Graduate » Current Students » Courses Descriptions (2017-18)

REAL ANALYSIS

J. De Simoi

Measure Theory**: **Lebesgue measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem.

Functional Analysis**: **Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p-spaces, Holder and Minkowski inequalities.

**Textbook:**

Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley 2nd edition, 1999

**References:**

Elias Stein and Rami Shakarchi: Measure Theory, Integration, and Hilbert Spaces

Eliott H. Lieb and Michael Loss: Analysis AMS Graduate Texts in Mathematics, 14 (either edition)

H.L. Royden: Real Analysis, Macmillan, 1988.

A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis, 1975.

REAL ANALYSIS II

J. Arthur

Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, topological vector spaces, Schwartz space, distributions.

Functional Analysis: The main topic here will be the spectral theorem for bounded self-adjoint operators, possibly together with its extensions to unbounded and differential operators.

**Textbook:**

G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley.

**Reference**:

E. Lorch, Spectral Theory.

W. Rudin, Functional Analysis, Second Edition, Indian Edition (if available; the book is hard to get, although there is a pdf on line).

COMPLEX ANALYSIS

E. Bierstone

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- Review of elementary properties of holomorphic functions. Cauchy’s integral formula, Taylor and Laurent series, residue calculus.
- Harmonic functions. Poisson's integral formula and Dirichlet's problem.
- Conformal mapping, Riemann mapping theorem.
- Analytic continuation, Monodromy Theorem, Riemann surfaces.
- Modular functions and the Picard Theorems.
- Other topics are possible, like product theorems, elliptic functions, and non-isolated removability theorems.

**Recommended prerequisites: **A first course in complex analysis and a course in real analysis. Measure theory is not required.

**Main References: **

L. Ahlfors: Complex Analysis, 3rd Edition

Stein and Shakarchi: Complex Analysis

**Additional References:**

T. Gamelin: Complex Analysis

W. Rudin: Real and Complex Analysis, 2nd or 3rd edition

D. Sarason: Complex Function Theory

PARTIAL DIFFERENTIAL EQUATIONS I

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This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the field. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' basic bag of tools.

**Textbook:**

L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN-13 978-0821849743

**References:**

- R. McOwen, Partial Differential Equations, (2nd ed),

Hardcover: 2003 Prentice Hall ISBN 0-13-009335-1,

Paperback: 2002 Pearson ISBN-13 978-0130093356 - Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 978-1-4614-4808-2

PARTIAL DIFFERENTIAL EQUATIONS II

S. Alexakis

This course will consider a range of mostly nonlinear partial differential equations, including elliptic and parabolic PDE, as well as hyperbolic and other nonlinear wave equations. In order to study these equations, we will develop a variety of methods, including variational techniques, and fixed point theorems. One important theme will be the relationship between variational questions, such as critical Sobolev exponents, and issues related to nonlinear evolution equations, such as finite-time blowup of solutions and/or long-time asymptotics.

The prerequisites for the course include familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems.

Other topics in PDE will also be discussed.

**Textbook:**

L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN-13 978-0821849743

**Reference:**

- R. McOwen, Partial Differential Equations, (2nd ed),

Hardcover: 2003 Prentice Hall ISBN 0-13-009335-1,

Paperback: 2002 Pearson ISBN-13 978-0130093356 - Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 978-1-4614-4808-2

ALGEBRA I

A. Braverman

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Basic notions of linear algebra**:** brief recollection. The language of Hom spaces and the corresponding canonical isomorphisms. Tensor product of vector spaces.

Group Theory**:** Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.

Ring Theory**:** Rings, ideals, Euclidean domains, principal ideal domains, and unique factorization domains.

Modules**:** Modules and algebras over a ring, tensor products, modules over a principal ideal domain

**Recommended prerequisites **are a full year undergraduate course in Linear Algebra and one term of an introductory undergraduate course in higher algebra, covering, at least, basic group theory. While this material will be reviewed in the course, it will be done at "high speed", assuming that you have already some familiarity with the basics. You will be very well prepared indeed, if you have no difficulties reading and understanding the book, listed here under "Other References", M. Artin: Algebra that the author wrote for his undergraduate algebra courses at MIT.

**Textbooks:**

Lang: Algebra, 3rd edition

Dummit and Foote: Abstract Algebra, 2nd Edition

**Other References:**

Jacobson: Basic Algebra, Volumes I and II.

Cohn: Basic Algebra

M. Artin: Algebra.

ALGEBRA II

F. Murnaghan

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Fields**:** Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.

Commutative Rings**: **Noetherian rings, Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties. structure of semisimple algebras, application to representation theory of finite groups.

**Textbooks:**

Dummit and Foote: Abstract Algebra, 3rd Edition

**Other References:**

Jacobson: Basic Algebra, Volumes I and II.

Cohn: Basic Algebra

Lang: Algebra 3rd Edition

M. Artin: Algebra.

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Local differential geometry: the differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.

Differential forms: exterior algebra, forms, pullbacks, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.

**Prerequisites:** linear algebra; vector calculus; point set topology

**Textbook:**

John M. Lee: Introduction to Smooth Manifolds

TOPOLOGY II

A. Nabutovsky

Fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.

Homology: simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.

**Textbook:**

Allen Hatcher, Algebraic Topology

**Recommended Textbooks:**

Munkres, Topology

Munkres, Algebraic Topology

MATHEMATICAL PROBABILITYI

D. Panchenko

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The class will cover classical limit theorems for sums of independent random variables, such as the Law of Large Numbers and Central Limit Theorem, conditional distributions and martingales, metrics on probability measures. **References:**

Lecture notes and a list of recommended books will be provided.**Recommended prerequisite: **

Real Analysis I.

**Textbook:**

Durrett's "Probability: Theory and Examples", 4th edition

MATHEMATICAL PROBABILITY II

D. Panchenko

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The class will cover some of the following topics: Brownian motion and examples of functional central limit theorems, Gaussian processes, Poisson processes, Markov chains, exchangeability.**References: **

A list of recommended books will be provided.**Recommended prerequisites: **

Real Analysis I and Probability I.

