Abstract. Étale cohomology is a cohomology theory for schemes whose behavior resembles that of singular cohomology. It was introduced and worked out in the 1960s by Grothendieck and M. Artin in order to prove the Weil conjectures. In this mini-course, I will give an overview of some basic concepts and examples in étale cohomology. Topics covered will include:

1. Motivation: Weil conjectures, topological étale site;
2. Review of étale morphisms;
3. Étale topology and sheaves;
4. Stalks, pushforwards, and pullbacks of étale sheaves;
5. Computational methods: Čech theory, étale fundamental group, Kummer and Artin-Schreier sequences, applications;
6. If time permits, cohomology of curves.

Prerequisites. Math 632 or equivalent.

Abstract. Multiplier ideals are an important tool in complex algebraic/analytic geometry. They are often used to study e.g. the singularities of divisors, or metrics on holomorphic line bundles. In this mini course, I will give an overview of the analytic and algebraic theories, along with plenty of examples. Topics will include:

1. Analytic multiplier ideals;
2. Review of Q-divisors and log resolutions;
3. Algebraic multiplier ideals: basic properties, and examples;
4. Geometric properties: restrictions, inversion of adjunction;
5. Positivity and vanishing theorems;
6. Metrics on line bundles and their multiplier ideals;
7. If time permits, we will discuss either the Kodaira Embedding theorem or the Fujita Approximation theorem.

Prerequisites. Math 632 or equivalent.

Abstract. I will first begin by discussing heights on projective varieties and the use of metrized line bundles. I'll then talk about arithmetic intersection theory for surfaces and (time permitting) discuss some of the ideas of Gillet-Soule on arithmetic Chow groups. This mini-course will be very informal, with an emphasis on ideas rather than proofs.

Abstract. The classification of Riemann surfaces/algebraic curves was known since the 19th century, and first divides them into three types: those with positive, zero, and negative curvature. The Italian school of algebraic geometry focused on the question: how can we classify algebraic surfaces, or even higher dimensional varieties? Our goal in this mini-course is to discuss Mori's approach to this birational classification problem, known as the Minimal Model Program. We will focus on the following topics:

1. Mori's Bend and Break technique;
2. Cones of curves;
3. Mori's Cone and Contraction Theorems (for smooth varieties);
4. An overview of the Minimal Model Program, in particular the classification of surfaces;
5. A discussion of generalizations to the singular setting.

Prerequisites. Math 632.

Abstract. Information Theory began with two fundamental questions:

1. Are there limits to how much data can be compressed?
2. How efficiently can we transmit data reliably over unreliable channels?

At the heart of answering these two questions is the need to formulate a notion of the amount of information content in a set of data. The quantities that arise in this endeavor—entropy, mutual information, and relative entropy—turn out to be exceedingly useful, not just with regard to these foundational questions, but also in many other areas of mathematics, statistics, and computer science.

In this course, I will talk about data compression, channel capacity, and applications of information theory to statistical inference. If there is time, I may talk about gambling.

Abstract. Machine languages describe sequences of actions to be performed by a computer, such as moving bit patterns from memory to registers or performing arithmetic operations.

Functional programming languages describe functions in the category of sets. Compilers for functional programming languages translate such descriptions into machine code which computes the described function.

Haskell is a popular functional programming language with an interesting feature: on top of describing morphisms in the category of sets, you can also describe endofunctors which can be used to produce new morphisms from old morphisms. Despite what some people say, Haskell is a serious language with real applications outside academia. For example, Haskell plays a big role in Facebook's spam detection software.

In this mini-course, we shall explore Haskell by building a working interpreter for the language scheme (a variant of the Mathematica language). This is a great first project because you can use tools like monadic parsers and monad transformers in a concrete down to earth way.

Prerequisites. No prior Haskell experience is required and we can go slowly if people want to spend more time on the basics.

Abstract. Elliptic differential operators have played an important role in geometry and physics for the last 100 years, particularly in the last 30 years. They are some of the most well understood partial differential operators, as they are well behaved. The most studied one is the Laplacian. In 1911, Weyl showed that the eigenvalues of the Laplacian determine both the dimension and volume of a region. The subject of spectral geometry aims to find what geometric properties can be determined by the eigenvalues.

