#### Math 500. Abstract Algebra I

Professor Nathan Dunfield

Basic facts about groups, rings, vector spaces, such as those covered in Math 416 and Math 417 courses, are assumed. Instructors should not spend time on elementary material: the syllabus is quite full. Books that could be used include “Abstract Algebra” by Dummit and Foote, “Algebra” by Hungerford, and “Advanced Modern Algebra” by Rotman.

1. Group Theory. [Approximately 4.5 weeks]
(a) Isomorphism theorems for groups.
(b) Group actions on sets; orbits, stabilizers. Application to conjugacy classes, centralizers, normalizers.
(c) The class equation with application to finite p-groups and the simplicity of A5.
(d) Composition series in a group. Refinement Theorem and Jordan-Hölder Theorem. Solvable and nilpotent groups.
(e) Sylow Theorems and applications.
2. Commutative rings and Modules.[Approximately 5 weeks]
(a) Review of subrings, ideals and quotient rings. Integral domains and fields. Polynomial rings over a commutative ring.
(b) Euclidean rings, PID’s, UFD’s.
(c) Brief introduction to modules (over commutative rings), submodules, quotient modules.
(d) Free modules, invariance of rank. Torsion modules, torsion free modules. Primary decomposition theorem for torsion modules over PID’s.
(e) Structure theorem for nitely generated modules over a PID. Application to finitely generated Abelian groups and to canonical form of matrices.
(f) Zorn’s lemma and Axiom of Choice (no proofs). Application to maximal ideals, bases of vector spaces.
3. Field Theory. [Approximately 5 weeks]
(a) Prime fields, characteristic of a field.
(b) Algebraic and transcendental extensions, degree of an extension. Irreducible polynomial
of an algebraic element.
(c) Normal extensions and splitting fields. Galois group of an extension.
(d) Algebraic closure, existence and uniqueness via Zorn’s Lemma. Finite fields.
(e) Fundamental theorem of Galois theory.
(f) Examples of Galois extensions. Cyclotomic extensions.
(g) If time permits, application of Galois theory to solution of polynomial equations, symmetric functions and ruler and compass constructions.

#### MATH 502 INTRODUCTION TO COMMUTATIVE ALGEBRA

Professor S. P. Dutta

12:30 – 1:50 Tu-Th

This course is intended mainly for students who are going to specialize in Commutative Algebra, Algebraic Geometry, Algebraic K-theory and Algebraic Number Theory.

In this course we will mainly focus on Noetherian rings and modules. The topics will include: Primary decomposition, Artin-Rees Lemma, Flatness, Completion, Hilbert-Samuel Polynomial, Dimension Theory, Integral extensions, Going-up and Going-down theorems, Noether’s Normalization (its geometric interpretation), Regular rings and the notion of depth. We would also like to study Cohen- Macaulayness if time permits.

Prerequisite: Math500, 501

Recommended text: Commutative ring theory by H. Matsumura

#### MATH 518 differentiable manifolds

Professor Eugene Lerman

Prerequisites: Point set topology and linear algebra.

Course outline

1. Manifolds: Definitions and examples including projective spaces and Lie groups; smooth
functions and mappings; submanifolds; Inverse Function Theorem and its applications
including transversality; (co)tangent vectors and bundles; vector bundles; manifolds
with boundary; orientations.
2. Calculus on Manifolds: Vector fields, flows, and Lie derivative/bracket; differential forms
and the exterior algebra of forms; orientations again; exterior derivative, contraction,
and Lie derivative of forms; integration and Stokes Theorem, DeRham cohomology.

Recommended Texts

• Introduction to Smooth Manifolds by John M. Lee, Springer, ISBN: 978-1-4419-9981-8 (Print) 978-1-4419-9982-5 (Online) [free online access from a campus computer]

The course grade will be based on weekly homework (35%), a midterm (25%) and a final (40%).

#### MATH 520

Professor Eugene Lerman

Prerequisites: A course in differentiable manifolds such as Mathematics 518.

