math graduate courses, S’23

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.

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).

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, Lie groups , Universitext,
Springer-Verlag, Berlin, 2000.
• P. Olver, Applications of Lie groups to Differential Equations, 2nd Edition , GTM 107, Springer-Verlag, Berlin, 1993.
• J.-P. Serre, Lie algebras and Lie groups, Lecture Notes in
Mathematics,

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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 the
foundational theory, we will begin with operator space tensor
products, focusing on the minimal, projective, and Haagerup operator
space tensor products. In passing we will also discuss the theory of
operator systems. Such objects also arise naturally when considering
C-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

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 de€nition of quantum Markov
chains. We then review a structure theorem for quantum Markov chains,
the fundamental differences to classical Markov chains, and—time
permi‹ing—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 wri‹en
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: De€nition, properties, examples,
Holevo information and minimum output entropy
• antum relative entropy: De€nition 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 se‹ing,
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 Œeory 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. Su‹er.Approximate quantum Markov
chains. Vol. 28. SpringerBriefs in Mathematical Physics, 2018.
Available online at:https://arxiv.org/abs/1802.
• J. Watrous. Œeory of ‹antum Information. Cambridge University
Press, 2018. Available online at:
https://cs.uwaterloo.ca/~watrous/TQI
• M. M. Wilde. ‹antum Information Œeory. 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

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.