Summer minicourses 2019
The summer minicourses are a chance for Michigan graduate students to teach each other interesting math in a friendly, informal setting.
Generally, courses meet once a day, in the afternoon, for a week. All are welcome to attend, but keep in mind the target audience is graduate students in the mathematics department.
The lineup this year will appear below, though additions and changes are likely! You can show or hide all the abstracts for printing purposes.
If you're giving one of the minicourses and you'd like to post notes, just send them to me and I'll post them!
Topic  Speaker  Dates  Location and Time  Abstract  Notes  

Motivic integration  Devlin Mallory  May 13–May 17  (M–F)  EH 3088  1–2:30  
Abstract. Motivic integration was introduced by Kontsevich in 1995 as a tool to prove that birational Calabi–Yau manifolds have the same Hodge numbers (generalizing previous results using padic integration). Since then, it's found numerous applications in algebraic and arithmetic geometry, particularly towards the study of cohomological invariants and singularities. In this course, we'll introduce arc schemes and the Grothendieck ring of varieties before developing the basic theory of motivic integration; we'll then apply the theory to prove Kontsevich's result on Hodge numbers. Additional time permitting, we'll give applications to the study of Igusa zeta functions and Hodge spectra. Throughout, we will focus on examples and motivation rather than technical proofs; the only prerequisite is basic algebraic geometry. 

Homotopy theory and derived algebra via ∞categories  Montek Singh Gill  May 20–May 23  (M–Th)  EH 3088  1–3  
Abstract. I will describe foundational aspects of homotopy theory and derived algebra, from the perspective of ∞categories. First, I'll cover some basics of homotopy theory and that of ∞categories as a context in which to do homotopy theory, including the ∞category of spaces as the most important example. Next, I'll discuss ∞(co)limits, and the corresponding notion of homotopy (co)limits for spaces. Another formal framework in which to do homotopy theory is that of model categories, and I will describe the relation between these two frameworks. Next, I'll describe the idea behind derived/homotopy coherent/higher algebra, and how this leads us to stable ∞categories, as well as to the stable ∞category of spectra, the most important example of such categories. Finally, I'll discuss how to do more of the derived algebra via operads in spaces and via ∞operads. All throughout, there will be an emphasis on detail and precision. 

Combinatorial commutative algebra  Francesca Gandini  May 28–May 31  (Tu–F)  EH 3088  1–3  
Abstract. This minicourse is a gentle introduction to a modern field of mathematics via examples. Starting in the seventies with the work of Stanley and Hochster, new results in commutative algebra have been proved by studying associated combinatorial objects. Ever since, new synergies between algebraic and combinatorial objects have been established. Each day I will introduce one of the main four families of examples: monomial ideals, binomial ideals, determinantal ideals, and linear ideals. The course will emphasize applications of the results from the literature to specific computations, so that we can understand via concrete examples what we can find out about these algebraic objects using the combinatorial perspective. The course will be a mix of lecture and problem solving as each day we work through a worksheet. 

Hodge theory for combinatorial geometries 
Shelby Cox,
Will Dana,
Sameer Kailasa, Harry Richman, and Robert M. Walker 
June 3–June 7  (M–F)  EH 3088  1–2:30  
Abstract.
In 2015, Adiprasito, Huh, and Katz resolved a longstanding conjecture of Rota and Welsh stating that the coefficients of the characteristic polynomial of a matroid (for example, the chromatic polynomial of a graph) form a logconcave sequence. For matroids representable over a field, Huh and Katz had already proven this by associating an algebraic variety to a matroid and using inequalities from Hodge theory. Inspired by this, the general proof defines a "matroid Chow ring" and proves precise analogues of the Hodgetheoretic results without using algebraic geometry, creating an exciting new vantage point from which to study matroids.


Master course on algebraic stacks  Ruìān Chén  June 10–June 14  (M–F)  EH 3088  1–2:30  
Abstract.
This minicourse recaptures a part of “A master course on algebraic stacks” taught by Bertrand Toën in University of Toulouse in 2005 (hence the minicourse title). More specifically, we will take the perspective a homotopy theorist to try to understand stacks, or more specifically, stacks in groupoids.


Explicit class field theory for global function fields  Angus Chung  June 17–June 21  (M–F)  EH 3088  2–3  
Abstract.
Class field theory is the study of abelian Galois extensions. A great example would be cyclotomic extensions of Q. In fact, the KroneckerWeber theorem states that all finite abelian extensions of Q are contained in some cyclotomic extensions. So we can construct any abelian extensions of Q very explicitly. Another fantastic point of the theory is that we can obtain explicitly abelian extensions where only a certain set of primes ramify. Apart from Q, we only know how to construct all abelian extensions explicitly for imaginary quadratic field. This is by using theory of complex multiplication. The story is a mystery for other number fields.