Course outline

• Symplectic linear algebra.
• Basic examples of symplectic manifolds.
• Review of vector bundles.
• Lagrangian embedding theorem, applications.
• Classical Hamiltonian systems.
• Hamilton’s principle, Euler-Lagrange equations, Legendre transform, examples.
• Poisson brackets, complete integrability.
• A fast introduction to Lie groups and group actions.
• Symplectic group actions, symmetries of Hamiltonian systems.
• Moment map and its properties.
• Fiber bundles, basic forms, coisotropic reduction.
• Symplectic quotients.
• Completely integrable systems revisited, action-angle variables.
• Stratified spaces, singular quotients.
• Dirac structures, port-Hamiltonian systems
• Brief introduction to symplectic topology

#### MATH 526 Algebraic Topology II

Instructor: Vesna Stojanoska Email: vesna AT illinois.edu Office: #307, located at 805 W Pennsylvania Ave, Urbana

Course description: This is the second semester of the algebraic topology sequence, and for the most part will concentrate on studying singular cohomology, its structure and applications. The first part of the course will concentrate on the cup product in cohomology, Poincar´e duality, and various applications. Then we will study vector bundles, characteristic classes, and cohomology operations, and if time permits, we’ll cover some basics related to complex K-theory.

Prerequisites: MATH 525 or consent of instructor.

Textbooks: The main textbooks will be
Algebraic Topology, by Hatcher. (Free pdf version is available at http://www.math. cornell.edu/~hatcher/AT/ATpage.html)
Geometry and Topology, by Bredon. (Free pdf version is available through the library.
Characteristic Classes, by Milnor and Stasheff. (A free pdf version is available through the library.)
Algebraic Topology, by Switzer,
A Concise Course in Algebraic Topology, by May.

Assignments: There will be homework every 2-3 weeks, and a final project.

#### Math 531 : Analytic Theory of Numbers I

Instructor: Alexandru Zaharescu E-mail: zaharesc@illinois.edu Office Hours: Mondays 10 am – 1 pm

Course Description. The first part of the course will follow closely Professor Hildebrand’s lecture notes and the second part of the course will follow Davenport’s book.

Topics will include:

• Theory of arithmetic functions
• Elementary theory of primes
• General theory of Dirichlet series
• Properties of the Riemnann zeta function and proof of the Prime Number Theorem
• Dirichlet characters and Dirichlet L-functions
• Primes in arithmetic progressions

Prerequisite: MATH 448 and either MATH 417 or MATH 453.

Grading Policy: Comprehensive final exam: 35%; Two midterm exams: 2 × 25=50%; Homework: 15%.

Recommended Textbooks:

• A. J. Hildebrand, Introduction to Analytic Number Theory, available on Pro- fessor Hildebrand’s webpage.
• Harold Davenport, Multiplicative number theory. Third edition. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. xiv+177 pp. ISBN: 0-387-95097-4

#### Math 533: Characteristic Classes

Professor: G. La Nave

Course description: Vector bundles occur naturally in geometry and physics. From their abstract definition, it is clear that locally vector bundles of the same rank are all the same. So a local observer cannot distinguish a cylinder from a Moebius bundle. Characteristic classes, among other things, are a formidable cohomological tool to just do that. They have many applications in algebraic topology, algebraic geometry, arithmetic geometry, geometric analysis, and mathematical physics. They can distinguish between the tangent bundles of two different smooth structures on the same topological manifold, they can be used to count the number of lines lying in a cubic surface in complex projective space, or decide whether a given 4-manifold admits an almost complex structure. They also feature prominently in String Theory and in the topological classification of matter In this course we will learn, amongst other things

• Smooth manifolds and their tangent bundles
• Vector bundles, clutching functions, and principal bundles
• Classifying spaces and homotopy theory of bundles
• The Splitting Principle and the Projective Bundle Theorem
• Definitions of Stiefel–Whitney and Chern classes
• Homology of Grasmannians
• Applications of Stiefel–Whitney classes to manifolds
• The Thom Isomorphism Theorem and the Gysin Sequence
• Characteristic classes as obstructions
• K–theory: the Chern Character, the Bott Periodicity Theorem
• Connections, curvature, and Chern–Weil Theory
• Pontryagin classes, transversality, and cobordism
• The Atyiah-Singer index theoremOther possible topics include: Adams Spectral sequences, Supersymmety and supersymmetric proof of Atyiah-Singer Theorem, topological classification of matter and enumerative geometry.

