##### Math 500

This is an introduction to Abstract Algebra, at the graduate level. Assuming only a basic knowledge of groups and rings, we will give a comprehensive introduction to group, ring, and field theory. Highlights include the class equation, the Sylow theorems, the basic theory of PID’s and UFD’s, the basic theory of algebraic field extensions, and the fundamental theorem of Galois theory.

##### Math 519

course description:

• Operations on vector bundles, covariant derivative, curvature.

• Fiber bundles, principal bundles, associated bundles.

• Frame bundle of a vector bundle.

• Connections on principal bundles as distributions and as Lie

algebra-valued 1-forms. Induced covariant derivatives on the associated bundles.

• Curvature. Chern classes. Chern-Weil homomorphism.

• Parallel transport. Diffeological spaces, sheaves of sets on a site, thin fundamental groupoid and parallel transport.

• Sheaves of groupoids, descent, 1-stacks.

• Lie groupoids and geometric stacks over the site of manifolds (if time permits).

##### Math 522

Lie Groups and Lie Algebras

Lie groups and Lie algebras were discovered by the Norwegian

mathematician Sophus Lie in the second half of the 19th Century. His

motivation was to design a Galois Theory for differential equations,

which would allow to determine if one could solve a differential

equation, as well as to find systematic methods of integration. Soon

after, it was realized that Lie groups and algebras play a fundamental

role in many other areas of mathematics. Nowadays, they are recognized

as a basic tool in mathematics and its applications. This course will

provide a solid introduction to the basic of Lie groups and Lie

algebras, as well as some applications. Students taking this course

are assumed to know abstract algebra and differential geometry at the

level of Math 500 – Abstract Algebra I and Math 518 – Differentiable

Manifolds.

• **Lecturer:** Rui Loja Fernandes

• **Email:** ruiloja@illinois.edu

• **Office:** 366 Altgeld Hall

• **Web page:** https://faculty.math.illinois.edu/~ruiloja/

• **Prerequisites:** Math 500 and Math 518, or equivalent.

Syllabus:

• **Lie Groups and their Lie algebras:** Lie groups, Lie algebras,

homomorphisms, subgroups, coverings, 1-connected Lie groups,

exponential map, closed subgroups, Lie group-Lie algebra

correspondence. Adjoint and coadjoint representations. Lie’s Third

Theorem.

• **Proper Actions:** Proper actions, quotients, principal bundles and

homogenous manifolds. Bochner Linearization theorem, slices, tube

theorem, orbit spaces, orbit types and stratification, principal and

regular orbit types, blowups and desingularization.

• **Compact Lie groups and Their Representations:** Adjoint action,

centralizers, root and root spaces, maximal tori, orbit structure,

Weyl group, Stiefel diagram, integration and the Weyl integration

formula, Schur’s lemma, characters, G-types, Peter-Weyl Theorem.

• **Applications to Symmetries of Differential Equations:** jet

spaces, prolongation, symmetry groups of differential equations;

calculation of symmetry groups.

Textbooks:

• F. Warner, ** Foundations of Differentiable Manifolds and Lie Groups** , GTM 94, Springer- Verlag, Berlin, 1983.

• J.J. Duistermaat and J.A.C. Kolk,

**, Universitext,**

*Lie groups*Springer-Verlag, Berlin, 2000.

• P. Olver,

**, GTM 107, Springer-Verlag, Berlin, 1993.**

*Applications of Lie groups to Differential Equations, 2nd Edition*• J.-P. Serre,

**Lecture Notes in**

*Lie algebras and Lie groups,*Mathematics,

##### Math 525

ALGEBRAIC TOPOLOGY I

Igor Mineyev. Spring 2023. MWF 3:00-3:50 p.m., Altgeld 147.

All information will be on class website

Textbook:Algebraic topologyby Allen Hatcher. Freely available online at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

The tentative syllabus.This is the official syllabus, we will roughly

follow it in the course.

• Fundamental group and covering spaces.