**Textbook:**

Durrett's "Probability: Theory and Examples", 4th edition

TOPICS IN REAL ANALYSIS: DIMENSION PROPERTIES OF DETERMINISTICS AND STOCHASTIC FRACTAL SETS

I. Binder

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The course will start with the discussion of various notions of the dimension for Fractal sets. After investigating various techniques for computing these dimensions, we will study dimensional properties of various Fractal sets arising in Complex Dynamics (Julia sets and

Conformal Cantor sets) and Probability (Brownian motion, Schramm Loewner Evolution). After that, we will go further and discuss multifractal properties of various measures arising in Complex Dynamics, and of general planar harmonic measure.

**Prerequisite(s):**

Core graduate courses in Complex Analysis and Probability, or equivalent

**Textbook(s):**

1. "Fractal sets in Probability and Analysis" by C. Bishop and Y. Peres

2. "Thermodynamic Formalism and Holomorphic Dynamical Systems" by M. Zinsmeister

INTRODUCTION TO LINEAR OPERATORS

G. A. Elliott

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The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced). The intention will be to discuss a number of the topics in Pedersen's textbook Analysis Now. Students will be encouraged to lecture on some of the material, and also to work through some of the exercises in the textbook (or in the suggested reference books).

**Prerequisites:**Elementary analysis and linear algebra (including the spectral theorem for self-adjoint matrices).

**Textbook:**

Gert K. Pedersen, Analysis Now

**References:**Paul R. Halmos, A Hilbert Space Problem Book

Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras

Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS

G. A. Elliott

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.

Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.

Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)

The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classication of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras---both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras---what Bratteli called AF algebras---is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

Students will be encouraged to prepare oral or written reports on various subjects related to the course, including basic theory and applications.**Prerequisites: **

An attempt will be made to supply the necessary prerequisites when needed (rather few, beyond just elementary algebra and analysis).**Textbook:**

Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras**Recommended References:**

Edward G. Effros, Dimensions and C*-algebras

Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Kenneth R. Davidson, C*-Algebras by Example

TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS I: INTRODUCTION TO MEAN CURVATURE FLOW

R. Hashlhofer

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A surface moving so as to decrease its area most efficiently is said to evolve by mean curvature flow. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. Mean curvature flow and its variants have many striking applications in geometry, topology, material science, image processing and general relativity.

In this course we will provide a general introduction to mean curvature flow and some of its applications. In the first half of the course we will focus on the evolution up to the first singular time, which can be described in the classical framework of smooth differential geometry and partial differential equations. A central challenge is then to extend the solutions beyond the first singular time and to analyze the structure of singularities. This is crucial for the most striking applications, and will be our focus for the second half of the course.

**Prerequisite(s):**

Some basic background in geometry is required, i.e. you should be familiar with basic concepts such as surfaces, manifolds, Riemannian metrics and the second fundamental form. Some prior knowledge of PDEs is also helpful, but not strictly required.

**Reference(s):**

K. Ecker: Regularity Theory for Mean Curvature Flow, Birkhauser, 2004

R. Haslhofer, B. Kleiner: Mean curvature flow of mean convex hypersurfaces, CPAM, 2016

T. Ilmanen: Elliptic regularization and partial regularity for motion by mean curvature, Memoirs of the AMS, 1994

B. White: Lecture notes on Mean Curvature flow, available at https://web.math.princeton.edu/~ochodosh/MCFnotes.pdf

TOPICS IN ALGEBRA I: SYMMETRIES - FINITE GROUPS

(D. Bar-Natan, K. Rafi)

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The aim of the course will be to give a modern treatment of the classification of finite subgroups of various linear groups such as SO(3), O(3), SU(2), GL(2,C), their projective counterparts, and of some related arithmetic groups such as GL(2,Z), SL(2,Z). Especially a comprehensive classification of the finite subgroups of GL(2,C) is surprisingly difficult to find in the literature, but recent developments in Singularity Theory, such as the McKay correspondence, allow for a very succinct and "visual" classification.

Mathematical tools to be developed:

Some Euclidean and spherical geometry,

Root systems and reflection groups,

Linear representation theory of groups,

Geometry and algebra of quaternions,

Clifford Algebras and Pin and Spin groups.

The McKay Correspondence and Coxeter-Dynkin diagrams,

Goursat's Lemma for subgroups of products.**Prerequisites: **

a good undergraduate algebra course or, concurrently, the graduate core course in algebra.**References: **

There is no comprehensive textbook covering the material of this course.

Useful sources for the content matter:

--- A good textbook for the basic material, especially algebraic treatment of Euclidean geometry and finite symmetries, is:

Artin, M.: Algebra (2nd Edition).

Addison Wesley 2010. 560 pp. ISBN: 978-0132413770

--- Classification of finite subgroups of SO(3), SU(2) and arithmetic of quaternions:

Conway, J.H. and Smith, D.A.: On Quaternions and Octonions:

Their Geometry, Arithmetic and Symmetry.

A.K.Peters, Natick, Massachussetts, 2003. 159 pp. ISBN 1-56881-134-9

--- An excellent source book for complex numbers, quaternions, and beyond:

H.-D. Ebbinghaus et al.: Numbers Graduate Texts in Mathematics 123 Corrected 3. printing (or later)

Springer-Verlag New York Inc.1995 418 pp. ISBN 0-387-97497-0

--- Geometry and finite symmetry groups, especially those under consideration here:

Coxeter, H.M.S.: Regular Polytopes, Methuen & Co. Ltd, London, First published in 1948, xx + 321 pp.

and

Coxeter, H.M.S.: Regular Complex Polytopes, Cambridge University Press, Cambridge, First published in 1971, xiv + 210 pp.

Some course material is also covered on various web sites that we will discuss during the course.