Refences:

Pre-requisists are a working knowledge of homology and cohomology and rudiments of homotopy theory

Grades Will be based on Homework

#### MATH 540

XIAOCHUN LI

This course is mainly about the theory of functions on Rn, and it is designed for first-year graduate students in mathematics. The following topics are planed to be covered:

• Abstract measure theory, Lebesgue measure, and measurable functions
• Lebesgue theory of integration and convergence theorems
• Differentiation of functions, functions of bounded variation
• Lp spaces
• Hilbert spaces and Fourier series (if time permits)

Lectures: TBA

Textbook: G. B. Folland, Real Analysis, John Wiley & Sons

Grading: Homework (10%), 1 or 2 midterms (40%), and a final (50%). No make-up exams.

Exams:The midterm exam will be a 50-minute exam in class. The final will be a 3-hour exam.

#### Math 542 Complex Variables I, Fall 2023

M. Burak Erdo˘gan

This is a standard introductory course in complex analysis. Topics will include:

1. Complex number system. Basic definitions; topology of the complex plane; Riemann sphere, stereographic projection.
2. Differentiability. Basic properties; Cauchy-Riemann equations, analytic functions.
3. Elementary functions. Fundamental algebraic, analytic, and geometric properties. Basic conformal mappings.
4. Contour integration. The Cauchy integral theorem; consequences.
5. Sequences and series. Uniform convergence; power series.
6. The local theory. Zeros, Liouville’s theorem, Maximum modulus theorem, Schwarz’s Lemma.
7. Laurent series Classification of isolated singular points; Riemann’s theorem, the Casorati- Weierstrass theorem.
8. Residue theory. The residue theorem, evaluation of improper integrals; argument prin- ciple, Rouche’s theorem, the local mapping theorem.
9. The global theory. Winding number, general Cauchy theorem and integral formula; simply connected domains.
10. Uniform convergence on compacta. Ascoli-Arzela theorem, normal families, theorems of Montel and Hurwitz, the Riemann mapping theorem.
11. Infinite products. Weierstrass factorization theorem.
12. Runge’s theorem. Applications.
13. Harmonic functions. Basic properties; Laplace’s equation; analytic completion; the Dirichlet problem.

Prerequisites: Math 446 and Math 447, or equivalents.

Textbook: An Introduction to Complex Function Theory, B. Palka.

#### Math 547

Time and place: MWF 12-12:50, 1038 FLB

Instructor: T. Oikhberg, oikhberg@illinois.edu Office: Computer Applications Building (CAB) 33

This course is mainly concerned with applications of Asymptotic Geometric Analysis (that is, the theory of Banach spaces – or convex bodies – of finite, but “large,” dimension) to Quantum Information Theory. Our main guide will be Aubrun and Szarek, Alice and Bob meet Banach, which is available electronically at the UI library (http://www.ams. org.proxy2.library.illinois.edu/books/surv/223/), so no purchase is necessary. Other books or papers may be introduced as needed.

We shall begin by developing the asymptotic geometry tools (Part 2 of our book). The topics include:

• Convexity: an introduction (see Chapter 1 of the book).
• Banach-Mazur distances. John’s ellipsoid and the distance to the Euclidean space.
• Volume inequalities, such as Brunn-Minkowski.
• Concentration inequalities. Khintchine-Kahane Inequality.
• Applications of concentration inequalities, such as Dvoretzky Theorem about “almost Euclidean” sections of convex bodies.We then turn our attention to quantum information, and introduce some basic notions (see Part 1 of our book):
• Channels and completely positive maps. Stinespring Represen- tation Theorem.
• Quantum entanglement. Entanglement witnesses.Thus fully prepared, we move on to Part 3, and apply asymptotic geometry to quantum information theory. Topics to be covered include (tentatively):
• Entropy and its additivity properties.
• Use of Dvoretzky Theorem to disprove the additivity of entropy.
• Time permitting: The “size” of the set of separable states.
• Time permitting: Random states, and (in)effectiveness of Positive Partial Transpose in detecting entanglement.

