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 in-person, 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 line-up 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) PDF

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) PDF

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 algebro-geometric 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.

\ell-adic cohomology Zheng Yang June 27 - July 1 1 - 2:30 East Hall 2058 (MWF) / 4088 (TTh) and Zoom (link sent through email) PDF

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.

Moreover, Weil noted that if you could define a cohomology theory mimicking certain natural topological properties, then his conjectures (minus the Riemann hypothesis) would more or less follow formally. Such a cohomology theory was realized by Grothendieck and Artin in the 1960s in the form of \ell-adic cohomology, which is loosely speaking a 'repackaging' of etale cohomology. We will introduce and develop the theory of etale cohomology and apply it to the study of \ell-adic cohomology, focusing on certain key properties of etale cohomology most relevant to proving the Weil conjectures. We will mostly follow the exposition in Milne's "Lectures on Etale Cohomology," among other references.

[For a more detailed description of the course, click the "Notes" link to the right.]

D-modules and representation theory Brad Dirks July 4 - July 8 1 - 2:30 East Hall 2058 (MWF) / 4088 (TTh)

Abstract.

Crystalline cohomology Gleb Terentiuk July 11 - July 15 1 - 2:30 TBD

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Thermodynamic formalism Yuping Ruan July 18 - July 22 1 - 2:30 East Hall 2058 (MWF) / 4088 (TTh)

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Resonances and/or trace formulas in hyperbolic dynamics Katia Shchetka July 25 - July 29 1 - 2:30 East Hall 2058 (MWF) / 4088 (TTh)

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Introduction to motivic homotopy theory Jack Carlisle August 1 - August 5 1 - 2:30 TBD

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TBD Swaraj Pande August 8 - August 12 TBD East Hall 2058 (MWF) / 4088 (TTh)

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Statistical physics of neural networks Sameer Kailasa August 15 - August 19 1 - 2:30 East Hall 2058 (MWF) / 4088 (TTh)

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Branched covers of the sphere and plane Malavika Mukundan August 22 - August 26 TBD TBD

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The 2021, 2020, 2019, 2018, 2017, and 2016 schedules and abstracts are still available.