tutorial: numerical algebraic geometry · homotopy continuation singular isolated solutions...
Post on 25-Sep-2018
223 Views
Preview:
TRANSCRIPT
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Tutorial: Numerical Algebraic GeometryBack to classical algebraic geometry...
with more computational power andhybrid symbolic-numerical algorithms
Anton Leykin
Georgia Tech
Waterloo, November 2011
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Outline
Homotopy continuationpredictor-corrector numerical methods, Newton’s method, (global)homotopy continuation scenarios
Singular isolated solutionsregularization of singular solutions, deflation, dual spaces/inversesystems
Positive dimensionwitness sets, numerical irreducible decomposition, numericalprimary decomposition
Certified homotopy trackingnumerical zeros, α-theory of Smale, heuristic vs. rigorouspath-tracking
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Computational algebraic geometry
What is the game?• Level 0: Given a system of polynomial equations in K[x1, ..., xn]
with finitely many solutions,
SOLVE.
( K could be Q, Z/pZ, R, C, ... )• Level 1+: Describe positive-dimensional solutions (curves,
surfaces, ...)Classical methods “generalize” linear algebra:• Gröbner basis: a generalization of Gaussian reduction;• Resultant: a generalization of determinant.
These methods are symbolic.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Linear Algebra Numerical Linear Algebray yAlgebraic Geometry Numerical Algebraic Geometry
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Applications
Robotics: Stewart-Gough platforms.
Griffis-Duffy platform: the solutioncontains a curve of degree 28.
`1
`2
`3`4
s1
s2
p���3
Enumerative algebraic geometry:solutions of Schubert problems.
... control theory, optimization, computer vision, math biology, realalgebraic geometry, algebraic curves ...
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Polynomial homotopy continuation
• Target system: n equations in n variables,
F (x) = (f1(x), . . . , fn(x)) = 0,
where fi ∈ R = C[x] = C[x1, ..., xn] for i = 1, ..., n.• Start system: n equations in n variables:
G(x) = (g1(x), . . . , gn(x)) = 0,
such that it is easy to solve.• Homotopy: for γ ∈ C \ {0} consider
H(x, t) = (1− t)G(x) + γtF (x), t ∈ [0, 1].
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Exampletarget start
f1 = x41x2 + 5x21x32 + x31 − 4 g1 = x51 − 1
f2 = x21 − x1x2 + x2 − 8 g2 = x22 − 1
Start solutions→ target solutions:
H(x, t) = 0 impliesdx
dt= −
(∂H
∂x
)−1∂H
∂t.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Exampletarget start
f = −x2 + 2 g = x2 − 1
The solution of the homotopy equation
H(x, t) = (1− t)g(x) + tf(x) = (1− 2t)x2 − 1 + 3t = 0
is singular for t = 1/3.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Randomization
• Note: the complement of a complex algebraic variety isconnected.
space of polynomials
f
g
• For all but finite number of γ ∈ C the homotopy
H(x, t) = (1− t)G(x) + γtF (x).
is regular for 0 ≤ t < 1.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Global picture
Optimal homotopy:
• the continuation paths are regular;• the homotopy establishes a bijection
between the start and target solutions.
Possible singular scenarios:
non-generic diverging paths multiple solutions
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Numerical algebraic geometry
• Sommese, Verschelde, and Wampler, Introduction to NumericalAG (2005)
• Sommese and Wampler, The numerical solution of systems ofpolynomials (2005)
Software:• PHCpack (Verschelde);• HOM4PS (group of T.Y.Li);• Bertini (group of Sommese);• NAG4M2: Numerical Algebraic Geometry for Macaulay2 (L.).
and more, e.g.: Maple’s ROOTFINDING[HOMOTOPY].
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Possible improvements• Parallel computation:
Paths are mutually independent⇒ linear speedups.
• Minimize the number of diverging paths:
• Total degree: Number of start solutions =product of degrees of equations (Bézoutbound).
• Polyhedral homotopies: Number of startsolutions = mixed volume of sparse system(BKK bound).
