Summer Mini-Courses 2018

Abstract. I plan on generalising a bunch of concepts that are fairly standard in standard (co)homology, like what does it mean for a manifold to be oriented, the Thom isomorphism and characteristic numbers for a manifold to the setting of generalised cohomology theories.

The class officially start on Tuesday (5/15), but on Monday (5/14) I plan on revising some concepts about spectra since we will use them a lot. This review of spectra is gonna be very concrete: a bunch of definition that we need, some examples and very few proof/constructions. I just want you to know what are they and the main definitions (like: what is the fundamental group of a spectrum, what is a ring spectrum, what is the homology theory defined by a spectrum…) Let me know if you are interested so we can schedule a meeting on Monday.

I will use some spectral sequences tools and I plan on dedicating Tuesday afternoon to them. I’m aware that one cannot pretend to learn sseq in one hour, but I’ll try to do all the computations and if needed I’m more than happy to schedule additional afternoon meetings to work out some examples/details. Again, just let me know.

REFERENCES: I will mainly base my class on Kochman’s book: Bordism, Stable Homotopy and Adams Spectral Sequences, while the introduction to spectra is taken from Rudyak’s On Thom spectra, Orientability, and Cobordism.

Abstract. An introduction focused on computing and applying characteristic classes. Covering Stiefel-Whitney, Chern, Pontrjagin, and Wu classes. There will be a problem session in the afternoon.

Abstract. When describing scattering of Quanta, we want to assign a scattering amplitude for different physical regimes. The Lippman-Schwinger equations do the job.

Abstract. This course runs Thursday-Wednesday. We will start with an introduction to Riemannian geometry and then cover some topics in geometric analysis. We will use a reading course style and take turns presenting material. We might meet twice per day at the beginning to quickly cover the background material in Riemannian geometry, but anyone who is well-versed in this material is free to skip it.

If you plan to attend, please email Dan Weser, and he will send you the course materials and have you fill out a sign-up sheet for presenting topics.

Here is an idea of the material to be covered:

• Riemmannian geometry:
  • Metrics, vectors, and one-forms
  • The musical isomorphisms
  • Inner product on tensor bundles
  • Connections on vector bundles
  • Curvature in the tangent bundle
  • Sectional curvature, Ricci tensor, and scalar curvature
  • Covariant derivatives of tensor fields
  • Double covariant derivatives
  • Commuting covariant derivatives
  • Gradient and Hessian
  • Bianchi Identity
  Reference: Jeff Viaclovsky. "Math 865, Topics in Riemannian Geometry". Lecture notes, Fall 2011. [Lectures I-II]
• Geometric analysis:
  • First and second variational formulas for area
  • Volume comparison theorem
  • Laplacian comparison theorem
  Reference: Peter Li. "Geometric Analysis". Cambridge University Press, 2012. [Chapters I-II, IV]

Abstract. We will go over some basics of symmetric spaces, with a lecture each morning and an exercise session each afternoon. The audience is not required to have a strong understanding of Riemannian geometry and Lie theory.

The main topics I hope to get to are:

• Definitions of symmetric spaces: Lie theoretic vs. geometric
• Relationship to general nonpositively curved manifolds
• Cartan fixed point theorem
• The symmetric space associated to SLn and in what sense it is "universal"
• Flats, parallel sets, the boundary at infinity, Tits geometry
• Horocycles and Busemann functions
• Cartan decomposition
• Discrete subgroups and Anosov representations

Abstract. The Riemann-Roch theorem is a celebrated result in algebraic geometry, whose history is closely intertwined with the development of the field itself. Every generation of mathematics eventually turns to the RR theorem, and reinterprets or generalizes it in accordance to contemporary trends.

We will first review the classical curve case of the RR theorem, and snap a brief look at how the story works for algebraic surfaces. The majority of the course will be devoted to the Grothendieck-Riemann-Roch theorem, an arbitrary-dimensional relative generalization. This will serve as an excuse for us to learn some intersection theory, after which we will state the GRR theorem and give a reasonably complete proof.