LIE GROUPS AND HAMILTONIAN PDEs

B. Khesin

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This is a course on infinite-dimensional Lie Groups and how they are used to study Hamiltonian PDEs with a particular emphasis on equations appearing in fluid dynamics. Short course syllabus:

I. Introduction and main notions (Lie groups and Lie algebras, central extensions, Euler-Poincare equations for Lie groups, bihamiltonian systems)

II. Geometry of infinite-dimensional Lie groups and their orbits.

1. Affine Kac--Moody Lie algebras and groups.

2. The Virasoro algebra and group. The KdV equation.

3. Groups of diffeomorphisms. The hydrodynamical Euler equation.

4. Groups of (pseudo)differential operators. Integrable KP-KdV hierarchies.

III. Applications to PDEs in geometric fluid dynamics (ideal hydrodynamics and optimal mass trasport, Otto's calculus, compressible fluid dynamics, Madelung transform, structures on and dynamics of vortex sheets, Fisher-Rao metric).

**Prerequisite:**

A basic course (or familiarity with main notions) of symplectic geometry would be helpful.

**Textbook(s):**

B. Khesin and R. Wendt "The geometry of infinite-dimensional groups,''

Ergebnisse der Mathematik und Grenzgebiete 3.Folge, 51, Springer-Verlag (2008), xviii+304pp, see http://www.math.toronto.edu/khesin/papers/Lecture_notes.pdf

INTRODUCTION TO COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY

P. Milman

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We will study the geometry of algebraic varieties. The course will cover affine and projective varieties, Zariski topology, dimension, smoothness, elimination (via resultants) etc. Topics will include classic results of Nullstellensatz, Bezout Theorem, Weierstrass Preparation, Puiseux Factorization, Chow Theorem and (time permitting) Chow Ideal Theorem and its application to Hironaka Valuation Criterion with proofs from the recommended books below (placed on reserve in the Math. Department's library) and/or printed lecture notes. All necessary results and concepts used will be introduced and explained.

**Prerequisite:**

Familiarity with the basic undergraduate program of the first 3 years in math.

**References:**

1. David Mumford "Algebraic Geometry I . Complex Projective Varieties"

2. M.F. Atiyah and I.G. Macdonald "Introduction to Commutative Algebra"

TOPICS IN ALGEBRAIC GEOMETRY: TORIC VARIETIES AND NEWTON POLYHEDRA

A. Khovanskii

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Toric varieties connect algebraic geometry with the theory of convex polyhedra. This connection provides an elementary way to see many examples and phenomena in algebraic geometry. It makes algebraic geometry much more computable and concrete.

I am also going to presents main ideas of the theory of Newton{Okounkov bodies, tropical geometry and the theory of Grobner bases. These theories relate algebraic geometry with convex geometry, piecewise linear geometry and geometry of lattice.

**Reference(s):**

1. G.Kempf, F.Knudsen, D.Mamford, B.Saint-Donat. \Toroidal Embeddings", Springer Lecture Notes 339, 1973.

2. K.Kaveh, A.Khovanskii. \Newton{Okounkov bodies, semigroups of integral points, graded algebras and intersection theory", Annals of Mathematics, V. 176, No 2, 925{978, 2012.

ADVANCE TOPICS IN ALGEBRAIC GEOMETRY: ALGEBRAIC D-MODULES

A. Braverman

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This is an introduction to the subject of algebraic D-modules. Specific topics include:

1) Modules over the algebra of differential opeators on an affine space, Bernstein inequality, holonomic modules, applications to analytic continuation of distributions with respect to a parameter

2) D-modules on affine varieties, inverse and direct images

3) D-modules on general varieties

4) Holonomic modules on general varieties

5) D-modules with regular singularities and Riemann-Hilbert correspondence

**Prerequisite:**Very basic knowledge of algebraic geometry.

AUTOMORPHIC FORMS AND REPRESENTATION THEORY II: THE TRACE FORMULA

J. Arthur

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The principle of functoriality was introduced by Langlands as a conjecture in 1970. It is the fundamental pillar of the Langlands program, and as such, will have enormous implications for number theory and representation theory. However, we are still a long way from any general proof.

Langlands later (around 2000) made an extended proposal, which he called Beyond Endoscopy, for using the trace formula to attack the functoriality conjecture. The proposal will not be easy to carry out- it will probably require the efforts of many people- but it has already led to new questions that are very interesting in their own right.

The course will be an introduction to Beyond Endoscopy. There is as yet no monograph on the subject, but I shall nonetheless attempt to give a broad overview on the trace formula, and how it is expected to relate eventually to functoriality.

**Prerequisites:**

Core courses in analysis and algebra or their equivalents, basic theory of Lie (and/or algebraic groups), and perhaps also a course in algebraic or analytic number theory. The course will be a natural successor to this year's course MAT1197HS on the trace formula and last year's course MATH1196HS on representations of real groups, but in general, I will try to be flexible about prerequisites.

**References:**S. Gelbart, Lectures on the Arthur-Selberg trace formula, University Lecture Series, AMS, 1995.

J. Arthur, An introduction to the trace formula, Clay Math Proceedings, Vol. 4, 2005, 1-263.

R. Langlands, Beyond Endoscopy, in Contributions to Automorphic Forms, Geometry and Number Theory, Johns Hopkins Univ. Press, 2004, 611-697.

J. Arthur, Problems Beyond Endoscopy, to appear in Proceedings of Conference in Honor of the 70th Birthday of Roger Howe.

ANALYTIC NUMBER THEORY

J. Friedlander

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1. Introduction to some of the famous problems and theorems of the subject

2. Simple tools from elementary number theory, algebra and analysis

3. Dirichlet's theorem on primes in arithmetic progressions

4. Prime Number Theorem

5. Prime number theorem for arithmetic progressions

6 An introduction to sieve methods

7. A selection, if time permits, of some subset of the following topics:

a) further zeta-function theory

b) L-functions and character sums

c) exponential sums and uniform distribution

d) Hardy-Littlewood-Ramanujan method

e) further theory of prime distribution

**Prerequisites:**

1. A half-year course in complex variables such as MAT 334. (This is the most important prerequisite.)

2. A course in groups, rings, fields, such as MAT 347.

3. A half year course course in introductory number theory such as MAT 315.

4. A commitment to attend all lectures.

**Reference(s):**A) General Analytic Number Theory:

1. H. Davenport, Multiplicative number theory, 3rd ed. (revised by H.L. Montgomery) Graduate Texts in Mathematics, Vol. 74 Springer-Verlag 2000

2. H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, Vol. 53. 2004

3. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory I.Classical theory, Cambridge Studies in Advanced Math, 97, Cambridge 2007.

