The 3-dimensional mirror symmetry is a certain powerful set of ideas originating in modern high energy physics that equates seemingly disparate computations in two mirror quantum field theories. These physical ideas can be projected to mathematics in many different ways, and various mathematical implications, generalizations, and analogies of the fundamental physics predictions are currently being studied by many groups of researchers around the world. The goal of my two lectures will be to give a gentle brief (and, as a result, one-sided) introduction to this subject with a view towards applications in enumerative geometry and, time permitting, automorphic forms.
The past decade has seen the emergence of surprising new connections between the real-world physics of elementary particle scattering processes, and new mathematical structures in combinatorics, algebra and geometry. These ideas provide, in a number of examples, a different starting point for conceptualizing this basic physics, where the fundamental principles of spacetime and quantum mechanics are not taken as primary, but instead emerge from a more primitive mathematical rubric. In these lectures I will illustrate these ideas in their most transparent setting, showing how the simplest particle scattering amplitudes are determined by avatars of famous polytopes known as (generalized) associahedra. I will also show how the combinatorial relationships captured by the facets of these polytopes remarkably admit a "curvy" realization in terms of "binary geometries", with the physical interpretation of generalizing particles to strings. The presentation will be elementary and essentially entirely self-contained, no previous knowledge of the relevant physics or mathematics will be assumed or needed.
A 1939 paper by Oppenheimer and Snyder (O-S) showed that,
according to an extreme situation of Einstein’s general relativity
theory (GR), a spherically symmetrical body of material without
pressure (“dust”) could collapse under gravity to a singularity where
the density and the space-time curvatures diverge to infinity.
However, the artificiality of the lack of pressure in the O-S model
and, more importantly, the assumption of exact spherical symmetry,
led most astrophysicists to disregard the appearance of the singularity,
even in the unlikely event of such an extreme situation of
gravitational collapse actually arising in nature.
Nevertheless, from around 1960, the discovery of quasars?celestial
bodies that can emit radio signals of an intensity that can exceed that
of a thousand galaxies, yet having a diameter less than that of our
solar system?showed that the scales of collapse envisaged in the O-S
model would actually be relevant. Yet the presence of the central
singularity was not considered to be relevant by most astrophysicists,
because of the artificiality of the assumption of spherical symmetry
(and of dust) that are made in the O-S model.
However, in 1964, I was able to prove a “singularity theorem” which
demonstrated, in effect, that in a collapse situation as extreme as that
of the O-S model, singularities are inevitable, irrespective of these O-
S symmetry and “dust” assumptions, so long as the local energy flux
never becomes negative. This led to the apparent inevitability of a
black hole, where the actual space-time singularity remains unseen,
hidden by the black hole’s visual horizon.
The mathematical techniques that I had developed, were then
extended by Stephen Hawking, so as to apply to the singularity that
describes the Big Bang, showing that this, also, cannot be evaded by
the introduction of irregularities that spoil the symmetry. There is a
time-symmetry in all these mathematical procedures, showing that the
classical theory of GR cannot evade these singular blemishes, either
in the past or the future. But what about the actual physics of the
situation? It is the common view that we must bring in quantum
theory (QT), and turn to a quantized version of GR?i.e. quantum
gravity?which should tell us how the physical world must actually
operate under these very extreme circumstances.
Yet, there is something very odd about the expected nature of these
two types of singularity: future and past, this being profoundly related
to the origin of the 2 nd Law of thermodynamics (which asserts, in
effect, that the randomness in the universe is relentlessly increasing
with time). We seem to need a profound time-asymmetry in the
combining of QT with GR if this is to explain the time-asymmetry of
the singularities, where the Big-Bang singularity is enormously
constrained. It can be argued that any standard quantization of GR
cannot resolve this issue without, perhaps, an accompanying
“gravitization of QT” in which the fundamental “measurement
problem” of QT is also resolved. Such a theory is currently
fundamentally lacking, however.