(1) Definition of the fundamental group. (2) Covering spaces and

lifts of maps. (3) Computing the fundamental group via covering

spaces. (4) Applications, such as the Fundamental Theorem of Algebra

and the Brouwer fixed point theorem in 2d. (5) Deforming spaces:

retraction and homotopy equivalence. (6) Quotient topology and cell

complexes. (7) Homotopy extension property and applications to

homotopy equivalence. (8) Fundamental groups of CW complexes. (9)

Van Kampen’s Theorem. (10) Covering spaces and subgroups of the

fundamental group. (11) Universal covers. (12) The definitive

lifting criterion, classification of covering spaces. (13) Covering

transformations and regular covers.

• Homology.

(14) Delta complexes and their cellular homology. (15) Singular

homology. (16) Homotopic maps and homology. (17) The long exact

sequence of the pair. (18) Relative homology and excision. (19)

Equality of cellular and singular homology. (20) Applications, such

as degree of maps of spheres, invariance of dimension, and the Brouwer

fixed point theorem. (21) Homology of CW complexes. (22) Homology

and the fundamental group: the Hurewicz theorem. (23) Euler

characteristic. (24) Homology with coefficients. (25) Intro to

categories and axiomatic characterization of homology theories. (26)

Further applications, such as the Jordan curve theorem, wild spheres,

invariance of domain.

##### Math 527

Homotopy Theory

Spring 2023 (1–2 MWF, in 347 Altgeld)

• **Instructor:** Charles Rezk

• **Office:** 257 CAB

• **Email:** rezk@illinois.edu

• **Webpage:**

http://faculty.math.illinois.edu/rezk/

Course description:

This is a course on topics in algebraic topology and homotopy theory,

organized as literature seminar in the style of the famous “Kan

seminar”. The goals are:

- To give students who have had some grounding in the foundations literature.
- To give students experience in the process of quickly reading and assimilating research papers.
- To give students experience in giving mathematical talks.

Although the topics of the course are drawn from algebraic topology

and allied fields, it may be useful to people interested in other

areas.

The course will be organized as follows.

• Each student will give talks on three papers, chosen by me and the

student in consultation. They may be chosen to align with the

students particular interests. I’ll provide a list of papers as

suggestions, but papers outside of my list are possible as well.

• Students are expect to attend all talks. Only students in the class

are allowed as audience members for talks (no auditors).

• Everyone not speaking is expected to have skimmed the paper before

the talk on it, and to submit a “reading response”. This is meant to

be a short note where you can express your ideas on what the paper

is about, describe how it fits with other things you may know, what

may have seemed surprising about the paper, and to ask any questions

you might have.

• It is typical (and highly recommended) for students to organize

informal practice talks to other members of the class, before giving

the final version of their talk in class.

Prerequisites: Math 525 and/or 526, or instructor consent.

##### Math 530

Algebraic Number Theory Instructor: Alexandru Zaharescu

Prerequisite:MATH 500 or equivalent

Grading Policy: Comprehensive final exam: 40%; One midterm exam: 30%; Homework: 30%.

Recommended Textbook:

Daniel A. Marcus, Number Fields, Springer-Verlag, 1987, NY.

We will discuss material from the first five chapters, and then, as

time permits, present other selected topics.

##### Math 532

Sieve Methods

Instructor: Kevin Ford

Time: MWF 2–2:50

Prerequisites:Basic analytic number theory will be very helpful

(equivalent of Math 531). Some knowledge of basic probability will be

helpful but not necessary.

Recommended Texts: Sieve methods, H. Halberstam and H.-E. Richert,

Dover paperback edition (highly recom- mended for purchase). Opera de

Cribro, J. Friedlander and H. Iwaniec, AMS, 2010.

I will also post extensive online lecture notes

Course Description. A sieve is a tool for bounding the number of

integers from a finite set which remain when those divisible by a set

of primes are excluded. Several different sieves will be covered in

this course, each tailored to certain types of applications. We will

take a probabilistic viewpoint, interpreting various results in terms

of certain random variables, and discussing the role of independence

vs. dependence. Here are some sieves and applications we will learn

about:

• The “small sieve”, and applications to prime/almost prime values of

polynomials, k-tuples of primes, primes in polynomials (The

Brun-Titchmarsh inequality), primes in short intervals.