Algebraic Ktheory  Shubhankar Sahai  June 24–June 28  (M–F)  EH 3088  12:30–2  
Abstract.
Algebraic KTheory has its roots in Grothendieck's proof of the celebrated GrothendieckRiemannRoch (GRR), a generalisation of earlier analytical results of Hirzebruch. In his proof, Grothendieck, among other things , developed the algebraic K^0group of coherent sheaves on a scheme. Following his success, Atiyah and Hirzebruch developed topological Ktheory, a generalised Eilenberg Steenrod cohomology theory, by applying the K^0functor to topological vector bundles. Using results of Bott on the periodicity of certain homotopy groups, they were able to extend the topological K^0functor to a sequence of functors K^i's which satisfied the Eilenberg Steenrod axioms. Coupled with the AtiyahHirzebruch spectral sequence (AHSS), this theory is extremely powerful and has far reaching applications to Index Theory, Stable Homotopy theory, etc.


Elasticity and geometry  Ian Tobasco  June 24–June 28  (M–F)  EH 3088  2–3  
Abstract. This oneweek minicourse is an introduction to elasticity theory  the study of deformable bodies  and geometry with a particular emphasis on the recent geometric rigidity theorem of Friesecke, James, and Muller. After introducing the basic concepts of strains and displacements and reviewing John’s counterexample to LinftyLinfty rigidity, we prove L2L2 rigidity following FJM. Time permitting, we explain the use of this rigidity theorem to derive Kirchhoff’s plate theory as a Gammalimit and other recent developments along these lines. The course is aimed at graduate students having some basic familiarity with Sobolev spaces, though we will spend some time reviewing the basics in the first lecture. No knowledge of elasticity theory will be assumed. 

The Atiyah–Singer index theorem  Shubhankar Sahai  July 1–July 5  (M–W, F)  EH 3088  1–2:30  
Abstract. The Atiyah–Singer Index Theorem is an immensely powerful theorem relating the analytical index (the difference of the dimension of the kernel and cokernel of the operator) to the topological index (the integral of the chern character and the Todd Class appropriately defined). In this course we will present the KTheoretic proof of the same by roughly following Gregory Landweber's article 'KTheory and Elliptic Operators', which itself is a distilled account of the original papers of Atiyah and Singer  'The Index of Elliptic Operators I and III'. We would like to note that the proof will be presented from a topological perspective and analytical facts will be blackboxed. However we will introduce the requisite KTheory and therefore the prerequisites for the course are just some familiarity with Algebraic Topology and Topological Vector bundles. Time remaining we may prove equivariant versions of the same. 

Bruhat–Tits buildings (for GL(n) and SL(n))  Yiwang Chen  July 8–July 12  (M–F)  EH 4096  1–2  
Abstract.
We will discuss some topics about BruhatTits building that are normally used in the study of padic groups and their representation theory.
Rough schedule that I am now having in mind as follows:
Monday, a brief discussion of padic numbers and some properties, then define the padic group we concerned throughout the minicourse and examine its structure.
Tuesday, define parahoric subgroups, and how they reduce many problems in representation theory to the case of a finite group.
Thursday, define the Bruhat–Tits building.
Friday, construct an important class depth zero supercuspidal representations.


The Fargues–Fontaine curve  Shubhodip Mondal  July 22–July 26  (M–F)  EH 4096  4–5  
Abstract. In this minicourse we will define the Fargues–Fontaine curve. We will discuss geometric structures such as vector bundles on this curve. We will discuss geometric interpretations of comparison theorems in padic cohomology theories in terms of the curve. Also, we will mention geometric interpretations of problems in padic galois representation theory in terms of this curve. Prerequisites on perfectoid rings will be discussed briefly when needed. 

A Ktheoretic approach to the representation theory of finite groups  Attilio Castano  July 29–August 2  (M–F)  EH 4096  3–4  
Abstract.
Abstract: A celebrated theorem of Atiyah and Segal provides a comparison map from the ring of complex representations Rep(G) of a finite group, to the 0th complex Ktheory KU^0 (BG) of a certain space BG. This comparison map is not an isomorphism, but rather it presents KU^0(BG) as the completion of Rep(G) with respect to a certain augmentation ideal. In this mini course, we will investigate a way in which one can “decomplete” KU^0(BG) in order for it to have all the information about the representation theory of G.


Abel's theorem on complex abelian integrals  Jason Liang  August 12–August 16  (M–F)  EH 3088  9–10  
Abstract. 

Translation surfaces  Mark Greenfield
and
Bradley Zykoski 
August 19–August 23  (M–F)  EH 4096  1–3  
Abstract. Translation surfaces are downtoearth geometric objects that you can easily draw, and are right now receiving a lot of attention from dynamicists, algebraic geometers, and lowdimensional topologists. In this course we hope to:

The 2018, 2017, and 2016 schedules and abstracts are still available.