B) More specialized texts:

4. J. Friedlander and H. Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, Vol 57, 2010.

5. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed, (revised by D. R. Heath-Brown) Clarendon Press, Oxford 1986.

6. R. C. Vaughan, The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Mathematics,Vol. 125, Cambridge 1997

TOPICS IN NUMBER THEORY: INTRODUCTION TO SHIMURA VARIETIES

S. Kudla

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This course will provide a basic introduction to the theory of Shimura varieties. The classical modular curves are quotients $\Gamma\back \H$ of the upper half plane

$$\H = \{ z = x+iy\mid y>0\}$$

by subgroups $\Gamma\subset \SL_2(\Z)$ of finite index. Their rich number theoretic content arises from the fact that they can be viewed as moduli space of elliptic curves with some additional structure. In particular, they and various line bundles over them can be defined over number fields, i.e., have 'canonical models'. This fact provides fundamental rationality properties of modular forms, the sections of such bundles, and these, in turn,

have many important arithmetic applications, for example to the special values of L-functions.

Classical generalizations of the modular curves include Hilbert modular surfaces, Siegel modular varieties, complex ball quotients, etc., and, as complex manifolds, all have the form

$$\Gamma\back D$$

where $D$ is a bounded hermitian domain associated to a reductive group $G$ and $\Gamma\subset G$ is an arithmetic subgroup. By a result of Baily-Borel $\Gamma\back D$ is the set of complex points of an algebraic variety defined over $\C$. The theory of Shimura varieties provides a beautiful and deep description of the arithmetic aspects of such varieties and the vector bundles over them. Because of their 'explicit' construction and connection with automorphic forms, such varieties provide a fruitful testing ground for many conjectures about the arithmetic of algebraic varieties which are generally inaccessible. Moreover, the Langlands program and conjectural theory of motives suggest that Shimura varieties should be, in some sense, universal.

For example, every elliptic curve over $\Q$ arises from a modular curve.

Topics will include:

- modular curves as moduli space for elliptic curves

- basic facts about abelian varieties and period matrices

- Siegel modular varieties

- hermitian symmetric domains

- arithmetic groups

- locally symmetric varieties and the Baily-Borel theorem

- Hodge structures

- period domains

- moduli spaces of abelian varieties

- examples, PEL Shimura varieties

- complex multiplication

- canonical models

- Deligne's formulation

- Hodge type Shimura varieties

- automorphic vector bundles

- arithmetic automorphic forms

**References: **

We will loosely follow various sources:

P. Deligne, *Vari\'et\'es de Shimura: interpr\'etation modulaire, et techniques de construction de mod\'eles canoniques*, in ÔAutomorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2Õ, Proc. Symp. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I.(1979), pp. 247Ð289.

J. Milne, *Introduction to Shimura varieties*, Clay Math. Proc. vol 4, 2005,

J. Milne, *Shimura varieties and moduli*, www.jmilne.org/math/xnotes/svh.pdf

G. Shimura, \textsl{Moduli of abelian varieties and number theory}, in 'Algebraic Goups and Discontinuous Subgroups', (Proc. Sympos. Pure Math., Boulder, Co., 1965),

Proc. Symp. Pure Math., IX, Amer. Math. Soc., Providence, R.I.,(1966), pp. 312--332.**Prerequisites:**

Familiarity with Lie groups and Lie algebras, basic algebraic number theory and basic algebraic geometry

** **

COMBINATORIAL THEORY

A. Nabutovsky

A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.

**Prerequisites: **

Linear algebra, elementary number theory, elementary group and field theory, elementary analysis.

TOPICS IN ALGEBRAIC TOPOLOGY: CALABI-YAU MANIFOLDS

M. Gualtieri

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This course is an introduction to Calabi-Yau manifolds, a central topic in algebraic geometry, mathematical physics, and string theory. This course is aimed at beginning graduate students but will also contain more advanced material not covered in our other courses. We will cover all the basics of Calabi-Yau manifolds, providing a foundation for research in the subject. The following are the main topics:

i) Complex manifolds, holomorphic vector bundles, Chern classes

ii) Kahler metrics, the Yau theorem

iii) Toric geometry, hypersurfaces, and the construction of Calabi-Yau manifolds

iv) The Batyrev construction

v) Mirror symmetry and enumerative geometry

vi) Special Lagrangian submanifolds and their moduli

vii) Donaldson-Thomas theory and the moduli of sheaves on a Calabi-Yau

We will use texts and papers by Candelas, Batyrev, Joyce, Donaldson, and Thomas. Evaluation will be based on class participation and the presentation of a review of a research article.

TOPICS IN GEOMETRY:

The course will be an introduction to the geometry of hyperbolic surfaces. Topics to be covered include:

the hyperbolic plane (models, isometries, trigonometry), Teichmüller space and moduli space, Fenchel-Nielsen coordinates, the collar lemma, the Bers constant, Mumford's compactness criterion, Hurwitz's automorphisms theorem, the Nielsen realization problem. Along the way, we will learn the answer to the ultimate question of life, the universe, and everything.**References:**

Benson Farb and Dan Margalit, A Primer on Mapping Class Groups, Princeton Univesity Press.

John Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1, Matrix Editions.

Peter Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser.

TOPICS IN GEOMETRY: CHARACTERISTIC CLASSES

V. Kapovitch

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Vector Bundles, Classifying spaces, Stiefel-Whitney, Chern and Pontriagin classes, Them isomorphism, Cobordism rings, Them spaces and transversality, Hirzebruch signature formula, Connections, curvature and transversality.

**Textbook:**

"Charactersitic classes" by Milnor and Stasheff

INTRODUCTION TO NONCOMMUTATIVE GEOMETRY

G. A. Elliott

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Some of the most basic objects of study in Connes's non-commutative geometry---for instance, the non-commutative tori---will be considered from an elementary point of view. In particular, various aspects of the structure and classication of these objects will be studied, and a comparison made between the properties of these objects and the properties of the underlying geometrical systems. Some indication will be given of their use in index theory.**Prerequisite: **

The Spectral theorem.**References:**

M. Khalkhali, Basic Noncommutative Geometry (EMS Series of Lectures in Mathematics, 2010.)

A. Connes, Noncommutative Geometry, Academic Press, 1994.

J. Gracia-Bondia, J.C. Varilly, and H. Figueora, Elements of Noncommutative Geometry, Birkhauser, 2000.

Y. Kawahigashi and D.E. Evans, Quantum Symmetries on Operator Algebras, Oxford University Press, 1998.

M. Rordam, F. Larsen, and N.J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000.

G.K. Pedersen, Analysis Now, Springer, 1989.