• The “large sieve” and applications to the least quadratic nonresidue

modulo a prime, and to the distribution of primes in progressions

(Bombieri-Vinogradov theorem)

• Sieving for integers lacking large prime factors (“smooth” numbers),

application to large gaps between primes

• prime factors of shifted primesp± 1 , application to the

distribution of values of Euler’s function

• The Kubilius model of integers, application to integers with a

prescribed number of prime factors, the Erdos-Kac ̋ theorem

• The Maynard-Tao sieve with applications to bounded gaps between

primes and to large gaps between primes.

• The random sieve, connection to large gaps between primes and the

Hardy-Littlewood primek-tuples conjec- tures.

• The asymptotic sieve for primes using Type-I and Type-II

information.

Grades.The course grade will depend on homework assignments, which

will be given periodically.

##### Math 540

Real Analysis I

MWF 9:00–9:50am

Contact info: Instructor: M. Burak Erdogan

E-mail: berdogan@illinois.edu

Office: 41 CAB Webpage: <https://faculty.math.illinois.edu/∼berdogan/>

This is an introductory course in measure theory. Topics to be covered include (not necessarily in this order):

- Measures on the line: Abstract measure theory, outer measure,
- Lebesgue measure on the real line, measurable sets, Borel sets, Cantor sets and functions, non-measurable sets. (Optional: Baire’s category theorem.)
- Measurable functions: Structure of measurable sets, approximation of measurable functions by simple functions, Littlewood’s three principles, Egorov and Lusin’s theorems.
- Integration: Lebesgue theory of integration, convergence theorems (Monotone Convergence, Fatou’s Lemma, Dominated Convergence), comparison of the Riemann and Lebesgue integrals, modes of convergence, approximation of integrable functions by con- tinuous functions. Product measures, the general Fubini-Tonelli theorem, the convolution product.
- Differentiability: Functions of bounded variation (structure and differentiability), absolutely continuous functions, maximal functions, fundamental theorem of calculus.
- Lp spaces: Jensen’s inequality, Holder and Minkowski’s inequalities, class of Lp functions, completeness, duals ofLpspaces, inclusions ofLpspaces.
- Hilbert spaces and Fourier series: Elementary Hilbert space theory, orthogonal projections, Riesz representation theorem, Bessel’s inequality, Riemann-Lebesgue lemma, Parseval’s identity, completeness of trigonometric spaces.

Textbook: G. B. Folland, Real Analysis, John Wiley & Sons. Additional references are Royden’s Real Analysis and Rudin’s Real and Complex Analysis.

Prerequisites: Ma 447 or equivalent. Basic set theory, see chapter 0 of Folland’s Analysis.

Grading: Weekly homework assignments (30%), a midterm exam (25%), and a final (45%). The letter grades will be assigned according to the scale: 87 – 100: A-, A, A+, 75 • 87: B-, B, B+, 65 – 75: C-, C, C+, 55 – 65: D-, D, D+, Below 55: F.

Homework Assignments: There will be weekly homework assignments. They will be posted on the course webpage in canvas. You are encouraged to discuss the homework problems with each other but should write the solutions by yourself in your own words. The worst two homework grades will be dropped.

##### Math 545

Xiaochun Li

This course is an introduction to modern harmonic analysis.The

following classic topics are planed to be covered.

• Marcinkiewicz interpolation; Approximation to the identity; Lp

theory of Fourier transforms;

• The theory of Calder ̈on-Zygmund singular integrals;

• Littlewood-Paley theory (continuous version and discreteversion);

Multiplier;

• BMO and Carleson measure; T1 theorem;

• Lp(p 6 = 2) unboundedness of the disk multiplier;

• Oscillatory integrals

Lectures: TR 11:00–12:20 in Altgeld.

References:There is NO textbook for the course. The following

references are used.

