1: An Invitation to Geometry - Mathematics

1: An Invitation to Geometry - Mathematics

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How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

-- Albert Einstein

Out of nothing I have created a strange new universe.

-- János Bolyai

An Invitation to Geometry and Topology via G2

The aim of the research school will be to give a thorough introduction to G2 geometry, starting from fundamental material and progressing through to recent breakthroughs and current research in which the UK plays a leading role. The school will also introduce participants to topics of broader interest in algebra (e.g. representation theory), analysis (e.g. elliptic regularity), geometry (e.g. holonomy) and topology (e.g. characteristic classes). The course will also indicate some connections beyond mathematics to contemporary theoretical physics (M-theory)

The three main courses are:

  • Special holonomy. (Robert Bryant, Duke)
  • Calibrated submanifolds. (Jason Lotay, UCL)
  • G2manifolds. (Johannes Nordström, Bath)

There will be three guest lectures by:

  • Nigel Hitchin (Oxford) The variational approach to G2 geometry
  • Bobby Acharya (KCL) Theoretical physics and its connections with G2geometry
  • Mark Haskins (Imperial) Recent advances in research in G2 geometry

These lecture courses will be supplemented by tutorial sessions.

Applications: Applications should be made using the registration form available via the Society’s website. Research students and post-docs in mathematics and in theoretical physics are particularly encouraged to apply.

The closing date for applications is Monday May 12 th 2014. Numbers will be limited and those interested are advised to make an early application.

*All applicants will be contacted within two weeks after the deadline information about individual applications will not be available before then*

All research students and early career researchers will be charged a registration fee of £150. There will be no charge for subsistence costs.

Other participants will be charged a registration fee of £250 plus the full subsistence costs (£350) £600 in total.

Some contribution to travel costs will be available for both UK-based and overseas-based participants.

Dr. Edward Burger is a professor mathematics at Williams College in Williamstown, MA. He received his BA from Connecticut College and his PhD from University of Texas at Austin.
He has received numerous awards including: the Nelson Bushnell Prize, for Scholarship and Teaching, Williams College, being listed among the top 100 best Math Teachers in the "100 Best of America", Reader's Digest's Annual Special Issue. He has also received the Award of Excellence, for "educational mathematics videos that break new ground", from Technology & Learning magazine.
His research interests include Algebraic Number Theory, Diophantine Analysis, padic Analysis, Geometry of Numbers, and the Theory of Continued Fractions.

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Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics

Combinatorial reciprocity is a very interesting phenomenon, which can be described as follows: A polynomial, whose values at positive integers count combinatorial objects of some sort, may give the number of combinatorial objects of a different sort when evaluated at negative integers (and suitably normalized). Such combinatorial reciprocity theorems occur in connections with graphs, partially ordered sets, polyhedra, and more. Using the combinatorial reciprocity theorems as a leitmotif, this book unfolds central ideas and techniques in enumerative and geometric combinatorics.

Written in a friendly writing style, this is an accessible graduate textbook with almost 300 exercises, numerous illustrations, and pointers to the research literature. Topics include concise introductions to partially ordered sets, polyhedral geometry, and rational generating functions, followed by highly original chapters on subdivisions, geometric realizations of partially ordered sets, and hyperplane arrangements.


Advanced undergraduate students and graduate students learning combinatorics instructors teaching such courses.

Reviews & Endorsements

All in all, the authors gather numerous topics in a single impressive work and organize the material in a very engaging and unique fashion.

Algebraic Geometry in the Time of COVID

Here is where things stand, as of evening Friday July 3, 2020.

For those already involved, here is where everything is (as of today).

  • You’re free just to watch, but participants are (intended to be) organized into small working groups, often with strangers, so they can work on problems and discuss the materials. The working groups are assembled in to groupoids . There are some shepherds who are helping to keep things organized and there is one shepherd associated to each groupoid,
  • The notes we’ll base things on are here. When referring to something in the notes, please don’t refer to page numbers (which migrate over different versions), but instead a section number (e.g. 9.1.2, second paragraph) which are now quite stable, and ideally even the date of the version.
  • On wordpress (here): announcements. You can get email updates by “following” this site (there should be a button for that a little bit to the right and above this).
  • The pseudolectures will be on Saturdays, 8-9 am Pacific time. They will appear in this youtube channel: . (If you can’t access youtube, please let me know, and you can watch on zoom with the shepherds.) After the pseudolectures, I’ll take down the youtube link, and then soon repost an edited version (cutting out the beginning and end irrelevant bits). The slides for the pseudolectures will be posted in approximately realtime on dropbox. (For example, the slides for the second pseudolecture are here: Links for everything will be posted in zulip (see below) in the #agittoc announce stream.
  • A number of things are happening zulip — in the “working group” streams, working groups are doing what they feel like together. (Some people prefer not to be in working groups.) In the “groupoid” streams, people discuss what they feel like in larger fora. In another stream, people ask questions and discuss while watching the pseudolectures.
  • There are other places and resources available for use, that some people are taking advantage of — some are using Erik Demaine’s coauthor for discussions and for working on problems. Some people are discussing on the Algebraic Geometry Syndicate discord (in the #agittoc-general channel).
  • Summary : There are a lot of things to potentially play with in this experiment, but all you need to is to “follow” this wordpress site (for announcements), and be on zulip to talk with your working group and groupoid, and to get links for the pseudolectures.