DIFFERENTIAL TOPLOGY

D. Faifman

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Smooth manifolds, Sard's theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

o Manifolds: Embedded and intrinsic definitions, equivalence.

o Charts and coordinates, tangent space.

o Smooth maps, immersions, submersions, diffeomorphisms, smooth embeddings.

o Regular values and Sard's lemma, transversality. Homotopy and stability.

o Manifolds with boundary, oriented manifolds. Intersection index, Lefschetz number.

o Whitney's immersion and embedding theorems.

o Borsuk-Ulam and Brouwer theorems, the Euler characteristic and Hopf-Poincare theorem.

o Additional topics: differential forms and integration, linking numbers and Hopf invariant, elements of Morse theory, Frobenius integrability theorem, topological dimension.

**Prerequisites: **

Introduction to Topology course (MAT327H) and Analysis (MAT257Y).

TOPICS IN DIFFERENTIAL GEOMETRY: LIE GROUPOIDS AND LIE ALGEBROIDS

E. Meinrenken

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Lie groupoids, and their infinitesimal counterparts, the Lie algebroids, arise in many areas of differential geometry, such a Poisson geometry, index theory, foliation theory, or symmetries of differential equations. In recent years, there has been a lot of progress towards understanding these structures, such as the Crainic-Fernandes integrability obstructions and various results on 'multiplicative structures' on Lie groupoids.

Tentative topics include:

1. Lie groupoids: Definitions, examples, basic properties

2. Lie algebroids: Definitions, examples, basic properties

3. The Crainic-Fernandes theorem

4. Cohomology of Lie groupods and Lie algebroids, van Est map

5. Double structures

**Prerequisite(s):**Introduction to Topology course (MAT327H) and Analysis (MAT257Y).

**Reference(s):**

Main textbook: Victor Guillemin and Alan Pollack, Differential Topology.

Other references:

John W. Milnor, Topology from the Differential Viewpoint.

John M. Lee, Introduction to Smooth Manifolds.

B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry - Methods and Applications. Part II. The Geometry and Topology of Manifolds.

Antoni A. Kosinski, Differential Manifolds.

John W. Milnor, Morse Theory.Crainic-Fernades: Lectures on Integrability of Lie Brackets. See http://msp.org/gtm/2011/17-01/gtm-2011-17-001s.pdf

Moerdijk-Mcrun: Introduction to Foliations and Lie Groupoids. Cambridge University Press

TOPICS IN DIFFERENTIAL GEOMETRY

K. Rafi

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This is a continuation of the course "Hyperbolic surfaces" offered in fall 2017 by Maxime Fortier Bourque. It goes somewhat deeper into the study of deformation theory of hyperbolic structures on surfaces. As a first goal, we will cover the recent paper of Mirzakhani "Counting Mapping Class group orbits on hyperbolic surfaces". We start with the needed background material such as, geodesic currents, Thurston measure on PML, Weil-Petersson symplectic form and Thurston metric on Teichmüller space and then cover the basic arguments of the paper. We then discuss some implications and related results including some counting problems in Teichmüller space.

**Prerequisite(s):**Basic hyperbolic geometry and some familiarity with Teichmüller space.

**Textbook(s):**"Closed curves on surfaces", an unfinished monograph by Francis Bonahon

**Reference(s): **

- "The geometry of Teichmüller space via geodesic currents", by Francis Bonahon
- "Counting Mapping Class group orbits on hyperbolic surfaces" by Maryam Mirzakhani
- "Counting Curves in Hyperbolic Surfaces" by Viveka Erlandsson and Juan Souto

INTRODUCTION TO DIFFERENTIAL GEOMETRY: RIEMANNIAN GEOMETRY

R. Rotman

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The topics include:

Riemannian metrics, Levi-Civita connection, geodesics, isometric embeddings and the Gauss formula, complete manifolds, variation of energy.

It will cover chapters 0-9 of the "Riemannian Geometry" book by Do Carmo.

SINGULARITY THEORY

P. Milman

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Topics (provisional):

1. Brief introduction to resolution of singularities by blowing up of smooth centers (so called classical desingularization).

2. Nash desingularization algorithm (for binomial varieties in 'essential' dimension 2 - conjectural in higher dimensions).

3. Applications of the classical desingularization

(a) to a proof of the Chow type theorem for coherent ideals and to the Valuation Criterion of Hironaka (algebraic geometry);

(b) to an estimation of the 'cuspidality' of boundary of a domain by the order of vanishing of the Jacobian of its desingularization map (classical geometry);

(c) to a proof of validity of the Sobolev-Gagliardo-Nirenberg type inequalities on domains with a singular boundary (analysis).

4. Whitney's classification of topologically stable maps of R^2 to R^2 that started "Singularities of Smooth Maps Theory" (time permitting):

(a) C∞ preparation and division Theorems;

(b) Malgrange's version of Preparation Theorem;

(c) Whitney's topological classification of stable R^2 to R^2 maps.

**References:**

1. Original research papers papers and slides style lecture notes (both provided along).

2. Lectures on Resolution of Singularities, János Kollár Princeton University Press, 2007 (by Kollar) and

3. Differentiable Germs and Catastrophes, Cambridge University Press, 1975 (by Th. Broecker).

**Prerequisites: **

Familiarity with the basic undergraduate program in analysis, geometry and commutative algebra. All necessary results and concepts used will be introduced and explained.

COMPLEX MANIFOLDS

J. Bland

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An introduction to complex manifolds: vector bundles, complex line bundles, Hermitian connections, curvature, Kahler metrics, Kodaira embedding theorem, Hodge theory.