Javier Duoandikoetxea,Fourier analysis, Graduate studies in math., Vol 29. AMS,

E. M. Stein, Singular Integrals and Differentiability Properties of Functions ,Princeton Univ. Press, Princeton, 1970

E. M. Stein,Harmonic analysis: Real variable methods, Princeton Univ. Press, Princeton, 1993

E. M. Stein and G. Weiss,Introduction to Fourier Analysis in Euclidean Spaces, Princeton Univ. Press, Princeton, 1971

Grading: Homework (100%) Prerequisites: Solid knowledge of real analysis.

##### Math 553

Partial Differential Equations

Basic introduction to the study of partial differential equations;

topics include: the Cauchy problem, classification, canonical forms,

the method of characteristics, the wave equation, the heat equation,

Laplace’s equation, Sturm-Liouville problems and separation of

variables, harmonic functions, potential theory, Fourier series, the

Dirichlet and Neumann problems, and Green’s functions.

Prerequisite: Consent of instructor.

##### Math 561:

Theory of Probability I (Spring 2023)

Goals and topics: This is the first half of the basic graduate course in probability theory. The goal of this course is to understand the basic tools and language of modern probability theory. We will start with the basic concepts of probability theory: random variables, distributions, expectations, variances, independence and convergence of random variables. Then we will cover the following topics:

basic limit theorems (law of large numbers, central limit theorem and large deviation principle);

martingales and their applications. If time allows, we will give a

brief introduction to Brownian motion and Stein’s method for normal

approximation.

Logistics Website <https://faculty.math.illinois.edu/~psdey/Math561SP23.html> Textbook I will post pdf lecture notes for each class. Richard Durrett: Probability: Theory and Examples (Free Online edition v5). We will cover the first four chapters. It is okay to use another edition for studying. Some other relevent books: P. Billingsley Probability and Measure (3rd Edition). Chapters 1-30 contain a more careful and detailed treatment of some of the topics of this semester, in particular the measure-theory background. Recommended for students who have not done measure theory. Prerequisite The prerequisite for Math 561 is Math 540 – Real Analysis I. We will review measure theory topics as needed. Math 541 is nice to have, but not necessary.

Exams and Grades Homework Policy Homework will be assigned weekly on Thursdays on Canvas, to be submitted at the start of next Thursday lecture or earlier in Canvas. Solving a lot of problems is an extremely important part of learning probability. You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately. Late homework will not be graded. If for some reason you’ve done a homework but can’t turn it in online, send it via email before class. Because of this strict policy on late homework, I will drop your lowest score. Please talk to the instructor in cases of emergency. Homework Philosophy Mathematics is something that you learn by doing: doing homework problems, and explaining them to each other. If, after thinking and talking about homework problems, you get stuck or have questions, I will be happy to help. You’ll have a high probability of doing well in this class by combining all of these resources: classes, textbook, homework, o!ce hours, and discussions with classmates. Exams There will be one in-class midterm exam (tentatively) Tuesday, March 21, 2023. It will be technically comprehensive, but emphasizing recent material up to the most recent graded and returned homework assignment. Exam problems will be similar to homework problems.

The final take home exam will cover the most important topics of the whole course. It will be assigned on the last day of the class and will be due on (tentatively) Tuesday, May 9, 2023.

##### Math** **583

**Partially Ordered Sets** are part of Extremal

Combinatorics. In this course we spend in depth understanding of some

of their main methods, in particular how to apply modern powerful

tools, such as Szemer ́edi Regularity Lemma and the hypergraph

container method. The class will have about 4 homework assignments

and students are supposed to write some class notes (average student

needs to write note about two classes of the semester).

##### Math 586

Algebraic combinatorics

Instructor:Alexander Yong

Meeting times:MWF 11-11:50pm.

Course description: This is a graduate course on algebraic combinatorics. The course aims to provide preparation to conduct research in this field. The intended audience for this course will be PhD students in com- binatorics as well as students in representation theory, algebra and geometry where methods from algebraic combinatorics arise. This course will be taught from a combinatorial perspective. Exceptions such as representation theory of the symmetric group and GL(n) will only assume modest background in graduate level algebra. Grading:Based on homework exercises and a final presentation.