The final grade will be based on homework (there will be 5 to 8 assignments throughout the semester).

#### Math 554: Linear Analysis and Partial Differential Equations

Time: Tu-Th, 9:30 – 10:50 am

References:

1. Math 554 Notes by E. Kirr, posted online;
2. Partial Differential Equation by Lawrence C. Evans (recommended);
3. Equations of Mathematical Physics by V.S. Vladimirov (recommended);
4. Sobolev Spaces by R. A. Adams (recommended);
5. Semigroups of linear operators by A. Pazy (recommended).

Syllabus: We will basically cover Chapters 5-7 from the reference  but use mostly my notes. We will start by relating certain partial differential equations with physical phenomena. Next, we will introduce Sobolev spaces and review certain properties of Banach, Hilbert spaces, as well as of linear operators defined on them. Then, we will use this framework to study both weak and classical solutions of the following second order equations on bounded domains: elliptic equations (Lax-Milgram Theorem, reg- ularity, maximum principles, spectral properties i.e., eigenvalues and eigenfunction); parabolic equations (energy estimates, regularity, maximum principle); hyperbolic equations (energy estimates, propagation of singularities).
We will also develop the semigroup theory and show how it applies to linear evo- lution equations (such as the parabolic and hyperbolic ones). The theories and the applications encountered in this course will create a strong foundation for studying nonlinear equations and nonlinear science in general.

Prerequisites: Real Analysis (Math 447 or equivalent) and Ordinary Differential Equa- tions (Math 489 or equivalent) or consent of the instructor. Measure Theory (Math 540), Functional Analysis (Math 541) and familiarity with partial differential equa- tions (Math 553) are helpful but not required.

Homework: Assignments will be given approximately every two weeks. Solutions should be written up independently and turned in by the deadline for each assignment.

#### MATH 562: PROBABILITY THEORY II

Instructor. Partha Dey, psdey(at)illinois.edu

Times and Places. TR 9:30–10:50 am at G30 Literatures, Cultures and Linguistics Building.

Course Website. go.illinois.edu/math562. Course info also available at the Canvas site. Contact. By email from your @illinois email, psdey(at)illinois.edu, with “Math 562” in the subject.

Office: 35 Computer Application Building. Office Hours. TBA. Ask questions immediately before, during and after class. Otherwise schedule a time by email.

Textbook. Lecture notes and homework problems will be posted on the course website and they are the main texts of the course. We will mainly follow the textbook

Math 561 Probability Theory I – is a prerequisite for this course. However, if you have not taken Math 561, but are willing to invest some extra time to pick up the necessary materials from 561, you may register for this course. Check the course website for 561 lecture notes.
Math 540 Real Analysis I – we will review measure theory topics as needed.
Math 541 is also nice to have but not necessary.

Course Syllabus. This is the second half of the basic graduate course in probability theory. The goal of this course is to understand the basic theory of stochastic calculus. We will cover the following topics:

• Brownian motion;
• Continuous Time Matingales;
• Markov processes;
• Stochastic Integrals;
• Ito’s formula;
• Representation of Martingales;
• Girsanov theorem and Stochastic Differential Equations.
• If time allows, we will give a brief introduction to mathematical finance.

Grading. 20% of your grade will be based on detailed LaTex scribe on a relevant topic, 40% will depend on biweekly homework assignments, and 40% will depend on a take home final exam.

Grading Policy. You are encouraged to work together on the homework and discuss them on Canvas, but I ask that you write up your own solutions and turn them in separately. Few problems will be assigned; emphasis will be placed on clear, concise, and coherent writing.

#### MATH 581 EXTREMAL GRAPH THEORY

TEXT: D. B. West, ”The Art of Combinatorics”, Volume I: Extremal Graph Theory. There also could be some handouts. Specialized texts covering the topics of the course will be listed on the course web site and will be on reserve in the library.

TOPICS: This is a companion course to Math 582 (Structure of Graphs). The two courses are independent and discuss advanced material in graph theory. Extremal Graph Theory includes topics drawn from the following.