**Prerequisites:**A good background in differentiable manifolds including the de Rham complex of differential forms, Stoke's thereom, Frobenius integrability.

A good background in complex analysis in one variable.

**References:**Kodaira: ''Complex manifolds and Deformation of complex structures''

Griffiths and Harris: ''Principles of Algebraic Geometry''

SET THEORY

S. Todorcevic

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This will be a descriptive set theory course specializing on recent connections between descriptive set theory and certain areas of Ramsey theory and topological dynamics. We will explore first the close connections between the theory of computations of Ramsey degrees on one side and the representation theory of universal minimal flows of topological groups on the other. Then we will study dynamics of isometry groups and its close relationship with the approximate Ramsey property of finite metric structures. If time permits, we will apply this theory and give a representation of the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex and other related metric structures.

**Prerequisite(s):**MAT409 or MAT 309

**Textbook(s):**1. A. Kechris, Classical Descriptive Set Theory. Graduate Text in Mathematics, Vol 156, Springer 1995

2. A.Pestov, Dynamics of infinite-dimensional groups, Amer. Math. Society, 2006.

TEACHING LARGE MATHEMATICS CLASSES

J. Repka

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The goals of the course include techniques for teaching large classes, sensitivity to possible problems, and developing an ability to criticize one's own teaching and correct problems.

Assignments will include such things as preparing sample classes, tests, assignments, course outlines, designs for new courses, instructions for teaching assistants, identifying and dealing with various types of problems, dealing with administrative requirements, etc.

The course will also include teaching a few classes in a large course under the supervision of the instructor. A video camera will be available to enable students to tape their teaching for later (private) assessment.

TOPICS IN GEOMETRIC ANALYSIS: CONVEXITY

A. Burchard

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The main topic of this course would be on geometric inequalities, starting from Brunn-Minkowski theory and the isoperimetric inequality, and then turning to analysis in high dimensions, as presented in the textbook of Artstein-Avidan, Giannopoulos, and Milman, referenced below.

**Prerequisite(s):**Basic analysis and functional analysis; some probability (at the level of MAT 1000/1001 which could be taken at the same time. Some background (undergrad-level) in Probability would be useful.

**Textbook(s):**

Asymptotic Geometric Analysis, Part I by

Shiri Artstein-Avidan: Tel Aviv University, Tel Aviv, Israel,

Apostolos Giannopoulos: University of Athens, Athens, Greece,

Vitali D. Milman: Tel Aviv University, Tel Aviv, Israel

TOPICS IN GEOMETRIC ANALYSIS: OPTIMAL TRANSPORTATION, GEOMETRY, AND DYANMICS

R. McCann

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This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, and nonlinear partial differential equations.The basic problem is to find the most efficient structure linking two continuous distributions of mass --- think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.

Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Amp\'ere), geometric inequalities with sharp constants, periodic orbits for dynamical systems, long time asymptotics in kinetic theory and nonlinear diffusion, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, dynamical systems and fluid mechanics, not to mention physics, economics, and geometry. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provides a framework for adapting powerful ideas from Riemannian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.

**Prerequisite(s):**

(Corequisite: MAT 1000F or equivalent)

**Textbook(s):**F Santambrogio. Optimal transport for applied mathematicians. Birkhauser 2015.

**Reference(s):**Ambrosio, Gigli and Savare: 'Gradient Flows in Metric Spaces and in the Space of Probability Measures'.

Birkhauser 2005 (and subsequent works by Gigli).

D Burago, Y Burago and S Ivanov ''A course in metric geometry'' AMS 2001.

McCann and Guillen 'Five lectures on optimal transport' In Analysis and Geometry of Metric Measure Spaces

G. Dafni et al, eds. Providence: Amer. Math. Soc. (2013) 145-180

Villani 'Topics in Optimal Transportation', AMS GSM \#58 2003

Villani 'Optimal Transport: Old and New', Springer-Verlag 2009.

ASYMPTOTIC AND PERTURBATION METHODS

C. Sulem

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- Local Methods.
- Classification of regular and singular points of linear ODEs. Approximate solutions near regular, regular-singular and irregular singular points; irregular point at infinity.
- Asymptotic series.
- Some examples of nonlinear differential equations.

- Asymptotic expansion of integrals.
- Laplace method.
- Method of stationary phase.
- Steepest descent.

- Perturbation methods.
- Regular perturbation theory.
- Singular perturbation theory.

- Global Analysis.
- Boundary layer theory.
- WKB theory : Formal expansion, conditions for validity, shortwave asymptotics: geometrical optics and semi-classics.
- Bohr-Sommerfeld approximation.
- Multiple scale analysis for ODEs: Resonance and secular behavior, damped oscillator.
- Multiple scale analysis for PDEs.

** Prerequisites:**

2nd year calculus

ODEs and PDE courses

Complex variables

**Textbooks:**

Advanced Mathematical Methods for Scientists and Engineers; Asymptotic Methods and Perturbation Theory, by Carl M. Bender and Steven A. Orszag, 1st edition, McGraw-Hill, 1978, 2nd ed., Springer 1999.

Applied Asymptotic Analysis by Peter Miller, AMS, Grad. Studies in Math, Vol 75.

T

M. Sigal

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In this course we study partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. We will touch upon questions of existence, long-time behaviour, formation of singularities, pattern formation. We will also address questions of existence of static, traveling wave, self-similar, topological and localized solutions and their stability.

Specifically we consider Allen-Cahn equation (material science), Ginzburg-Landau equation (condensed matter physics -superfluidity and superconductivity ), Cahn-Hilliard (material science, biology), Mean curvature flow and the equation for minimal and self-similar surfaces (geometry, material sciences), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), Gross-Pitaevskii equation (Bose-Einstein condensation) and Chern-Simmons equations (particle physics and quantum Hall effect).

