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Graduate Courses 2016/17

Please note: all the other courses are described in the catalog.  To view graduate courses offered in previous years, please visit the archived pages at

FALL 2016 WINTER 2017 SPRING 2017

510 de Rham Cohomology

V. Vologodsky (13:00)

511 Intro to Complex Analysis I

J. Isenberg (12:00)

512 Intro to Complex Analysis II

J. Isenberg (12:00)

513 Intro to Analysis I

H. Lin (9:00)

514 Intro to Analysis II

H. Lin (9:00)

515 Intro to Analysis III

P. Gilkey (9:00)

531 Intro to Topology I

R. Lipshitz (13:00)

532 Intro to Topology II

R. Lipshitz (13:00)

533 Intro to Differential Geometry

M. Warren (14:00)

541 Linear Algebra

E. Eischen (12:00)

544 Intro to Algebra I

N. Addington (11:00)

545 Intro to Algebra II

N. Addington (11:00)

546 Intro to Algebra III

E. Eischen (11:00)

561 Intro Methods of Statistics I

E. Eischen (9:00)

D. Levin (12:00)

562 Intro Methods of Statistics II

D. Levin (12:00)

563 Intro Methods of Statistics III

D. Levin (12:00)

 607 Theoretical Neuroscience

Y. Ahmadian (10:00)


 607 Symmetric Functions and Fock Space

B. Young (10:00)

607 Combinatorial Number Theory

S. Akhtari (14:00)

607 Moduli Spaces of Sheaves

N. Addington (12:00)

607 Rational Homotopy Theory

V. Vologodsky (12:00)

607 Toric Varieties

N. Proudfoot (15:00)

607 Floer Homology, Symplectic Geometry, and Low-Dimensional Topology

R. Lipshitz (14:00)

607 A First Course on Yang-Mills Theory

J. Isenberg (14:00)

616 Real Analysis I

C. Phillips (9:00)

617 Real Analysis II

C. Phillips (9:00)

618 Real Analysis III

C. Phillips (9:00)

634 Algebraic Topology I

B. Botvinnik (13:00)

635 Algebraic Topology II

B. Botvinnik (13:00)

636 Algebraic Topology III

B. Botvinnik (13:00)

637 Differential Geometry

M. Warren (10:00)

638 Differential Geometry

P. Gilkey (10:00)

639 Differential Geometry

P. Gilkey (10:00)

647 Abstract Algebra I

A. Kleshchev (11:00)

648 Abstract Algebra II

A. Kleshchev (11:00)

649 Abstract Algebra III

A. Kleshchev (11:00)

681 Advanced Algebra

B. Elias (11:00)

682 Advanced Algebra

B. Elias (11:00)

683 Advanced Algebra

B. Elias (11:00)

684 Operator Algebras

H. Lin (12:00)

685 Operator Algebras

H. Lin (12:00)

 690 K-theory

C. Phillips (13:00)

 691 Characteristic Classes

V. Vologodsky (13:00)


D. Dugger (8:30-1100TR)

Fall Seminars

 Yashar Ahmadian (10:00): 607 Theoretical Neuroscience

Our cognitive functions and  consciousness  emerge out of the collective behavior of billions of nonlinear neurons in our brains, interacting through trillions of plastic synapses. Despite great advances in our understanding of information processing at the single neuron level, many aspects of the collective dynamics and computation in large neuronal networks remain puzzling. To understand the emergence of cognitive functions out of the perplexing complexity of the brain we need to go beyond word theories and develop suitable mathematical theories. Theoretical or Computational Neuroscience provides quantitative theories aiming to connect the biological circuitry of nervous systems to their higher level  functions such as sensory perception, learning and memory, motor control and decision making. In this course we will explore successful examples of such theories and models.

In parallel, I will introduce techniques and concepts from dynamical systems theory, statistical physics, information theory, and statistical learning theory that have proved useful in theoretical neuroscience.

I aim to make this course accessible to students in mathematics, physics and computer science, as well as to biology students who are quantitatively inclined.

Prerequisites: calculus, elementary linear algebra, and elementary probability theory.

I plan to cover the following specific topics (those in parenthesemay or may not be covered depending on the pace of progress and feedback from class):

– Single neuron models

– Neural coding and selectivity

– The efficient coding hypothesis

– Feedforward networks

– (Statistical learning and generalization)

– Models of plasticity

– Unsupervised learning and generative models

– Associative memory in attractor networks

– (Theories of memory capacity)

– Recurrent networks 1: from attractors to chaos

– Recurrent networks 2: excitation and inhibition

– Models of decision making

– (Integrated Information Theory of consciousness)


 Shabnma Akhtari (14:00) 607 Combinatorial Number Theory

The aim of this course is to study some combinatorial problems in number theory.  Broadly, given a sufficiently large set of integers A (or more generally a subset of some abelian group) we are interested in understanding additive patterns that appear in A.  An important example is whether A contains non-trivial arithmetic progressions of some given length k.  The goal is that at the end of the term students are prepared to understand the proof of the Green-Tao Theorem (2003) which states that the rational primes contain arbitrarily long non-trivial arithmetic progressions.