Below I give a list of specific topics we will cover (time

permitting): “`

I. Symmetric functions

monomial elementary, homogeneous, power sum bases and theirtransitions

Representation theory of the symmetric group, characters and the Murnaghan-Nakayama rule

Lindstr ̈om-Gessel-Viennot lemma

Schur polynomials

Schur polynomials as characters ofGLnrepresentations

II. Young tableaux:

The hook-length formula

RSK correspondence

The Littlewood-Richardson rule

Jeu de taquin

Symmetric group combinatorics including Bruhat order, reduced words,

and the Edelman-Greene correspondence

III. Polynomials and more:

• Schubert polynomials and an introduction to Schubert calculus

• Quasisymmetric functions

• Macdonald polynomials

• Combinatorial commutative algebra

• Special topics (based on current trends in the subject), e.g., root-system combinatorics, matroids, theory of polytopes, complexity theory.

Textbooks will be

• H. S. Wilf,Generating Functionology, available for free down- load

at https://www.math.upenn.edu/~wilf/DownldGF.html

• R. P. Stanley,Enumerative combinatorics, vol. 1, 2nd edition,

Cambridge University Press, 1997.

• R. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge

University Press, 1999.

• W. Fulton,Young tableaux, Cambridge University Press, 1997.

##### Math 595 Integrable Combinatorics

Spring 2023, Mon and Wed, 11:00 to 12:20, Math Dept.

Instructor: Philippe Di Francesco

This topics course aims at introducing various integrable structures

arising from physical/combinatorial problems. These involve

combinatorial objects such as: triangulations, trees, tilings, vertex

models, alternating sign matrices, plane partitions and

networks. Integrability arises both as a consequence of the symmetries

of the problem and the possibility of introducing parametric

deformations that preserve them. It provides powerful algebraic and

analytic tools for exact enumeration and more.

Informal plan of the course:

Introduction to statistical physics

Introduction to the combinatorics of paths and matchings

Integrable models I: Lorentzian Triangulations

Integrable models II: Lozenge tilings and Plane partitions

Discrete integrable systems and Cluster algebras: classical andq-combinatorics

Integrable lattice models III: Tilings of the Aztec diamond

Integrable lattice models IV: General theory-example of the six vertex model and connection to Alternating Sign Matrices

Integrable lattice models V: Twenty vertex model and domino tilings

Limit shapes: (a) multivariate asymptotics and (b) the tangent method

The course is largely self-contained. No prerequisites. We include

quick introduction to all the necessary concepts and basic tools of

statistical physics. Combinatorial methods will be developed when

needed (generating functions, infinite matrices, continued fractions,

determinants, decorated trees, (non-intersecting) lattice paths,

networks, cluster algebras, saddle-point approximation, etc.). General reading material: Lecture note for the International Congress of Mathematicians 2018

(P. Di Francesco): https://arxiv.org/abs/1711. Slides by D. Bressoud:

http://www.macalester.edu/~bressoud/talks/2009/asm-ASU.pdf Method of assessment: Class participation, and response to open homework problems.

##### Math 595 Integrable systems

J. Palmer The focus of this course will be on thesymplectic

theory of integrable systems, and their interplay withsingular

Lagrangian fibrationsandgroup actions. An integrable system is,

roughly, a dynamical system which has the maximum number of quantities

preserved by the system. For instance, a mass hanging from a string

(the “spherical pendulum”) is an integrable system because both total

energy and angular momentum are conserved.

The space of possible states of the system, called thephase space,

naturally has the structure of a symplectic manifold. Each conserved

quantity corresponds to an associated symmetry, i.e. a Lie group

action on the phase space of the system. Integrable systems are thus

intimately connected with several areas, including:

• symplectic geometry;

• symplectic and Hamiltonian group actions;

• (singular) Lagrangian fibrations;

• dynamical systems and geometric mechanics.

We will build the theory from scratch, starting with an introduction

to the study of integrable systems and the relevant symplectic

geometry. Nevertheless, by the end of the semester we will be

studyingmodern techniquesandcurrent results, including:

• invariants of integrable systems;

• the classification of singularities in integrable systems;

• toric and semitoric integrable systems;

• deformations and bifurcations of integrable systems;

• classifications of integrable systems and torus actions;

• almost toric fibrations;

• (equivariant) symplectic capacities.