Certain geometric elliptic operators are closely linked with the Chern-Gauss-Bonnet, Hirzebruch signature, and Hirzebrch-Riemann-Roch theorems. The Atiyah-Singer index theorem generalizes these theorems to all elliptic operators and is one of the greatest theorems of the 20th century.

The aim of these lectures is to introduce elliptic operators and the basics of both the local and global theory. Putting the local and global pictures together in the correct way will give both Weyl's result and the Atiyah-Singer index theorem. The schedule is as follows:

1. Introduction to elliptic operators: definitions and important examples. Sobolev spaces will be introduced as a black box.
2. Elliptic regularity and the fundamental elliptic inequality.
3. Global theory and the Hodge theorem.
4. Local theory and Weyl's law.
5. Introduction to the Atiyah-Singer index theorem, and the idea of the heat kernel proof.

Prerequisites. Some familiarity with the basics of Riemannian geometry. Some knowledge of functional analysis will also be helpful.

For Friday, knowledge of cohomology, vector bundles, and characteristic classes. Some knowledge of K-theory, de Rham cohomolgy, the curvature characterization of characteristic classes, and familiarity with one of the Chern-Gauss-Bonnet, Hirzebruch signature, or Hirzebruch-Riemann-Roch theorems will be helpful.

Abstract. This week I am offering a mini-course that will give an introduction to quantum mechanics. The main goal of the course is to give an idea of what quantum mechanics is about. I will introduce and demonstrate applications of the basic tools underlying a typical quantum mechanical problem: Hilbert spaces, Hamiltonians, quantization, and Schrödinger's equation. The culmination of the mini-course will be a quantum mechanical treatment of the dynamics of the hydrogen atom, where I will use these tools to predict the colors of light emitted by a heated hydrogen atom.

I will point out the representation theory of SO(3, R) and SL(2, C) as it comes up. For instance, the finite irreducible representations of SL(2, C) are classified by their dimension, which physically corresponds to fundamental particles having spin. Spin 1/2 particles (i.e., matter) correspond to the fundamental two-dimensional representation, spin 1 particles correspond to the three-dimensional representation, and so on. I also hope to have enough time to introduce the Dirac equation, the generalization of Schrödinger's equation, which will provide the appropriate context for the previous two sentences.

Abstract. The goal of this mini-course is to understand the recent work of June Huh and collaborators, proving the log-concavity (or unimodality) of sequences coming from matroids. A matroid is a combinatorial object that generalizes the following situations: 1) a (finite) simple graph and its cycles, and 2) a finite set of vectors / hyperplanes and their incidence relations. In the case of hyperplane arrangements, log-concavity is proved using tools from intersection theory and Hodge theory (on a compactification of the complement of the hyperplanes). This argument is adapted for general matroids by defining an analogous "Chow ring" and proving that the desired properties hold.

As an "extended abstract" I recommend reading the introduction of Hodge Theory for Combinatorial Geometries, or Matt Baker's blog post.

Prerequisites. I will assume you know the definition of a matroid, or have read the definition from this blog post.

Abstract. Geometric group theory is less a subject than a collection of mathematical threads unified by a general philosophy: that the study of of group actions on spaces can provide a bridge between the algebraic properties of the group and the geometric / topological properties of the space.

In this minicourse we will see how this works in the illustrative case of Gromov-hyperbolic groups. These are groups which are "negatively curved", in a narrow but precise sense involving slim triangles. It was a profound insight of Gromov that this slim triangles condition captures the essence of negative curvature; many results about word-hyperbolic groups are inspired by the geometry of negatively-curved Riemannian manifolds.

Two highlights that will be covered in the minicourse: (1) Gromov-hyperbolic groups have word problem solvable in linear time, and (2) Gromov-hyperbolic groups satisfy a Tits alternative: any subgroup is either virtually cyclic, or contains a free group on two generators.

Prerequisites. Some prior exposure to (the ideas of) hyperbolic geometry would be helpful for motivational reasons, but is not essential.

Abstract. This course will cover the basics of quantum computing, which can be described (albeit glibly) as the use of the C-linearity of quantum mechanics to achieve speed-ups over classical algorithms. Quantum computing is a relatively new field; the concept was introduced in the early 80s by Feynman and Manin (among others), but its true potential became clear even more recently with Shor's discovery of an effective quantum factoring algorithm.

