Sasakian manifolds is replaced by a join construction that produces a 2m+2n+1 dimensional Sasakian manifold from Sasakian manifolds of dimensional 2m+1 and 2n+1.

We are interested in eventually classifying up to homeomorphism or diffeomorphism the results of applying the join construction to a circle bundle over a closed surface of genus g and to a 3-sphere with a weighted circle action. We will describe results of the first step in a classification, namely, the calculation of homotopy and homological invariants. 

Authors: Candelario Castaneda and Ross Staffeldt

 August 25

Speaker:  Prasit Bhattacharya, NMSU

Equivariant Orientation Theory for Disconnected Base Spaces

Unlike classical homotopy theory, equivariant orientation neither guarantees a Thom class nor a Thom isomorphism, which is a primary tool in many geometric applications. In this talk, I will introduce a notion of homogeneity of an equivariant vector bundle which guarantees an equivariant Thom class, and hence an equivariant Thom isomorphisms, for equivariant vector bundles. I will also introduce a G-equivariant first Stiefel-Whitney class.

 August 18

Speaker:  Elizabeth Gasparim

Homological Mirror Symmetry for Adjoint Orbits

I will present the Homological Mirror Symmetry Conjecture of Kontsevich as a duality between symplectic andalgebraic geometry, and illustrate it with examples coming from Lie Theory.

Previous Seminars


Spring 2023

 April 28

Speaker:  Inkang Kim, Korean Institute of Advanced studies, Seoul

Signature, Toledo Invariant, and the Surface Group Representations in Hermitian Semi-simple Lie Groups

We invent a new invariant called Rho invariant to give a modern interpretation of the signature of flat bundles over the surface with boundary, which echoes the Atiyah-Patodi-Singer signature formula of manifolds with boundary. The integration part is the Toledo invariant, and the correction term is Rho invariant. This is a joint work with P. Pansu and X. Wan.

 April 14

Speaker: Franjo Šarčević (University of Sarajevo, Bosnia)

Title:  r-Immersions from the Point of View of Functor Calculus

The calculus of functors - also known as Goodwillie calculus - deals with the questions of approximating functors between two categories. It has three main branches, depending on which categories one takes and what conditions one sets for functors. In this talk, I will present the idea of manifold calculus of functors and, especially, the main results obtained in the functor calculus for r-immersions of a manifold M in R^n. Here, the space of r-immersions of M in R^n is defined to be the space of immersions of M in R^n such that at most r − 1 points of M are mapped to the same point in R^n. This is mostly a joined work with G. Arone.

 April 7

Speaker:  Morgan Opie (University of California at Los Angeles

Classifying topological vector bundles on complex projective spaces

Abstract:  Given two complex topological bundles over $\mathbb CP^n$, one can ask whether the bundles are topologically equivalent. The first test is to compare their Chern classes, since equivalent bundles must have the same Chern data. The converse fails in general, which leads to the following question: given $n$ and $k$ positive integers, what invariants beyond Chern classes are needed to distinguish complex rank $k$ topological bundles on $\mathbb CP^n$, up to topological equivalence? 
In this talk, I will discuss the subtleties of using methods from stable homotopy theory to answer this question. I'll start by explaining how Atiyah--Rees classified all complex rank 2 topological vector bundles on $\mathbb CP^3$ via an invariant valued in the generalized cohomology theory of real K theory. I will then discuss my work classifying complex rank 3 topological vector bundles on $\mathbb CP^5$ using a generalized cohomology theory called topological modular forms. As time allows, I will discuss work in progress extending this work to other ranks and dimensions.

 March 3

Speaker:  Luis Valdez-Sanchez (University of Texas, El Paso

Genus One Hyperbolic Knots in the 3-sphere

Each knot in the 3-sphere bounds infinitely many compact, connected, orientable surfaces embedded in the 3-sphere, called Seifert surfaces for the knot, and the smallest genus among the Seifert surfaces bounded by the knot is the genus of the knot.

In this talk we restrict our attention to genus one hyperbolic knots in the 3-sphere.

We will see that the number of `disjoint and nonparallel' Seifert tori bounded by such knots is at most 5 and discuss the construction of various examples of knots bounding maximal disjoint collections of Seifert tori.

