Summer minicourses 2022
The summer minicourses are a chance for Michigan graduate students to teach each other interesting math in a friendly, informal setting.
Courses will meet daily for a week, usually in the afternoon. Most courses will take place inperson, across two rooms: East Hall 2058 on Mondays, Wednesdays, and Fridays, and East Hall 4088 on Tuesdays and Thursdays.
A few courses will instead be online. I'll email the Zoom meeting IDs to UM math graduate students. Email me if you're not on that list but would like to participate!
The lineup this year appears below. Check back for updated information and abstracts as the courses draw closer! All times are in EST.
If you're giving one of the minicourses and you'd like to include notes, slides, or any other information, just send them to me and I'll add them!
Topic  Speaker  Dates  Time  Modality & Location  Abstract  Notes  

A quick tour towards local Langlands correspondence  Guanjie Huang  June 6  June 10  1  2:30  Zoom (link sent through email)  
Abstract. Local Langlands correspondence (LLC) is a conjectural parametrization of complex representations of reductive groups over local fields, by Langlands parameters, which are analogies of Galois representations. In these talks I will give the basic definition of these objects, explain the expected properties of LLC, and work explicitly with the case of GL2. If time permits, I will also introduce some recent progress in LLC and the technique used in the proof. Familiarity with basic Lie theory and algebraic number theory will be helpful. 

Curves on surfaces  Shend Zhjeqi  June 13  June 17  11  12:30  Zoom (link sent through email)  
Abstract. The concept of moduli is to analyze families of objects with some fixed invariants/properties. In particular, one of the problems is the construction of the moduli space itself. In this minicourse we will construct the moduli of line bundles on a surface (under the right algebrogeometric adjectives), which is called the Picard scheme. In the course of doing so, we will also construct the moduli of curves (divisors) on a surface. Using the same arguments one can in general show the existence of the moduli of sheaves of ideals/ moduli of subvarieties known as the Hilbert schemes, and the same construction of the Picard scheme works in any dimension. Also, we will make some remarks on the smoothness of those schemes. A first course in algebraic geometry is helpful (basics of schemes and cohomology). We will follow Mumford's book "Lectures on Curves on an Algebraic Surface". 

Grothendieck duality  Andy Jiang  June 20  June 24  1  2  East Hall 2058 (MWF) / 4088 (TTh)  
Abstract. I will give an introduction to Serre duality and Grothendieck duality of coherent sheaves from a category theoretic point of view, following some recent advances. Prerequisites are basic algebraic geometry and category theory, though I will also recall the relevant categorical inputs to the theory. 

\elladic cohomology  Zheng Yang  June 27  July 1  1  2:30  East Hall 2058 (MWF) / 4088 (TTh) and Zoom (link sent through email)  
Abstract. Let X be a smooth projective variety over F_q (for q a prime power), so X is globally cut out by a finite set of homogeneous polynomials with coefficients in F_q. One important arithmetic question is to determine how many simultaneous solutions exist in F_{q^n} for each n. One can put this data together into a zeta function Z_X(t) \in Q[[t]] whose logarithmic derivative is precisely the 'naive' generating function for N_n. Information about the zeros and poles of Z_X(t) can yield information about the growth rate of its logarithmic derivative in a precise way, and vastly generalizing this, Weil conjectured very special structural properties for Z_X(t) informed by certain computations he worked out for curves. 

Dmodules and representation theory  Brad Dirks  July 4  July 8  1  2:30  Zoom (link sent through email)  
Abstract. The representation theory of semisimple Lie algebras over the complex numbers has many success stories: all finite dimensional representations decompose into irreducibles (which we can classify), we can classify all simple Lie algebras, there are wellknown character formulas for a large class of representations. This theory is ubiquitous in mathematics, it has interactions with algebraic geometry, number theory, analysis, combinatorics. 

Crystalline cohomology  Gleb Terentiuk  July 11  July 15  1  2:30  East Hall 2058 (MWF) / 4088 (TTh)  
Abstract. Roughly speaking, the theory of crystalline cohomology provides a lift of the de Rham cohomology of a variety over a field of positive characteristic to characteristic 0. We will discuss it as well as its relation to etale cohomology, de RhamWitt complex, and briefly how it fits together into a big picture of prismatic cohomology. 

Thermodynamic formalism  Yuping Ruan  July 18  July 22  1  2:30  East Hall 2058 (MWF) / 4088 (TTh)  
Abstract. Thermodynamic formalism has been developed since G. W. Gibbs to describe the properties of certain physical systems. The underlying mathematical structure is of great interest to a number of areas, e.g. the study of Anosov diffeomorphisms and flows in differential dynamical systems. In this minicourse we will give a quick introduction to thermodynamic formalism for symbolic dynamical systems, which consists of the following two parts:
References:


Trace formulas and RuellePollicott resonances (Ruelle spectrum) in hyperbolic dynamics  Katia Shchetka  July 25  July 29  1  2:30  East Hall 2866 (MWF) / 4088 (TTh)  
Abstract. We will develop functional and spectral approach to hyperbolic dynamics. We will study RuellePollicott resonances (Ruelle spectrum), trace formulas, dynamical zeta functions, and anisotropic Sobolev spaces. Finally, we will present some applications, e.g. mixing properties and counting periodic orbits. If time permits we will discuss semiclassical (microlocal) analysis of geodesic flow on a constant negative curvature surface. 

Introduction to motivic homotopy theory  Jack Carlisle  August 1  August 5  1  2:30  Zoom (link sent through email)  
Abstract. This minicourse will introduce participants to the basic definitions and constructions of motivic homotopy theory. We will pay special attention to complex and real motivic homotopy theory, which are intimately related to classical and C_2equivariant stable homotopy theory. Basic knowledge of algebraic topology and algebraic geometry will be helpful, and all are encouraged to participate. 

Branched covers of the sphere and plane  Malavika Mukundan  August 8  August 12  1  2:30  East Hall 2866 (MWF) / 4088 (TTh)  
Abstract. See the PDF to the right. 

Statistical physics of neural networks  Sameer Kailasa  August 15  August 19  1  2:30  East Hall 2866 (MWF) / 4088 (TTh)  
Abstract. This minicourse will survey several topics at the intersection of statistical physics and theoretical neuroscience, with a focus on highlighting contemporary works. I plan to discuss models of associative memory (e.g. the Hopfield model), techniques of statistical field theory and applications to random recurrent neural networks, and theories of excitationinhibition balance in cortical networks. Prerequisites are limited to probability and willingness to tolerate clearly divergent integrals. 
The 2021, 2020, 2019, 2018, 2017, and 2016 schedules and abstracts are still available.