The course will be relatively self-contained, but familiarity with elementary ordinary and partial differential equations and Fourier analysis will be assumed.**Prerequisites: **

Elementary ordinary and partial differential equations, Fourier analysis, Elementary analysis and theory of functions or physics equivalent of these.**Textbook: **

The instructor's notes**Recommended books: **

R. McOwen, Partial Differential Equations, Prentice Hall, 2003

J. Ockedon, S. Howison, A. Lacey, A. Movchan, Applied Partial Differential Equations, Oxford University Press, 1999

Peter Grindrod Patterns and Waves: Theory and Applications of Reaction-diffusion Equations (Oxford Applied Mathematics & Computing Science) 1996

GENERAL RELATIVITY

S. Aretakis

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General Relativity is a geometric theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It describes the evolution of gravitational physical systems such as planetary systems, galaxies, black holes and ultimately the universe as a whole.

This course will strive in a very intuitive, but still rigorous, way to uncover the subtle connections of Einstein's theory with geometry and analysis. We will prove purely mathematical theorems which have direct consequences for fundamental phenomena of our natural world. Moreover, this course will provide students with the background needed for studying recent research papers in the field and for initiating their own research projects.

The following topics will be covered:

A. Geometry: Introduction to Riemannian, Lorentzian and Null geometry,

B. Einstein equations and fundamental predictions: gravitational waves, black holes, trapped surfaces,

Penrose incompleteness theorem

C. Black hole dynamics: Schwarzschild and Kerr families

**Prerequisites:**

Some familiarity with basic differential or Riemannian geometry is desirable but not required. All students should be familiar with advanced multivariable calculus.

**References:**

We will mainly follow the lecturer's ''Lecture notes on general relativity'' (available online).

Robert Wald's "General relativity",

Hawking and Ellis' ''The large scale structure of spacetime''.

(View Timetable)

The goal of this course is to explain key concepts of Quantum Mechanics and to arrive quickly to some topics which are at the forefront of active research. In particular we will present an introduction to quantum information theory, which has witnessed an explosion of research in the last decade and which involves some nice mathematics.

We will try to be as self-contained as possible and rigorous whenever the rigour is instructive. Whenever the rigorous treatment is prohibitively time-consuming we give an idea of the proof, if such exists, and/or explain the mathematics involved without providing all the details.

**Prerequisites:**

* Some familiarity with elementary ordinary and partial differential equations

* Knowledge of elementary theory of functions and operators would be helpful

Syllabus (not all topics would be covered):

* Schroedinger equation

* Quantum observables

* Spectrum and evolution

* Atoms and molecules

* Density matrix and open systems

* Quasiclassical asymptotics

* Perturbation theory

* Adiabatic theory and Berry phase

* Self-consistent approximations

* Bose-Einstein condensation

* Open systems and Lindblad evolution

* Quantum entropy

* Quantum channels and information processing

* Quantum Shannon theoroms

**References:**

S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2011

For material not contained in this book, e.g. quantum information theory, we will try to provide handouts and refer to on-line sources.

**Additional reference:**

L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008.

TOPICS IN MATHEMATICAL PHYSICS: PARTIAL DIFFERENTIAL EQUATIONS OF QUANTUM PHYSICS

M. Sigal

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In this course we cover several fundamental equations of quantum physics: the Schrödinger equation, which lies at the foundation of Quantum Mechanics, the Chern-Simons, Landau-Lifshitz and Hartree-Fock equations playing an important role in condensed matter physics, the Ginzburg-Landau equations of superconductivity, and the Yang-Mills equations of particle physics.

Textbook:

Jürgen Jost, Riemannian Geometry and Geometric Analysis, Springer-Verlag Berlin Heidelberg 2011

COMPUTATIONAL MATHEMATICS: SHAMELESS MATHEMATICA

D. Bar-Natan

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Yes, we all dream of the day we will prove that powerful theorem, whose beauty and sophistication will leave our colleagues breathless. It will, of course, be a product or pure thought, affirming that our intellect rises far, oh so far, beyond everybody else's. Obviously, no computers will be used. We are artists and philosophers, not technicians.

As a temporary measure I have learned to work with computers, and I plan to share what I have learned with you. For me, computer-assisted mathematics is a powered exoskeleton (seen Avatar? Iron Man?) for the brain. It's still my inner powers that everybody should admire, yet they reach much farther now that I've learned how to integrate them so tightly with the machine. Learn that too and reach far! I often use the platform "Mathematica" (though not only), and hence that's what I'll teach (though perhaps not only).

About one third of the course will be a systematic overview of Mathematica following that or another textbook. The other two thirds will be divided into chapters, each about some (mathematical) real life problem that I have at some point encountered and solved with computers. The typical chapter will start with a mathematical introduction (sometimes deep and meaningful in itself). I will then pose a computational problem, and challenge you to solve it better than the solution that I found and will present a week or two later. Many (though not all) of the problems will involve algebraic computations in knot theory, as this is what I know best. There will also be graphics, and some interaction with the web and with TeX.

See the web page of the 2016 "Shameless Mathematica" class at http://drorbn.net/16-1750.

This course is designed to provide upper level undergraduates and beginning graduate students with an understanding of how mathematical modeling is the link between mathematics and the rest of the world. Nature is highly complex and mathematical modeling can help to unlock the secrets to problems in engineering, physics, astronomy, biology, medicine, chemistry, economics, and the social sciences among other fields. Modeling is a way of structuring questions about the world so that mathematical techniques may provide insight. It is as much an art as a science.

This course will cover a broad spectrum of modeling approaches including optimization, dynamical systems, and stochastic processes. Some topics covered include sensitivity analysis, model robustness, multi-variable optimization, linear programming, operations research, compartment models, neuronal models, bifurcation theory, renewal theory, Markov chains, queuing theory, diffusion, and some stochastic models in biology. Regular computer lab sessions will be used to teach visualization and simulation of the mathematical models studied.

A substantial component of the course grade will be a modeling project with a student-selected topic. Evaluation of the project will be based on a research report and a presentation. Regular written homework and two midterm exams will also be part of the course grade.

**Prerequisites:**

In order to be successful in this course, it will be necessary to have a solid foundation in ordinary differential equations (MAT244) and linear algebra (MAT223,MAT224). A course in probability or some experience with Matlab or Python would be helpful.