We'll begin with the basic formalism of quantum mechanics and circuits, before moving on to some basic examples of algorithms, including the Deutsch–Jones, Bernstein–Vazirani, and Simon's algorithms, pointing out the speedups these quantum algorithms offer over any possible classical counterpart. We'll then cover Shor's factoring algorithm in depth, which demonstrates the ability of quantum computers to break RSA encryption, which is believed to be classically impossible. Time permitting, we may cover additional topics, including Grover's search algorithm, quantum walks, or quantum magic square games.

Prerequisites. We'll assume only a knowledge of elementary linear algebra and a willingness to gloss over the details of the physics involved.

Abstract. I'm giving a minicourse this week on some standard ADE classifications. For those that aren't in the loop:

Definition: An ADE classification is a "natural" bijection between the simply-laced Dynkin diagrams and the isomorphism classes of some family of things.

There are several such classifications that are well-known and they get into some really interesting mathematics. We'll be taking the following route:

I will give as many proofs as possible and indicate references when I'm unable to do so. Here's the tentative schedule:

Hope to see you there!

Abstract. An algebraic group is, roughly, a group as an algebraic geometer would want to see it. Because algebraic geometers love schemes, this means an object that is both a group and a scheme. I will discuss the basics of affine group schemes in this mini-course. In particular, we will see how to break down an algebraic group into smaller pieces in a standard way, involving

• unipotent groups
• tori (split or non-split)
• semisimple groups
• finite etale groups.

Prerequisites. I will assume you have seen the definition of an affine scheme (or, a commutative ring).

Abstract. In this course, we will power through topics in toric algebraic geometry that can be discussed almost exclusively in Math 614/Math 631 vernacular. The tentative topics list is a "mash-up" of seminar talks I've given in the past:

• First steps: recipes for normal toric varieties.
• Toric Divisor Theory up through a "Eilenberg-Maclane" type construction.
• "Under the hood" of the affine story: neat connections between commutative algebra and convex geometry (e.g., symbolic powers of monomial ideals and F-singularities).
• An instance of Applied AG: an examples-oriented approach to Bernstein's Theorem (BKK Bound on solutions) in low dimensions.

Prerequisites. Math 614/631 will suffice. Convexity notions will be defined as needed.

Abstract. The study of the absolute Galois group of number fields is a fundamental goal of number theory. Unfortunately the group is too complicated to understand as a whole, so we instead study its representations (aptly called Galois representations), particularly those coming from geometry. Langlands' reciprocity conjecture (for GL(n)) essentially says that the L-function of a finite (n-)dimensional complex representation of the absolute Galois group is equal to the L-function of some other arithmetic object called an automorphic form. The "modularity conjecture" of Taniyama–Shimura, which says that the L-function of every elliptic curve defined over Q comes from a modular form, is (a part of) the p-adic analogue of the reciprocity conjecture for GL(2).

We will first spend some time discussing how Fermat's Last Theorem can be reduced to a certain case of this "modularity" conjecture by work of Frey, Serre, Ribet, and others. Our main goal is to sketch Wiles' proof and introduce the techniques involved. Since much of the work is extremely technical, we will only highlight the main steps and use the same techniques to prove the analogous modularity conjecture for GL(1) instead (the Kronecker-Weber theorem). Topics will include:

1. Fermat's Last Theorem and Serre's Conjecture;
2. Complex and p-adic Galois representations;
3. Algebraic Hecke characters and the Kronecker-Weber theorem;
4. Deformation problems and representability;
5. R = T theorems and modularity lifting.

Prerequisites. A solid understanding of basic algebraic number theory (ramification, splitting of primes, decomposition and inertia groups, local fields, etc), basic representation theory, and 614/631/632 is necessary, as well as (infinite) Galois theory. Some familiarity with modular forms and elliptic curves (and the representations naturally attached to them) will be extremely useful to follow certain portions of Wiles' work.