Nevertheless, a deeper understanding of the special nature of the Big
Bang can be illuminated by examining it from the perspective of
conformal geometry, according to which the Big-Bang singularity
becomes non-singular, this being quite different from the situation
arising from the singularities in black holes. In conformal geometry,
big and small become equivalent, which can only hold for a
singularity of the type we seem to find at the Big Bang. This situation
is also relevant in relating the extremely hot and dense Big Bang to
the extremely cold and rarefied remote future of a previous “cosmic
aeon”, leading to the picture of conformal cyclic cosmology (CCC)
according to which our Big Bang is viewed as the conformally
continued remote future of a previous cosmic aeon. It turns out that
there are now certain strong observational signals, providing some
remarkable support for this highly non-intuitive but mathematically
consistent CCC picture.
It is well known that a generic function on a compact smooth manifold has only Morse critical points (i.e. locally equivalent to a non-degenerate quadratic form). Counting gradient lines between critical points (saddle connections) one obtains a differential on so called Morse complex, giving a way to calculate cohomology of the underlying manifold with integer coefficients. About 40 years ago S.P.Novikov proposed a generalization of Morse theory when the function is replaced by a closed 1-form.
I will talk about new developments in Morse-Novikov theory when the 1-form is real part of a holomorphic 1-form rescaled by a non-zero complex parameter t . A remarkable feature of the holomorphic situation is that for a generic value of the argument of t there is no saddle connections at all, and one obtains a canonical basis in Morse-Novikov cohomology, represented by "infinite chains" which are typically everywhere dense.
For countably many special values of the argument of t one obtains a canonical change of the basis, giving a new elementary example of so-called Wall-Crossing structure (originally discovered in much more complicated theory of Donaldson-Thomas invariants). In concrete terms, one obtains a relation between some explicit rational matrix-valued functions in several variables, and some counting problems in dynamical systems. I will also review the connection of new theory to resurgent series, like e.g. Stirling formula for the asymptotic of Gamma function at infinity.
The theory of complex multiplication gives an explicit construction of abelian extensions of quadratic imaginary fields (or more general CM fields) from the values of modular functions on modular curves (or more general higher dimensional Shimura varieties) at CM points. We will describe an approach to extending this theory beyond the setting of CM fields, in which modular functions are replaced by ``rigid meromorphic cocycles”.
The classification of algebraic varieties is at the heart of algebraic geometry. With roots in the ancient world the theory saw great advances in dimensions one and two in the 19th century and the first half of 20th century. It was only in the 1970-80’s that a general framework was formulated, and by the early 1990’s a satisfactory theory was developed in dimension 3. The last 30 years has seen great progress in all dimensions.
In this talk I will try to give a historical perspective and discuss the theory in general terms. I will explain how the theory is based on birational transformations and moduli considerations.
Boundedness and moduli spaces play important roles in algebraic geometry. The subject has evolved in different directions depending on the type of objects being parameterized. In this talk I will focus on boundedness and moduli of algebraic varieties. In particular, I will discuss Fano and Calabi-Yau varieties as well as minimal models of arbitrary Kodaira dimension.
The shift locus S_d is the space of conjugacy classes of degree d polynomials f(z) in one complex variable for which all the critical points tend to infinity under repeated application of f. When d=2 this is the complement of the Mandelbrot set. Although S_d is a very complicated space geometrically, it turns out one can get a surprisingly concrete description of its topology; for example, S_2 is homeomorphic to an open annulus (this is equivalent to the famous theorem of Douady-Hubbard that the Mandelbrot set is connected). I would like to discuss two very explicit ways to capture the topology of S_d, one via the combinatorics of laminations (Butcher paper) and one via algebraic geometry (Sausages). As a corollary of this explicit description one can show that S_d has the homotopy type of a complex of half its real dimension, and is a K(pi,1) if d is 3 or 4.