Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert’s Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.
Some exposure to advanced mathematical methods, particularly those pertaining to ring theory, fields extensions, and point-set topology.
Yes math fans, as previously hinted at in prior conversations, we’ll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won’t happen until July 31, 2017.
While it’s not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we’ll post them here first. As of this writing, there is no officially announced textbook for the course, but we’ve got some initial guesses and the best are as follows (roughly in decreasing order):
Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he’s that good. This class, my 22nd, will be the start of my second decade of math with him.)
A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises.
(.pdf download) Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a coin flip, the spin of a roulette wheel, or something similar; in brief, an ‘objective’ device, one for which everybody agrees on the numerical values of the probabilities involved. Rather oddly, in spite of the long history of the theory of subjective probability, nobody seems to have examined the consequences of basing mixed strategies on ‘subjective’ random devices, i.e. devices on the probabilities of whose outcomes people may disagree (such as horse races, elections, etc.).
For a constant ϵ, we prove a poly(N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player NxN games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1−ϵ)-fraction of the players are ϵ-best replying.
John Nash’s notion of equilibrium is ubiquitous in economic theory, but a new study shows that it is often impossible to reach efficiently.
There’s a couple of interesting sounding papers in here that I want to dig up and read. There are some great results that sound like they are crying out for better generalization and classification. Perhaps some overlap with information theory and complexity?
To some extent I also find myself wondering about repeated play as a possible random walk versus larger “jumps” in potential game play and the effects this may have on the “evolution” of a solution by play instead of a simpler closed mathematical solution.
Cotton twill hat features a full color embroidered Johns Hopkins lacrosse design showcasing our shielded blue jay. Unstructured low profile fit. Just the right wash; renowned perfect fit. Fabric strap closure with brass slide buckle. 100% cotton twill. Adjustable. Black. By Legacy.
I’d love to have a Johns Hopkins hat just like this with “Math” instead of “Lacrosse”. Surely the department has them made occasionally?
Songs about communication, telephones, conversation, satellites, love, auto-tune and even one about a typewriter! They all relate at least tangentially to the topic at hand. To up the ante, everyone should realize that digital music would be impossible without Shannon’s seminal work.
Let me know in the comments or by replying to one of the syndicated copies listed below if there are any great tunes that the list is missing.
I am totally stunned to learn that Maryam Mirzakhani died today, aged 40, after a severe recurrence of the cancer she had been fighting for several years. I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.
A nice obituary about a fantastic mathematician from a fellow Fields Prize winner.
Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.
This looks like a cool little paper.
Some thoughts after reading
And indeed it was. The opening has lovely long (though possibly incomplete) list of aspects of good mathematics toward which mathematicians should strive. The second section contains an interesting example which looks at the history of a theorem and it’s effect on several different areas. To me most of the value is in thinking about the first several pages. I highly recommend this to all young budding mathematicians.
In particular, as a society, we need to be careful of early students in elementary and high school as well as college as the pedagogy of mathematics at these lower levels tends to weed out potential mathematicians of many of these stripes. Students often get discouraged from pursuing mathematics because it’s “too hard” often because they don’t have the right resources or support. These students, may in fact be those who add to the well-roundedness of the subject which help to push it forward.
I believe that this diverse and multifaceted nature of “good mathematics” is very healthy for mathematics as a whole, as it it allows us to pursue many different approaches to the subject, and exploit many different types of mathematical talent, towards our common goal of greater mathematical progress and understanding. While each one of the above attributes is generally accepted to be a desirable trait to have in mathematics, it can become detrimental to a field to pursue only one or two of them at the expense of all the others.
As I look at his list of scenarios, it also reminds me of how areas within the humanities can become quickly stymied. The trouble in some of those areas of study is that they’re not as rigorously underpinned, as systematic, or as brutally clear as mathematics can be, so the fact that they’ve become stuck may not be noticed until a dreadfully much later date. These facts also make it much easier and clearer in some of these fields to notice the true stars.
As a reminder for later, I’ll include these scenarios about research fields:
A field which becomes increasingly ornate and baroque, in which individual
results are generalised and refined for their own sake, but the subject as a
whole drifts aimlessly without any definite direction or sense of progress;
A field which becomes filled with many astounding conjectures, but with no
hope of rigorous progress on any of them;
A field which now consists primarily of using ad hoc methods to solve a collection
of unrelated problems, which have no unifying theme, connections, or purpose;
A field which has become overly dry and theoretical, continually recasting and
unifying previous results in increasingly technical formal frameworks, but not
generating any exciting new breakthroughs as a consequence; or
A field which reveres classical results, and continually presents shorter, simpler,
and more elegant proofs of these results, but which does not generate any truly
original and new results beyond the classical literature.
I’ll bet I had a better Pi Day than you did today. I got to sit through three hours of a fascinating lecture on complex analysis covering Bernoulli numbers, Reimann’s Zeta Function, the Gamma Function, Euler’s Function, the duplication formula, the symmetrical form of the zeta functional equation, and Reimann’s Hypothesis. All replete with references to pi! Take that nerds!
The world’s foremost expert on pricing strategy shows how this mysterious process works and how to maximize value through pricing to company and customer.
In all walks of life, we constantly make decisions about whether something is worth our money or our time, or try to convince others to part with their money or their time. Price is the place where value and money meet. From the global release of the latest electronic gadget to the bewildering gyrations of oil futures to markdowns at the bargain store, price is the most powerful and pervasive economic force in our day-to-day lives and one of the least understood.
The recipe for successful pricing often sounds like an exotic cocktail, with equal parts psychology, economics, strategy, tools and incentives stirred up together, usually with just enough math to sour the taste. That leads managers to water down the drink with hunches and rules of thumb, or leave out the parts with which they don’t feel comfortable. While this makes for a sweeter drink, it often lacks the punch to have an impact on the customer or on the business.
It doesn’t have to be that way, though, as Hermann Simon illustrates through dozens of stories collected over four decades in the trenches and behind the scenes. A world-renowned speaker on pricing and a trusted advisor to Fortune 500 executives, Simon’s lifelong journey has taken him from rural farmers’ markets, to a distinguished academic career, to a long second career as an entrepreneur and management consultant to companies large and small throughout the world. Along the way, he has learned from Nobel Prize winners and leading management gurus, and helped countless managers and executives use pricing as a way to create new markets, grow their businesses and gain a sustained competitive advantage. He also learned some tough personal lessons about value, how people perceive it, and how people profit from it.
In this engaging and practical narrative, Simon leaves nothing out of the pricing cocktail, but still makes it go down smoothly and leaves you wanting to learn more and do more―as a consumer or as a business person. You will never look at pricing the same way again.