Abstract. The title is self-explanatory. Tentative Syllabus:

1. • Classical mechanics. I will use one simple example to show the equivalence of three different formulations.
     • Newton’s formulation
     • Lagrangian formulation: action principle
     • Hamiltonian formulation: Legendre transform of Lagrangian, Action principle from another perspective
       • Phase space geometry
       • Liouville’s Theorem
   • Statistical Mechanics
     • Canonical phase space, its measure
     • *optional Ergodic Theory, some related results
2. • Microcanonical Statistics Mechanics
     • Discrete phase space
     • Microcanonical entropy
     • Coupled system
     • Statistical Temperature
3. • Canonical Statistical Mechanics
     • Formalism
       • Partition Function, mean energy, energy variation
     • Continuous phase space
       • Why the statistical temperature is non-negative
4. • Free Energy
   • Maximum Entropy Principle
     • How to recover Microcanonical and Canonical distribution from maximum entropy principle
5. Optional (if time permits)
   • Variational Approximation of the Hamiltonian
   • Mean Field Theory
     • Multidimensional Ising Model
     • Phase Transition through one example

Prerequisites. I intend to keep the course basic. Calculus and any college level probability course would be sufficient. I will go over variational principle fast, so having seen functional derivative would help. No need to know much physics. I will try to give as many simple examples as possible.

Abstract. This mini-course will be an introduction to stability conditions on derived categories, wall-crossing and its applications to birational geometry of moduli spaces of sheaves and enumerative geometry on Calabi-Yau 3-folds (see details below). Schedule:

Lecture 1:

• Motivation
• Basic notions: derived categories, triangulated categories and Bridgeland’s notion of stability conditions on triangulated categories.

• Further discussion on stability conditions. I will give some examples of space of stability conditions.

• Application to birational geometry of moduli space of sheaves: Every Bridgeland stability condition specifies a moduli space of Bridgeland-stable objects. I will explain the relation between wall-crossing for Bridgeland-stability and minimal model program for the moduli space and discuss the result of Bayer-Macrí which shows that every minimal model of the moduli space of stable sheaves on a K3 surface appears as a moduli space of Bridgeland-stable objects on the K3 surface.
• Application to enumerative geometry on Calabi-Yau threefold: I will introduce Donaldson-Thomas (DT for short) invariants and Pandharipande-Thomas (PT for short) invariants of Calabi-Yau threefold. I will explain Toda’s proof of DT/PT correspondence by studying wall-crossing phenomena in the derived category.

Prerequisites. Basic knowledge of categories, schemes and coherent sheaves.

Abstract. The log canonical threshold is an invariant of singularities in algebraic geometry. Given a polynomial f in n variables such that f(0)=0, the log canonical threshold of f at the origin is the supremum over all real numbers c such that |1/f|^c is L² at the origin. Thus, this invariant measures the (possible) singularity of the hypersurface {f=0} at the origin. While the invariant was first studied from the analytic viewpoint in as far back as the 1950’s, it currently receives considerable interest in the field of birational geometry.

During this three days course we will discuss

• Basic properties of the log canonical threshold,
• Generic limits and the ACC Conjecture, and
• The space of R-valued valuations over a variety.

Prerequisites. Math 631 and 632. Attendance in Matt’s “Multiplier Ideals Course” will provide you with additional perspective/motivation, but is not necessary for this course.

Content. This course will be an introduction to various notions of singularities (i.e. how far a local ring is from being regular) in prime characteristic using the Frobenius endomorphism. The amazing fact is that regularity of a Noetherian ring in prime characteristic can be completely characterized by the Frobenius map! The hope is to at least define test ideals in a local setting, although I am not sure I will have enough time. This course actually started out with test ideals being the focus, but I think time will be better spent learning the basic theory.

Background. I will assume you have a good background in algebra. For example, you should have some idea of what the following words mean: regular local ring (very good idea of this one!), Cohen-Macaulay, Gorenstein, Noether Normalization, Cohen Structure Theorem in equal characteristic, (faithful) flatness, completion of a Noetherian local ring (so some version of Mel's 615 will be useful in addition to 614). If you have seen Tight Closure Theory it will help. But I will not talk about Tight Closure in this course.

We will try to blend local and global viewpoints, and familiarity with schemes, divisors on schemes (Weil and Cartier), and a basic understanding of the cohomology of sheaves will be important. In addition, knowledge of the notion of ampleness, Serre vanishing for ample line bundles, Serre duality, dualizing sheaves for projective varieties and applications of these concepts to curves (such as Riemann-Roch) will be useful.

*I like working with non-Noetherian rings. We will draw examples from valuation rings, and the basic theory will be developed without Noetherian hypotheses where appropriate.

**We will usually not work over algebraically closed ground fields. This will sometimes complicate statements of results, and may make the course more technical.

***The first couple of lectures will be gentle, and I will try to get a sense of the audience background to help prepare for future lectures.