The field of integrable systems is huge, and the study of such systems

goes back hundreds of years. The topics we concentrate on at the end

of the course will be partially up to the taste of the students.

The only assumed knowledge at the start of the course will be basic

facts about manifolds and differential geometry.

##### Math 595 Local Cohomology

Prof. S. P. Dutta

11 – 12 :20, Tu-TR

This course will be a study of Local Cohomology, introduced by

Grothendieck, with various applications. The main topics will include:

Cohen-Macaulay Rings and Modules, Injective Modules over Noethierian

rings, Gorenstein rings, local Cohomology – connection with dimension

and depth, local duality theorem of Grothendieck. Cohomology of

quasi-coherent and coherent sheaves, Serre’s Theorem on coherent

sheaves on projective spaces, classification of Line-bundles on Pn,

Hartshorne-Lichtenbaum Theorem and Faltings Connectedness Theorem.

Prerequisite: Math 502

Recommended Text: 1. Local Cohomology by R. Hartshorne; 2. Local

Cohomology by Brodmann and Sharp, Cambridge University Press.

##### Math 595 Operator Space Theory

ROY ARAIZA

The theory of operator spaces has a rich history with motivations

going back to foundational work of Ef- fros and Lance in the latter

part of the 20thcentury. These “quantized” Banach spaces arise in many

areas of mathematics including operator algebras, free probability

theory, and more recently, quantum information theory. Methods from

operator space theory have been used to solve many problems in

operator algebras, and such methods also led to a proof of arguably

the most famous problem in operator algebras, *Kirchberg’s conjecture*.

During the term we will cover the introductory theory of operator

spaces with an emphasis on its tensor product theory. We will start

the term with how such questions involving operator spaces arise

naturally when considering C*-algebras. The first ten weeks of course will cover the foundational theory of the sub- ject. Examples of such topics are completely bounded maps, Ruan’s representation theorem, and constructions and basic operations. After covering thefoundational theory, we will begin with operator space tensorproducts, focusing on the minimal, projective, and Haagerup operatorspace tensor products. In passing we will also discuss the theory ofoperator systems. Such objects also arise naturally when consideringC*-algebras, and are in the fact the natural objects to consider when

one talks about

*quantum channels.*Time permitting, we will also

discuss

*operator system tensor products.*

In the final six weeks of term we will look at applications of

operator spaces and operator systems, with a special emphasis on

applications in quantum information theory. Such applications we will

consider arise in the form of *quantum games* , and *quantum error correction.* Updates regarding the course can be found on my

websitehttps://math.illinois.edu/directory/profile/raraiza

Department of Mathematics, Illinois Quantum Information Science and

Technology Center, University of Illinois at Urbana- Champaign *Email address* :raraiza@illinois.edu

##### Math 595 Quantum channels

Spring term 2023 Lecturer: Felix Leditzky

The first part of the course is an introduction to the theory of

quantum channels in the nite-dimensional set- ting of quantum

information theory. We discuss the various mathematically equivalent

representations of quantum channels, focus on some important

subclasses of channels, and make various connections to

information-theoretic aspects of quantum information theory. In the

second part of the course we turn to the theory of quantum Markov

chains. We rst discuss the quantum relative entropy and its

fundamental property, the data-processing inequality, and give a proof

of this inequality that naturally leads to equality conditions and the

concept of recovery channels. Specializing this analysis to the

partial trace, we obtain the strong subadditivity property of the von

Neumann entropy, as well as a natural denition of quantum Markov

chains. We then review a structure theorem for quantum Markov chains,

the fundamental differences to classical Markov chains, and—time

permiing—discuss the notion of approximate quantum Markov chains.

PrerequisitesMATH 415 or MATH 416. A course in quantum mechanics or

quantum information theory such as ECE 404, Phys 486/487, Phys 513 is

useful, but not necessary.

Grading policy: there will be no homework assignments or wrien

exams for this course. Grading will be based on active class

participation.