 February 24

Speaker:  Urs Schreiber (New York University, Abu Dhabi

Introduction to Hypothesis H for Mathematicians

The key open question of contemporary mathematical physics is the elucidation of the currently elusive fundamental laws of “non-perturbative” states — ranging from bound states as mundane as nucleons but more generally of quarks confined inside hadrons (declared a mathematical “Millennium Problem” by the Clay Math Institute), over topologically ordered quantum materials (currently sought by various laboratories), all the way to the ultimate goal of fundamentally understanding background-free quantum gravity and “grand unification”. Now, the foremost non-perturbative effect in quantum physics is “flux quantization”; and I begin by explaining in detail how this finds its natural mathematical definition in cohesive homotopy theory (higher smooth stacks). By going through key examples— as a fun exercise in cohesive homotopy theory — I explain how to systematically derive from this:

  1. Magnetic flux quantization (experimentally seen in superconductors), and then by just the same logic:
  2. The widely expected Hypothesis K that “RR-flux” is quantized in topological K-theory, and in evident non-abelian generalization:
  3. Our novel Hypothesis H that “G-flux” is quantized in unstable Cohomotopy (i.e.: framed Cobordism). Depending on time and interest, I may close by indicating (A) how coupling to gravity enhances these flux quantization laws to *T*wisted & *E*quivariant & *D*ifferential (TED) refinements of these cohomology theories and (B) how Hypothesis H explains anyonic topological order controlled by KZ-monodromy in bundles of conformal blocks.
This is joint work with Hisham Sati. Talk slides will be available here:

 February 17

Speaker:  Julia Seminika, University of Muenster

Title:  Cut-and-paste Invariants of Manifolds, K-theory and Cobordisms

Abstract:  The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope P in R^n, one can cut P into a finite number of smaller polytopes and reassemble these to form Q. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut-and-paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste").

In this talk I will explain the construction that will allow us to speak about the "K-theory of manifolds" spectrum. The zeroth homotopy group of the constructed spectrum recovers the classical groups SK_n. I will show how to relate the spectrum to the algebraic K-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further I will describe the connection of our spectrum with the cobordism category.

 February 3

Speaker:  Luca Pol (University of Regensberg

Title:  Quillen stratification in equivariant homotopy theory

Abstract:  Quillen’s celebrated stratification theorem provides a geometric description of the Zariski spectrum of the cohomology ring of any finite group with coefficients in a field in terms of information coming from its elementary abelian p-subgroups. The goal of this talk is to discuss an extension of Quillen's result to the world of equivariant tensor-triangular geometry. For the category of equivariant modules over a commutative equivariant ring spectrum we obtain a stratification result in the terms of the geometric fixed points equipped with their Weyl-group actions for all subgroups, and hence a classification of localizing tensor ideals. Finally, I will apply these methods to several examples of interests such as Borel-equivariant Morava E-theory and equivariant topological K-theory. This is joint work with Tobias Barthel, Natalia Castellana, Drew Heard and Niko Naumann.

Fall 2022

 November 18

Speaker:  Clover May, Norwegian University of Science and Technology

A Structure Theorem for RO(G)-graded Equivariant Homology

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/2-coefficients.  The structure theorem describes the building blocks for the homology of C2-spaces 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

The Steenrod Algebra in Equivariant Algebraic Topology

In this talk, an extension from the elliptical to non-elliptical Student-t distribution is proposed to improve modelling capability. This extension is based on the mean-variance structure of the t and skew-t distributions. Although the probability density distributions of these non-elliptical 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

Higher Homotopy Associativity or A-infinity Structures

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 A-infinity operad, and of course, prove that S^7 cannot admit homotopy associative multiplications. 

 October 28

Speaker:  Tom Bachmann, University of Oslo, Germany

Real Étale Motivic Homotopy Theory Revisited

The real étale topology on schemes was invented by Coste--Roy and has been intensely studied (mainly by semialgebraic geometers) since. Motivic homotopy theory was invented by Morel--Voevodsky 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^1-invariant and satisfies real étale descent, then F is real étale locally constant.

 October 21

Speaker:  Jerry Lodder, New Mexico State University

A Cohomology from Tensors: Leibniz Cohomology

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 complex-hyperbolic 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 quaternionic-hyperbolic 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 Low-dimensional Topology

Abstract:  Taylor polynomials, sums of powers of (x-a), 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 low-dimensional 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.