Algebraic Geometry: Old and New - Next Einstein Initiative · •Higher-dimensional, more ... !"+$"+%"+&"+1=0 describes a Calabi-Yauthreefold. Robotics and algebraic equations •Robots

Algebraic Geometry: Old and New

Balázs Szendrői, University of Oxford

Future of Science, AIMS-Rwanda, Kigali, July 2019

Greek mathematics: conics as conic sections

Parabola Circle Ellipse Hyperbola

Descartes, Kepler 17th c.: equations of conics

• The cone has the following equation in coordinates (", $, %):

"2 + $2 = %2• The intersecting plane has equation

% = *" + +$ + ,• Combining the two equations, we get a quadratic

equation in coordinates (", $) in the plane:

-"2 + ."$ + /$2 + 0" + 1$ + 2 = 0

Equations of conics• Thus, conics can be described as plane quadrics: geometric objects

(curves) in the plane, given by quadratic equations!"2 + %"& + '&2 + (" + )& + * = 0

• After some algebra, we get one of the standard equations

Parabola Circle Ellipse Hyperbola

& = -". ".+&. = /. 0121 +

3141 = 1 01

21 −3141 = 1

Here (", &) are coordinates in the plane, and -, / are parameters.

Equations of conicsParabola Circle Ellipse Hyperbola

! = #$% $%+!% = '% ()*) +

+),) = 1 ()

*) −+),) = 1

Newton, late 17th c: cubic curves in the plane

−1 0 1 2−3

−2

−1

0

1

2

3

−1 0 1 2−3

−2

−1

0

1

2

3

−1 0 1 2−3

−2

−1

0

1

2

3

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

y2

= x3−

3

4x

y2

= x3−

3

4x +

1

4y

2= x

3

y2

= x3−

3

4x +

3

8

1

!" = $%

!" = $% − 34 $!" = $% − 34 $ +38

!" = $% − 34 $ +14

Geometric objects from polynomials• In general, given a polynomial f in variables (x,y), we get a plane curve

! ", $ = 0• We can think of (x,y) as being real numbers, but also• complex numbers ' + )*• rational numbers +/-• numbers - modulo some prime base p• …

• More variables: x,y,z,… - increase the dimension of our objects• More than one defining polynomial - increase their complexity

Quadric surfaces

After curves, the next simplest objects are quadric surfaces

!(#, %, &) = 0where ! is a generalquadratic polynomial.

Quadric surfaces: covered by families of lines

Cayley, late 19th century: cubic surfaces

• The equation of one cubic surface:

!"+ $"+ %" = 1

• It is not covered by lines. Rather, it contains exactly 27 lines.

Rational parametrization

• Let us go back to one of our simplest curves, the unit circle!" + $" = 1

• For any quantity ', the following formulas give a point of the circle:

! = )*+,)*-,, y = ")

)*-,

• Indeed, it is easy to check by direct algebra that)*+,)*-,

"+ ")

)*-,"= 1

Rational parametrization

• This map

! → !# − 1!# + 1 ,

2!!# + 1

is called a rational parametrization, and is a useful way to describe geometric objects that have them.• Quadrics, shapes given by one quadratic equation in any number of

variables, can always be rationally parametrized.• The covering by lines of the quadric surface was one such

parametrization.

Rational parametrization of cubics: the story

• Cubic curves in fact cannot be rationally parametrized (early 19th c.)• Cubic surfaces have interesting, nontrivial rational parametrizations

(late 19th c.)• 3-dimensional cubic geometries cannot be rationally parametrized

(Clemens-Griffiths 1972)• It has been an open conjecture for a long time that 4-dimensional

cubic geometries cannot be rationally parametrized either• Breaking News: Kontsevich (2020?) proves this conjecture.

An application of rational parametrization

• Integer Pythagorian triples: find integers (", $, %) so that

"2 + $2 = %2- a problem of Diophantus type!• In other words,

"%

*+ $

%*= 1

• We know the solution! Write , = -// to get"% =

,* − 1,* + 1 =

-* − /*-* + /* ,

$% =

2,,* + 1 =

2-/-* + /*

An application of rational parametrization

• So we deduce the solutions

! = #$ − &$ ' = 2#& z = #$ + &$

• For example, we get#, &) = 2,1 → (!, ', 0 = 3,4,5#, &) = 3,2 → (!, ', 0 = 5,12,13

• These formulae were known to the Greeks (and in fact earlier, to the Babylonians), but they were very likely derived in a very different way.

Cubic curves: group structure

• A cubic curve cannot be rationally parametrized.• But: the set of its points forms a group!• The group operations addition and

inverse are defined using simple geometry… • …but relatively complicated

polynomial equations.

Application of cubic curves: cryptography

• Information can be stored in the group structure of cubic curves…• …using a variant of the discrete logarithm problem. • This leads to Elliptic Curve Cryptography, and the need to find cubic

curves with large complexity.• One example:

!2 + $!= $& − 1895782483362476188247825431$+ 42810746555185028468846212199762991367145

Other geometries and their applications

• Higher-dimensional, more complex shapes are used in…• …variants of superstring

theory [cf. Jonathan’s talk]• For example, the equation!" + $" + %" + &" + 1 = 0

describes a Calabi-Yau threefold.

Robotics and algebraic equations

• Robots have rigid moving parts connected by joints• Idealized picture: triangular

platform• Geometry of this problem yields a

system of polynomial equations that describes its kinematics• Red curve: an algebraic set of

degree 40Picture created by Charles Wampler and Douglas Arnold

Polynomial equations in systems biology

Polynomial equations in systems biology

Theorem (Harrington et al): There are exactly 9 isolated stationary points in the Wnt shuttle model.

Method: Algebra of Grobner bases

Biological interpretation: The first model that explains multiple cellular decisions (division, movement, specialization)

Harrington et al, arXiv:1502.03188

Applied algebra and geometry: a rich fieldKinematics of robots

Algebraic systems biology

Elliptic curve cryptography

Power flow models

Topology of data

Some pictures and ideas from Seigal, arXiv:1603.08546