For those who want to get involved:

(1) “Follow” this wordpress pseudoblog for updates: There is no email list for this course instead, I’ll announce things here, and you can get email notifications by clicking “follow” (which should be a button you can see to your right).

(2) First form: Fill out the sign-up form, which is available here. (If you can’t access the form because you can’t access google where you live, just let me know.) This just lets me know a little bit about you, and how involved you might want to be.

(3) Second quick form: Fill out a second quick form here. (At some point I’ll just combine the two forms if I have a chance — which means in all honesty I won’t do it.) It asks for your AGITTOC#, but if you fill it out right away, just write down your name instead of the #, and I’ll figure it out.

When I get a chance, I will write to you about three things:

  • to tell you your “AGITTOC number”, which is basically a line in the spreadsheet I have of participants. (The “shepherds” also have access to this sheet.)
  • I’ll also send you an invitation to the Algebraic Geometry Syndicate discord, which you are welcome to accept, but needn’t (we’re not using it in an essential way right now.)
  • And I’ll invite you to zulip once there, you can temporarily subscribe to one of the groupoids.

When I have even more of a chance, I’ll add you to a working group, and let you know (and add you to the working group’s stream on zulip.)

Questions: If you have any questions, you can ask them either here in a comment to one of these posts, or else on the Algebraic Geometry Syndicate discord in the #agittoc-general channel, or else on zulip in your group or groupoid. (Probably zulip is the best). Then I or (hopefully) someone one else will answer them. (Others are usually much faster than I am at answering. And also please answer others’ questions!)

This is going to be a time-killer for me, so I’m going to concentrate on doing things in bulk and in batches. You can email me, but don’t expect a timely response.


[G] Adams: Lectures on Lie Groups — A clear and concise introduction to the theory of compact Lie groups. Bypasses Lie algebra theory.

[G] Almgren: Plateau’s Problem: An Invitation to Varifold Geometry — A short book on a single topic. Read after knowing some of the basics.

[U] Coxeter & Greitzer: Geometry Revisited — Coxeter and Greitzer give an exciting treatment of Euclidean geometry picking up where high school left off. The exercises are interesting and engaging like the text.

[U/G] Coxeter: Introduction to Geometry — This is geometry as it ought to be done.

[G] Dubrovin, Fomenko, Novikov: Modern Geometry — Methods and Applications. — 3 volumes. A comprehensive treatment of topology and geometry that never loses sight of visual intuition. Requires college-level mathematical maturity.

[U] Greenberg, Marvin: Euclidean and Non-Euclidean Geometry: Development and History — An excellent introduction with a lot of motivation and history.

[G] Hartshorne: Algebraic Geometry — A standard reference for the scheme-theoretic viewpoint of algebraic geometry. Contains many exercises, as notorious as they are pedagogically sound. Should be read in conjunction with a more classically-minded text, such as Shafarevich’s Basic Algebraic Geometry, Mumford’s Complex Projective Varieties, or Harris’s Algebraic Geometry: A First Course.

[G] Griffiths and Harris: Principles of Algebraic Geometry — Algebraic geometry from an entirely analytic point of view. Very classical, very beautiful, and very difficult fundamentally different in flavor, texture, and color from the other books on this list.

[U] Hilbert and Cohn-Vossen: Geometry and the Imagination — Hard to find, but great for getting geometric insight.

[U] Kedlaya: Packet on Euclidean Geometry — Has great problems and tools for Euclidean Geometry.

[U] Madsen and Tornheave: From Calculus to Cohomology – Has a very modern and sure-footed style, yet remains extremely accessible. Highly recommended as a first book on manifolds.

[G] Mumford: The Red Book of Varieties and Schemes — The only book that tells you why schemes are the correct objects to study in algebraic geometry. Lots of examples and explanation of very advanced topics from one of the best expositors ever.