• Optimal homotopies:
• Cheater’s homotopy;
• Special homotopies: e.g., Pieri homotopy.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Multiple solutions
In general, with probability 1, thepicture looks like this:
Singular end games [Morgan, Sommese, Wampler (1991)]:• power-series method;• Cauchy integral method;• trace method.
Deflation:• regularizes an isolated singular solution;• restores quadratic convergence of the Newton’s method.
How to describe a singularity?
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Cauchy integral endgame
• An implication of Cauchy residue theorem:Let y : U → C holomorphic on a simply connected U ⊂ C,a ∈ C, and C ⊃ C ' S1 be a contour winding I(C, a) timesaround a.Then
y(a) =1
2πiI(C, a)
∮C
y(z)
z − adt,
• H(x, t) = 0 defines (a possibly multivalued function) x = x(t) ina neighborhood of t = 1.
• Idea: as the homotopy tracker approaches a singular x∗ = x(1)use Cauchy integral to compute x∗ staying away from x∗.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Winding numbers
(1 2 3 4 5)(6 7 8)(9 10)
|1− t| = ε
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Cauchy integral endgame
1. Pick a point on x = x(t), a solution to H(x, t) = 0 for t = t ∈ R;let ε = 1− t.
2. Track the path
C = {x(1− εeiθI) | θ ∈ [0, 2π]},
where I > 0 is such that
x = x(1− εeiθI) ⇒ θ ∈ {0, 2π}.
3. Let y(z) = x(1− zI), then y(z) is holonomic for |z| < ε (if ε� 1).4. Find numerically the integral
x(1) = y(0) =1
2π
∫|z|=ε
y(z)
zdz =
1
2π
∫[0,2π]
x(1− εeiθI) dθ.
(Note: one may use samples made when tracking the path C.)
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Newton’s method: x(n+1) = x(n) −f(x(n))
f ′(x(n))
Example 1: f(x) = x(x− 1)3, x(0) = 0.1
x(1) = −0.05000000000000000000000000000000000000000000000000x(2) = −0.00625000000000000000000000000000000000000000000000x(3) = −0.00011432926829268292682926829268292682926829268293x(4) = −0.00000003919561993882928315798471103711494222972094x(5) = −0.00000000000000460888914457438597268761599543603706x(6) = −0.00000000000000000000000000006372557744092567103642
Example 2: f(x) = x2(x− 1)3, x(0) = 0.1
x(1) = 0.04000000000000000000000000000000000000000000000000x(2) = 0.01866666666666666666666666666666666666666666666667x(3) = 0.00905920745920745920745920745920745920745920745921x(4) = 0.00446662546689373374865785737653016156369492056043x(5) = 0.00221818070337351048684295922675846246257988477728x(6) = 0.00110537952927547542499858913840929687677679537995
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflation method
Let f(x) = (f1(x), ..., fN (x)), N ≥ n, fi(x) ∈ C[x] = C[x1, ..., xn].
Let A(x) =
(∂fi∂xj
)∈ CN×n be the Jacobian matrix.
Given: an approximation x(0) of an exact isolated solution x∗, whichis singular, i.e., corankA(x∗) = n− rankA(x∗) > 0.
Newton’s method in homotopy continuationloses quadratic convergence around x∗. Isthere a way to restore the convergence?
• Want: a symbolic procedure that “makes” x∗ regular.• Rules:
• New variables are allowed.• Assume that the numerical rank of A(x(0)) equals A(x∗).
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflation step: create an augmented system in C[x,a]1. Introduce n new variables a;2. Add equations coming from A(x)a = 0;
Example. Let f1 = x31 + x1x22, f2 = x1x
22 + x32, f3 = x21x2 + x1x
22 and
x∗ = 0. ∂1 ∂2
f1 3x21 + x22 2x1x2
f2 x22 2x1x2 + 3x22
f3 2x1x2 + x22 x21 + 2x1x2
[a1
a2
]= 0.
