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.)
Epigenetics refers to information transmitted during cell division other than the DNA sequence per se, and it is the language that distinguishes stem cells from somatic cells, one organ from another, and even identical twins from each other. In contrast to the DNA sequence, the epigenome is relatively susceptible to modification by the environment as well as stochastic perturbations over time, adding to phenotypic diversity in the population. Despite its strong ties to the environment, epigenetics has never been well reconciled to evolutionary thinking, and in fact there is now strong evidence against the transmission of so-called “epi-alleles,” i.e. epigenetic modifications that pass through the germline.
However, genetic variants that regulate stochastic fluctuation of gene expression and phenotypes in the offspring appear to be transmitted as an epigenetic or even Lamarckian trait. Furthermore, even the normal process of cellular differentiation from a single cell to a complex organism is not understood well from a mathematical point of view. There is increasingly strong evidence that stem cells are highly heterogeneous and in fact stochasticity is necessary for pluripotency. This process appears to be tightly regulated through the epigenome in development. Moreover, in these biological contexts, “stochasticity” is hardly synonymous with “noise”, which often refers to variation which obscures a “true signal” (e.g., measurement error) or which is structural, as in physics (e.g., quantum noise). In contrast, “stochastic regulation” refers to purposeful, programmed variation; the fluctuations are random but there is no true signal to mask.
This workshop will serve as a forum for scientists and engineers with an interest in computational biology to explore the role of stochasticity in regulation, development and evolution, and its epigenetic basis. Just as thinking about stochasticity was transformative in physics and in some areas of biology, it promises to fundamentally transform modern genetics and help to explain phase transitions such as differentiation and cancer.
This workshop will include a poster session; a request for poster titles will be sent to registered participants in advance of the workshop.
Adam Arkin (Lawrence Berkeley Laboratory)
Gábor Balázsi (SUNY Stony Brook)
Domitilla Del Vecchio (Massachusetts Institute of Technology)
Michael Elowitz (California Institute of Technology)
Andrew Feinberg (Johns Hopkins University)
Don Geman (Johns Hopkins University)
Anita Göndör (Karolinska Institutet)
John Goutsias (Johns Hopkins University)
Garrett Jenkinson (Johns Hopkins University)
Andre Levchenko (Yale University)
Olgica Milenkovic (University of Illinois)
Johan Paulsson (Harvard University)
Leor Weinberger (University of California, San Francisco (UCSF))
The equations of gauge theory lie at the heart of our understanding of particle physics. The Standard Model, which describes the electromagnetic, weak, and strong forces, is based on the Yang-Mills equations. Starting with the work of Donaldson in the 1980s, gauge theory has also been successfully applied in other areas of pure mathematics, such as low dimensional topology, symplectic geometry, and algebraic geometry.
More recently, Witten proposed a gauge-theoretic interpretation of Khovanov homology, a knot invariant whose origins lie in representation theory. Khovanov homology is a “categorification” of the celebrated Jones polynomial, in the sense that its Euler characteristic recovers this polynomial. At the moment, Khovanov homology is only defined for knots in the three-sphere, but Witten’s proposal holds the promise of generalizations to other three-manifolds, and perhaps of producing new invariants of four-manifolds.
This workshop will bring together researchers from several different fields (theoretical physics, mathematical gauge theory, topology, analysis / PDE, representation theory, symplectic geometry, and algebraic geometry), and thus help facilitate connections between these areas. The common focus will be to understand Khovanov homology and related invariants through the lens of gauge theory.
This workshop will include a poster session; a request for posters will be sent to registered participants in advance of the workshop.
Edward Witten will be giving two public lectures as part of the Green Family Lecture series:
March 6, 2017 From Gauge Theory to Khovanov Homology Via Floer Theory
The goal of the lecture is to describe a gauge theory approach to Khovanov homology of knots, in particular, to motivate the relevant gauge theory equations in a way that does not require too much physics background. I will give a gauge theory perspective on the construction of singly-graded Khovanov homology by Abouzaid and Smith.
March 8, 2017 An Introduction to the SYK Model
The Sachdev-Ye model was originally a model of quantum spin liquids that was introduced in the mid-1990′s. In recent years, it has been reinterpreted by Kitaev as a model of quantum chaos and black holes. This lecture will be primarily a gentle introduction to the SYK model, though I will also describe a few more recent results.