[G] Thurston: Three-Dimensional Topology and Geometry . — Visual intuition is stressed over rigor. Lots of great exercises.

[U] Yaglom: Geometric Transformations — A delightful book.


    (National Taiwan Normal University) (Universidade Federal de Ouro Preto) (Penn State University) (University of Tennessee Knoxville)
  • Leonardo Cavenaghi (University of Fribourg) (University of Göttingen) (University of Göttingen)
  • Juan Carlos Fernández (Universidad Nacional Autónoma de México)
  • Daniel Freese (Indiana University Bloomington) (University of British Columbia, Vancouver) (Massachusetts Institute of Technology) (University of Queensland) (Karlsruhe Institute of Technology) (Universidad Nacional del Sur) (Sorbonne University) (University of Science and Technology of China) (Lehman College)
  • Eduardo Longa (Universidade de São Paulo)
  • David Lundberg (Uppsala University) (National Center for Theoretical Sciences, National Taiwan University) (California State University, Fullerton) (Iowa State University) (Institut für Mathematik, Universität Würzburg) (University of Pennsylvania) (Rutgers University) (Northwestern University) (Universität Münster) (University of Queensland) (Michigan State University)
  • Celso Viana (Universidade Federal de Minas Gerais)
  • Gaoming Wang (Chinese University of Hong Kong) (National Cheng Kung University) (University of Science and Technology of China)
  • Albert Wood (National Taiwan University) (University of Sydney) (Indiana University Bloomington)

[2] The 2nd Geometric Analysis Festival (last updated on 2nd April 2021 )

G is for geometric shapes! We’re exploring hands-on math for the A-Z STEM series all this month! This geometric shapes math activity is perfect for preschool and kindergarten math play!

You can introduce shapes such as triangles, squares, pentagons, octagons, trapezoids and more while keeping the activity light and playful! What is STEM? Read all about STEM here.



I used our shapes printable packet to provide a variety of shapes to use. First I traced them onto the foam sheets. I made as many as I could so we would have lots of variety and colors! Plus, kids can fill in the names and count the sides and record them on the printable sheets!

Next I set out the foam shapes with popsicle sticks and magnetic numbers for an invitation to play and explore geometric shapes!

Geometric shapes are an important learning block for young kids. Learning about these shapes and their properties such as angles and sides will come in handy down the road, especially in geometry class! If you have pattern blocks you can also add them to the activity!


Geometric shapes, defined by Wikipedia: “Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles, squares, and pentagons.”

It’s easy to have a little hands on math fun with a homemade geometric shape activity. Our designs spread across the table. We did a lot of counting and picked out shapes from within the shapes.

He loved the 6 sided geometric shapes the best and made patterns and designs with both the foam geometric shapes and the popsicle sticks!

Geometric shapes are a huge part of our life! They are everywhere. We completed our geometric shapes activity morning with a walk around the house. In each room we pointed out the different shapes.

This list I found of everyday geometric shapes examples was very helpful for me to be able to pull out some items I wouldn’t have thought of to show every day geometric shapes! Also make sure to look for shapes out in the community.

Our Programs

The Department of Mathematical Sciences offers undergraduate, graduate, and accelerated degree programs that prepare students for professional careers as well as further studies in mathematics, statistics, computer science, and other related fields. The department also collaborates with the Teacher Preparation Program to offer an accelerated Bachelor of Arts in Mathematics-Teaching Option and Master of Arts in Teaching program which enables students to obtain New Jersey instructional certificates in five years.

Invitation to the Mathematics of Fermat-Wiles

Assuming only modest knowledge of undergraduate level math, Invitation to the Mathematics of Fermat-Wiles presents diverse concepts required to comprehend Wiles' extraordinary proof. Furthermore, it places these concepts in their historical context.

This book can be used in introduction to mathematics theories courses and in special topics courses on Fermat's last theorem. It contains themes suitable for development by students as an introduction to personal research as well as numerous exercises and problems. However, the book will also appeal to the inquiring and mathematically informed reader intrigued by the unraveling of this fascinating puzzle.

Assuming only modest knowledge of undergraduate level math, Invitation to the Mathematics of Fermat-Wiles presents diverse concepts required to comprehend Wiles' extraordinary proof. Furthermore, it places these concepts in their historical context.

This book can be used in introduction to mathematics theories courses and in special topics courses on Fermat's last theorem. It contains themes suitable for development by students as an introduction to personal research as well as numerous exercises and problems. However, the book will also appeal to the inquiring and mathematically informed reader intrigued by the unraveling of this fascinating puzzle.

Watch the video: Introduction to Geometry (June 2022).


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