**Textbook:**

"Mathematical Modeling" by Mark M. Meerschaert (4th ed.) ISBN13:9780123869128

**References:**

"Computational Mathematical Modeling: An Integrated Approach Across Scales" by Daniela Calvetti and Erkki Somersalo ISBN13:9781611972474

"Modeling and Simulation in Medicine and the Life Sciences" by Frank C. Hoppensteadt and Charles Peskin (2nd ed.) ISBN13:9780387215716

"Mathematical Physiology I: Cellular Physiology" by James Keener and James Sneyd (2nd ed.) ISBN13:9780387758466

"Mathematical Physiology II: Systems Physiology" by James Keener and James Sneyd (2nd ed.) ISBN13:9780387793870

TOPICS IN DYNAMICAL SYSTEMS: GEOMETRY OF NEUROSCIENCE

M. Marcolli

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This class will cover topics in geometry and topology applied to neuroscience. In particular, topics covered include: the geometry and topology of the visual cortex, geometry of segmentation and invariance, neural codes and neural rings, brain networks and topology, mathematical models of learning, including mathematical aspects of neural networks and deep learning, mathematical models of vision and language.

**Prerequisites:**

Some prior familiarity with differential geometry, real and complex analysis, and ordinary differential equations will be helpful.

**Textbook:** Neurogéomètrie de la Vision, Jean Petitot, Les Éditions de l'École Polytechnique, 2008.

Relevant research articles and other reading material will be posted on the class website as the class progresses.

INTRODUCTION TO HOLOMORPHIC DYNAMICS

G. Tiozzo

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We are planning to study the theory of dynamical systems in one complex variable.

The theory of iterations of polynomial maps on the plane goes back to the works of Julia and Fatou at the beginning of the 20th century, but it did not fully flourish until the 1980s with the introduction of computer simulations and the works of Douady, Hubbard, Sullivan, Thurston, Milnor, Lyubich, and several others. We will investigate the dynamical properties of iterations of polynomials and rational maps on the plane and discuss their classification and their parameter space.

The subject is by now classical, but still full of research questions. We will approach the topic from the beginning, assuming only fundamental knowledge of complex analysis and topology as provided by the core courses. Towards the end, we will also discuss the theory of entropy for real and complex polynomials, a topic I work on and which has been recently revisited by Thurston and still deserves to be fully explored. Topics

- Riemann surfaces and the Poincare' metric.
- The Julia set and the Fatou set.
- Local fixed point theory.
- Global fixed point theory.
- Structure of the Fatou set.
- Caratheodory theory and local connectivity.
- The Mandelbrot set.
- Combinatorial dynamics.
- Kneading theory.
- Entropy of unimodal maps.
- Hubbard trees.
- Core entropy.

**Prerequisite(s):**

Graduate complex analysis, real analysis, and topology.

**Textbook:**

J. Milnor, Dynamics in One Complex Variables

**Reference(s):**

J. Milnor, W. Thurston, On iterated maps of the interval

W. de Melo, S. v. Strien, One dimensional dynamics

C. T. McMullen, Complex Dynamics and Renormalization

MATHEMATICAL THEORY OF FINANCE

L. Seco

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Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.

**Prerequisites**:

APM 346H1, STA 347H1

READING IN PURE MATHEMATICS: VARIATIONAL THEORY OF MINIMAL SURFACES

A. Neves/ F. Marques

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Minimal surfaces are solutions of the most basic variational problem in submanifold theory, that of extremizing the area. These surfaces have been intensely studied for the past 250 years and have found striking applications in other fields of mathematics. We plan to discuss the basic variational theory of these objects, including the questions of existence and regularity. We will talk about unstable minimal surfaces obtained by min-max methods and their recent impact on the field. This includes in particular the solution to the Willmore Conjecture (1965).

Prerequisite: Riemannian geometry

Evolution equations arising in the modelling of life science problems is a fascinating subject area. It is at the very heart of the understanding of many important problems arising in biology, medicine, ecology, physics and mechanics. The goal of this course is to convince the audience of this and to show the important role of applying mathematics to real world situations, and explain, in particular, experimental findings.

I hope that this course will create additional interest in life-science problems among young researchers. Moreover, we aim to show how modelling real world situations (life science problems) has a crucial impact on mathematics, creating new challenging situations for mathematics itself, in particular for infinite-dimensional dynamical systems and their attractors.

Overview:**1. Introduction and course overview2. Mathematical modelling**2.1 What is mathematical biology?

2.2 Multiscale biological processes

2.3 Convergence of sciences: system biology, system medicine and mathematics

2.4 Verification of biological systems with diffusion, convection and interaction of species

**3. Functional spaces**3.1 Embeddings theorem

3.2 Fractal dimension

3.3 Kolmogorov entropy and its asymptotics

**4. Infinite dimensional dynamical systems**4.1 Global attractor and its generalisations

4.2 Reduction principle in dynamical systems

4.3 Biology enforce reduction principle to fail

**5. Mathematics itself benefits from biological modeling**5.1 An example

5.2 New classes of degenerate parabolic type equations

5.3 Effect of degeneracy on dynamical behavior of biological systems

**References:**

A.V. Babin and M.I.Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol.25, North-Holland, 1992.

M.A.Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics, Gakkotoscho International Series, Tokyo, vol.33 , 2010.

M.A.Efendiev, Evolution Equations Arising in the Modelling of Life Sciences, ISNM Series, Birkhäuser/Springer, vol.163, 2013.

M.A.Efendiev, Attractors for Degenerate Parabolic type Equations, American Mathematical Society; Serie Mathematical Surveys and Monographs, vol.192, 2013.

R.Temam , Infinite-Dimensional Dynamical Systems in Mechanics and in Physics, Springer, Applied Mathematical Sciences,68, 1988.

J.Wu , Theory and Applications of Partial Functional Differential Equations, Springer, Applied Mathematical Sciences, 119,

MEDICAL IMAGING

A. Nachman

(View Timetable)

We will present an elementary introduction to the revolutionary and important new theory of Compressed Sensing. We will fill in the basic mathematical prerequisites on Fourier Transforms and Wavelets. Other topics will depend on the interests of the class: we will choose between a detailed explanation of how MRI works, imaging electric properties of tissue, or present modern techniques in signal processing for denoising, segmentation and registration.

**Students requiring individual instruction in mathematical topics should consult with the Mathematics Graduate Office.**