If time permits, we will conclude the course with a vista into the land of derived algebraic geometry, and the light that this newer perspective shines on the RR theorem.

Anybody who would like to attend is warmly welcome to do so. Though some rudimentary knowledge of algebraic geometry will be assumed, a crash-course about the AG notions we will be using in the seminar will also be organized if there is interest.

Abstract. I will present a paper for constructing convex integration weak solutions to Euler Flows. Most likely I will present Phil Isett's Thesis paper on this topic but I am also considering presenting his 1/5 paper. I would like to introduce some of the new tools of modern convex integration for PDEs with more emphasis on how one improves on the regularity of a convex integration weak solution and less emphasis on constructing energy profiles. I will try to stay away from technical proofs and instead give intuition on how the general scheme works based on my own understandings of fluid equations. I will assume the audience is familiar with basic Littlewood-Paley Theory and applied math 2.

Abstract. The first three days of this course will be lectures on the basics of homological algebra with an eye towards computations:

• diagram chasing
• working with chain complexes
• computing homology and cohomology
• using the Kunneth and universal coefficient theorems
• computing Ext and Tor groups (including group cohomology calculations

On Thursday, Adrian will give a lecture on how to construct a spectral sequence from a filtration, and on Friday we will discuss the Serre and Eilenberg-Moore spectral sequences, which are also very useful for calculating homology / cohomology of topological spaces. There will be a problem session from 1-2pm.

Abstract. Over the course of this week we will explore the two intimately related ideas of descent and constructing new spaces from local models, such as constructing schemes (and their generalisations) from affine schemes, or manifolds from open subsets of Euclidean space. After proceeding systematically through the definitions of sites, sheaves, geometric sheaves, fibred categories, stacks, and geometric stacks, we will endeavour to treat as many examples of geometric stacks as possible at the end of the week.

Prerequisites: Other than categorical maturity it would be beneficial to be familiar with the basics of either differential or algebraic geometry, in order to understand examples.

Abstract. There are several things I have wanted to learn about principal bundles for their ubiquity in Physics: their abelianization on Riemann surfaces, the moduli space of flat connections, gauge transformations on connections, Higgs Bundles on Riemann surfaces and their moduli space, etc. I intend to cover some of these topics in a week long course.

Abstract.

• Monday: Vector bundles and definition of topological K-theory of a space.
• Tuesday: A proof of Bott Periodicity
• Wedensday: Cohomology properties of topological K-theory
• Thursday: Computations
• Friday: Arun on Spin^c Structures

Abstract. Depending on student interest, something like an introduction to cobordism theory, applications of the Thom isomorphism or Pontrjagin-Thom construction, etc.

Abstract. We will ask questions that people ususally ask about Dehn surgery, about cyclic branched covers.

• Monday:
• Lecture: Definitions, basic properties, and first constructions.
• Problem session: Montesinos construction.
• Tuesday:
• Lecture: Knot concordance implications for branched covers .
• Problem session: Knot concordance implications for branched covers.
• Wednesday:
• Lecture: Branched covers and knot invariants.
• Problem session: Branched covers and knot invariants.
• Monday:
• Lecture: Branched covers and knot invariants with an eye towards left orders.
• Monday:
• Lecture: Branched cover implications for knot concordance?

Abstract. Heegaard Floer homology is a package of powerful invariants of smooth 3-manifolds introduced by Ozsvath and Szabo in 2004. In this course, we will define the invariants for 3-manifolds including hat, plus, and minus flavors (well-definedness and invariance will not be proved). We will focus primarily on computing concrete examples, using key computational tools such as the surgery exact triangle, absolute gradings and d-invariants.. If we have time, we will say a word or two about knot Floer homology and bordered Floer homology.

The course website can be found here.