Matching and Independence: Bipartite matching, min-max relations, van der Waerden con- jecture; algorithms and applications (weighted matching, Menger’s Theorem, Hopcroft-Karp algorithm, randomized on-line matching, stable matching); factors (Tutte’s f -Factor Theorem, Edmonds’ Blossom Algorithm); independent sets and covers, dominating sets and hypergraph transversals.
Coloring: Vertex colorings (bounds, generalized colorings); structure of k-chromatic graphs (color-critical graphs, forced subgraphs); edge-colorings (Vizing’s Theorem and extensions); variations and generalizations (interval, list, circular colorings). Colorings of hypergraphs.
Perfect Graphs and Intersection Graphs: Perfect and imperfect graphs (Perfect Graph The- orem, partitionable graphs); classes of perfect graphs (chordal, interval, threshold, perfectly orderable, etc.).
Other Extremal Problems: Forbidden subgraphs (Tur´an’s Theorem, Erd˝os-Stone Theorem); graph decomposition (arboricity, fractional arboricity, paths and cycles); representation param- eters (intersection number, boxicity, interval number), etc.
Trees and Distance: Optimization with trees; diameter and distance; encodings and embed- dings.

COURSE REQUIREMENTS: There are no exams. There will be 5 problem sets, each requiring 5 out of 6 problems for 50 points total and 4 quizzes each worth 20 points. The problems in homeworks require proofs related to or applying results from class. Quizzes will ask for definitions, statements and applications of the course theorems. Roughly speaking, 85% of of 330 points suffices for an A, 66% for a B. Discussions between students about problems can help understanding. Collaborations should be acknowledged, and submitted homework should be written individually. Electronic mail is a good way to ask questions about homework problems or other matters.

PREREQUISITES: Math 580, or Math 412, or CS 473, or consent of instructor.

#### MATH 595 RZF, ADVANCED THEORY OF THE RIEMANN ZETA FUNCTION

INSTRUCTOR: ALEXANDRU ZAHARESCU

The Riemann zeta function, together with more general L-functions, play a central role in analytic number theory. In this course we will first have a quick review of basic properties of the Riemann zeta function following Davenport. Then we will follow the presentation in Chapter 5 of Iwaniec and Kowalski of the Riemann zeta function in the context of general L-functions. After that we will study some recent papers concerned with various deeper and more subtle aspects such as zeros of the Riemann zeta on the critical line, Jensen polynomials, Laguerre – P´olya inequalities, and nonvanishing of L-functions at the central point.

Prerequisite: MATH 531. Recommended Textbooks:

Harold Davenport, Multiplicative number theory. Third edition. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. xiv+177 pp. ISBN: 0-387-95097-4

Iwaniec, Henryk; Kowalski, Emmanuel Analytic number theory. Amer- ican Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp. ISBN: 0- 8218-3633-1

There will be no exams. Students registered for this course will be expected to give a couple of lectures on some topics related to the con- tent of the course. In addition some research projects will be suggested to students (or teams of students) enrolled in this class.

#### MATH 595 Group Cohomology

Instructor: ContactVesna Stojanoska Email: vesna AT illinois.edu Office: #307, located at805 W Pennsylvania Ave, Urbana

Course description:Cohomology of groups is a ubiquitous and informative invariant with applications in algebraic topology, number theory, representation theory, and any other area where group actions play a role. In this course, we will mostly focus on the cohomology of finite and profinite groups. We will devote a good portion of time on developing computational tools.Beyond the basics, such as definitions and interpretation of the lower-dimensional cohomology groups, topics will include:

• Tate cohomology, groups with periodic cohomology;
• The Hochschild-Serre spectral sequence, with interesting and explicit applications;
• Group actions on topological spaces, and equivariant cohomology;
• Duality in the continuous cohomology of some profinite groups, such as the absolute Galois groups of local and global fields, or the Morava stabilizer group.

Students will have opportunities for active engagement, through individual or group presentati- ons and/or projects.

Prerequisites: Some algebraic topology, especially homology and cohomology, such as covered in Math 525 and 526.

Useful texts: