This is the schedule for the summer 2018 mathematics graduate studentrun minicourses at UT Austin.
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Topic  Speaker(s)  Dates  Time and Location  Abstract  Notes 

Generalised (Cohomology, Orientation, Characteristic Classes)  Riccardo Pedrotti  May 1518  11am12pm RLM 10.176 

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. 

Characteristic Classes  Arun Debray  May 2125  11am12pm, 12pm RLM 10.176 

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

The LippmanSchwinger Equation  Michael Hott  May 2125  23pm RLM 10.176 

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

PDEs on Manifolds  Michael Hott, Max Stolarski, Dan Weser (Reading Seminar)  May 2425, 2830  1011am RLM 10.176 

Abstract. This course runs ThursdayWednesday. 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 wellversed 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 signup sheet for presenting topics. Here is an idea of the material to be covered:


Symmetric Spaces  Max Riestenberg  May 28June 1  11am12pm, 12pm RLM 10.176 

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:


RiemannRoch and Generalizations  Rok Gregoric  June 48  11am12pm, 12pm RLM 12.166 

Abstract. The RiemannRoch 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 GrothendieckRiemannRoch theorem, an arbitrarydimensional 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 crashcourse about the AG notions we will be using in the seminar will also be organized if there is interest. 

Holder Continuous Euler Flows in 3 dimensions  Andy Ma  June 48  23pm RLM 9.166 

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 LittlewoodPaley Theory and applied math 2. 

Intro to Homological Algebra/Spectral Sequences  Adrian Clough, Richard Wong  June 1115  11am12pm, 12pm RLM 10.176 

Abstract. The first three days of this course will be lectures on the basics of homological algebra with an eye towards computations:
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 EilenbergMoore spectral sequences, which are also very useful for calculating homology / cohomology of topological spaces. There will be a problem session from 12pm. 

Intro to Stacks  Adrian Clough  June 1822  11am12pm, 12pm RLM 9.166 

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. 

Aspects of Principal Bundles  Ali Shehper  July 2327  1011am, 12pm RLM 12.166 

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. 

Intro to K theory (and variants)  Ricky Wedeen (Reading Seminar)  July 30  Aug 3  34pm RLM 12.166 

Abstract.


Cobordism Theory/The Thom Isomorphism  Arun Debray  Aug 610  11am12pm, 12pm 

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

Cyclic branched covers of S^3  Lisa Piccirillo, Hannah Turner (Reading Seminar)  Aug 610  1011am, 11:30am1pm RLM 12.166 

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


Intro to Heegaard Floer homology  Jonathan Johnson  Aug 1317  10:30am12pm, 12pm RLM 12.166 

Abstract. Heegaard Floer homology is a package of powerful invariants of smooth 3manifolds introduced by Ozsvath and Szabo in 2004. In this course, we will define the invariants for 3manifolds including hat, plus, and minus flavors (welldefinedness 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 dinvariants.. 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. 