Table of contents

• Representations of quantum channels: Isometric picture, Unitary

evolution, Kraus representation, Linear operator representation

• Classes of quantum channels: Mixed-unitary and unital channels,

Entanglement-breaking channels, Symmetric and antidegradable

channels, PPT channels

• Unital channels and majorization: Majorization for real vectors,

Majorization for Hermitian operators, Schur-Horn eorem

• Covariant quantum channels: Denition, properties, examples,

Holevo information and minimum output entropy

• antum relative entropy: Denition and operational

interpretation, Joint convexity and data-processing inequality

• Equality conditions for data-processing: Petz’s proof of the

equality condition, Formulation in terms of recovery channels

• antum Markov chains: Strong subadditivity of von Neumann

entropy, antum conditional mutual information and its opera-

tional interpretations, Structure theorem for exact quantum Markov

chains

• Approximate quantum Markov chains: Classical vs. quantum seing,

Approximate recovery channels

Literature

• B. Ibinson et al. “Robustness of quantum Markov

chains”.Communications in Mathematical Physics277. (2008),

pp. 289–304. Available online at:https://arxiv.org/abs/quant-ph/

• M. A. Nielsen and D. Petz. “A Simple Proof of the Strong

Subadditivity Inequality”.antum Information & Computation5.6

(2005), pp. 507–513. Available online at:https : / / arxiv. org /

abs / quant – ph/

• D. Petz. “Monotonicity of quantum relative entropy

revisited”.Reviews in Mathematical Physics15. (2003),

pp. 79–91. Available online at:https://arxiv.org/abs/quant-ph/

• D. Petz.antum Information eory and antum

Statistics. eoretical and Mathematical Physics. Springer, 2008

• M. B. Ruskai. “Inequalities for quantum entropy: A review with

conditions for equality”. Journal of Mathematical Physics43.9

(2002), pp. 4358–4375. Available online

at:https://arxiv.org/abs/quant– ph/

• D. Suer.Approximate quantum Markov

chains. Vol. 28. SpringerBriefs in Mathematical Physics, 2018.

Available online at:https://arxiv.org/abs/1802.

• J. Watrous. eory of antum Information. Cambridge University

Press, 2018. Available online at:

https://cs.uwaterloo.ca/~watrous/TQI

• M. M. Wilde. antum Information eory. Cambridge University

Press, 2017. Available online at: https://arxiv.org/abs/1106.

• M. M. Wolf.antum Channels & Operations: Guided Tour. Lecture

notes. 2012. Available online at:

https://www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.

pdf

##### Math 595 Toric Varieties

Spring 2023 TH 12:30–1:50 Instructor: Sheldon Katz “` This first-half

minicourse provides an introduction to toric varieties. Toric

varieties provide a large class of complex algebraic varieties for

which abstract concepts can be made explicit and computations can be

performed algorith- mically. They are a standard and indispensible

tool for the working algebraic geometer. The minicourse will

introduce normal toric varieties, which can be de- scribed by fans,

and projective toric varieties, which can be described by

polytopes. For toric varieties which are both normal and projective,

the dic- tionary between the two descriptions will be explained. Toric

geometry will be used to describe line bundles and their cohomology,

differentials, Serre duality, the Mori cone, Chow groups, and more.

The main text will be Fulton’s minicourse Introduction to Toric

Varieties. We will cover the first three chapters of this text and

selected topics from the last two chapters. We will also cover

additional topics such as toric surfaces and mirror symmetry for

Calabi-Yau hypersurfaces in toric varieties, as time

permits. Supplementary material will be added including Chapter 3 of

Mirror Symmetry and Algebraic Geometryby Cox and Katz, and the

encyclopedic Toric Varietiesby Cox, Little, and Schenck.

Prerequisites: A first course in algebraic geometry: MATH 511 or MATH

512 or equivalent, with MATH 512 preferred. I will not be assuming

advanced knowledge of algebraic geometry, but students must be

comfortable with both affine and projective varieties. I will also be

assuming that you know what Spec and Proj are, so if you have taken

MATH 511 but not MATH 512, you should not take this class unless you

are able to read e.g. Chapter II.2 of Hartshorne’s Algebraic

Geometrybefore the beginning of the course.