Some specific topics to be covered:

  • Residue classes
  • Sums of squares
  • The weak prime number theorem
  • Multiplicative functions and Dirichlet series
  • Primes in arithmetic progressions
  • Sieve method and its application in establishing and upper bound for the number of primes in an arithmetic progression

Prerequisites: 500-level Algebra and Real Analysis; willingness to learn at least a little bit of Harmonic Analysis

References: Combinatorial and Analytic Number Theory, by Robert Tijdeman, Lecture Notes, available online.  Additive Combinatorics, by K. Soundarajan, Lecture Notes, available online.

 Nicholas Proudfoot(15:00) 607 Toric Varieties

Toric varieties are in some sense the easiest algebraic varieties to work with: they are defined by combinatorial input data, and anything that you want to compute (for example the Picard group and the ample cone) can be expressed in combinatorial terms. We will cover the basics of the subject, with an aim toward gaining a more concrete understanding of various fundamental notions in algebraic geometry through topic examples. It would be a nice follow-up to this year’s algebraic geometry course, but it could also be an accessible introduction to algebraic geometry for someone who is new to the subject.

Winter Seminars

James Isenberg (10:00) 607 A First Course on Yang-Mills Theory

A connection field A on a G-bundle over spacetime is called a Yang-Mills field if it satisfies the Yang-Mills equation D*F=0, where F is the curvature 2-form constructed from the connection A, where D is the covariant derivative based on A, and where * is the Hodge dual. Yang-Mills fields have played a fundamental role during the past 50 years in both physics and math. Physically, Yang-Mills fields (in quantized form) are a crucial element in the Standard Model of particle physics. Mathematically, the self-dual Yang-Mills fields – i.e., those connections with satisfy the condition *F=F – are very useful for exploring the topology of the manifolds on which such fields live.

In this course, after carefully defining Yang-Mills fields and discussing a bit of their history and a bit of their application in physics, we focus on two mathematical studies of Yang-Mills:

  1. The (Atiyah-Ward) representation of self-dual Yang-Mills fields as bundles over twistor space, and the (Witten-Green-Isenberg-Yasskin) representation of general Yang-Mills fields as bundles over ambitwistor space.
  2. The proof of long-time existence for Yang-Mills fields developed from initial data in Minkowski spacetime.

A student who chooses to take this course should know enough differential geometry to be familiar with vector bundles, and should know enough PDE theory to be familiar with the initial value problem for the standard wave equation.

Familiarity with twistors and ambitwistors is not assumed, nor is familiarity with proofs of long-time existence for solutions of hyperbolic PDE systems.  These topics will be introduced during the course.

 Nicolas Addington (12:00) 607 Moduli Spaces of Sheaves

Following Huybrechts and Lehn’s “Geometry of moduli spaces of sheaves.” What is a moduli space.  Many examples.  Stable and semi-stable sheaves. Sketch of th proof that the moduli space exists, a little bit of geometric invariant theory.  Brauer class obstructing the existence of the universal family; twisted sheaves.  Lots on moduli spaces of sheaves on K3 surfaces, which are compact hyperkaehler varieties.

 Robert Lipshitz (14:00) 607 Floer Homology, Symplectic Geometry, and Low-Dimensional Topology

In the first third of the class we will introduce some of the holomorphic curve invariants which provide key tools in symplectic geometry and parts of low-dimensional topology, and outline the analysis underlying their construction.  In the second, we will sketch some of the motivating problems in symplectic topology and applications of holomorphic curves to theses problems.  In the last third, we will discuss one way that symplectic techniques can be applied to 3- and 4-dimensional topology.

The course assumes a reasonable working knowledge of singular homology and a basic knowledge of the language of smooth manifolds (tangent spaces, vector fields, derivatives, differential forms).

Spring Seminars

Benjamin Young (10:00) Symmetric Functions and Fock Space

This class will be an introduction to the theory of symmetric functions, initially following the standard treatment (in, say, Stanley’s Enumerative Combinatorics 2, Chapter 7); we’ll mention links to the representation theory of the symmetric group and applications to the dimer model and various other enumeration problems in combinatorics.  We will also cover the bosonic/fermionic Fock space formalisms, which is an alternative approach to the theory.

Vadim Vologodsky (12:00) Rational Homotopy Theory

As the homotopy groups of spheres modulo torsion are very simple, it is reasonable to expect that there is an algebraic model for describing homotopy groups of any CW complex modulo torsion.  Indeed, the Rational Homotopy Theory, developed by Quillen and Sullivan, provides such a model. Informally, it describes the rational homotopy groups  (i.e. \pi_q(X) \otimes Q) of a simply-connected space in terms of the rational cohomology ring of this space equipped with  “higher operations” (so called Massey products).   In some cases, for example, for any H-space or for any smooth complex projective algebraic variety, the higher operations vanish and as a consequence the rational homotopy groups of such space can be computed just from its cohomology ring.

The precise statement of main result has the form of an equivalence of categories Ho_Q(Top_1) \cong Ho_Q(DGCA_1) between the rational homotopy category Ho_Q(Top_1) obtained from the category Top_1 of simply-connected spaces by formally inverting morphisms f: X–>Y which induce isomorphisms on rational homotopy groups and the homotopy category Ho_Q(DGCA_1)  of differential graded commutative algebras over the field of rational number obtained by inverting homomorphisms f: A–> B of  DGCA which induce isomorphisms on cohomology.   The algebraic constructions in rational homotopy theory proved to be instrumental for development of non-commutative algebraic geometry. Some of these applications will be explained in the course.

Prerequisites: 600 Algebra and 600 Topology sequences.