3. Compute the rank r of A(x∗); (r = 0 for our example)
4. Add n− r random linear equations.
5. Find the solution (x∗,a∗) of the augmented system; (8 equations)
6. Repeat if (x∗,a∗) is singular. (2 steps for the example)
Theorem (L., Verschelde, Zhao)The multiplicity of (x∗,a∗) in the augmented system is smaller thanthat of x∗ in the original system.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Multiplicity
Staircases for I = 〈f1, f2, f3〉, wheref1 = x31 + x1x
22, f2 = x1x
22 + x32, f3 = x21x2 + x1x
22.
6
-hh
hh
hh
x
xx
x1x22 + x32
x21x2 + x1x22
x31 + x1x22
xhx42
ω = (−1,−2)�
6
-hh
hh
hh
xx
xx21x2 + x1x
22
x1x22 + x31
x32 + x1x22
xh x41
ω = (−2,−1)�
Multiplicity is the number of integer points under the staircase.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Dual space (local inverse system)
• For x ∈ Cn, let ∆βx : R→ C be a linear functional,
∆βx(f) = (∂β · f)(x) =
∂|β|f
∂β(x), f ∈ R.
• For an ideal I, the dual space Dx[I] is the subspace ofSpanC{∆β
x} of the functionals that annihilate I.• Filter by order:
D(0)x [I] ⊂ D(1)
x [I] ⊂ D(2)x [I] ⊂ . . .
where D(d)x [I] is the set of functionals of order at most d.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Macaulay array• For I = 〈f1, . . . , fm〉, the deflation matrix A(d)
I (x) is a part of theMacaulay array, the infinite matrix with entries(
∂
∂xβ(xαfi)
)((i,α),β)
where |α| < d and |β| ≤ d.• For example, for I = 〈f1, f2〉 ⊂ C[x, y],
A(2)I =
id ∂x ∂y ∂2x ∂x∂y ∂2yf1 ∗ ∗ ∗ ∗ ∗ ∗f2 ∗ ∗ ∗ ∗ ∗ ∗xf1 ∗ ∗ ∗ ∂
∂x2 (xf1) ∗ ∗xf2 ∗ ∗ ∗ ∗ ∗ ∗yf1 ∗ ∗ ∗ ∗ ∗ ∗yf2 ∗ ∗ ∗ ∗ ∗ ∗
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Dual spaces and deflation• For x ∈ V (I), get D(d)
x [I] by computing kerA(d)I (x).
• For the running example,
D0[I] = Span{ ∆(3,0) −∆(2,1) −∆(1,2) + ∆(0,3),
∆(2,0), ∆(1,1), ∆(0,2),
∆(1,0), ∆(0,1), ∆(0,0) }.
The leading terms with respect toω = (2, 1) correspond to themonomials under the staircase forthe standard basis forω = (−2,−1).
6
-hh
hh
hh
xx
xx21x2 + x1x
22
x1x22 + x31
x32 + x1x22
xh x41
ω = (−2,−1)�
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflation continued...
Idea of proof of termination for deflation:• Deflation "deflates the staircase";• the multiplicity of becomes 1 after a finite number of steps.
Related work:• Dual spaces: Macaulay (1916), ..., Stetter (1993), Mourrain
(1997), Dayton and Zeng (2005), Krone (2011).• Deflation: Lyapunov (1900?) , ..., Ojika et al (1987), Lecerf
(2002), L. et al (2006), ..., Lihong Zhi et al (2010).
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Higher-dimensional solution sets• Let I = (f1, . . . , fN ) be an ideal of C[x1, . . . , xn].• Goal: Understand the variety
X = V(I) = {x ∈ Cn | ∀f ∈ I, f(x) = 0}.
• A witness set for an equidimensional component Y of X:• a generic “slicing” plane L with dimL = codimY• witness points wY,L = Y ∩ L• (generators of I)
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Numerical irreducible decomposition• Homotopy mapping wY,L → wY,L′ :
HL,L′,γ(x, t) =
(f(x)
(1− t)L(x) + γtL′(x)
), t ∈ [0, 1].
L
L’
• Monodromy action: a permutation on wY,Lis produced by homotopy HL,L′,γ followedby HL′,L,γ′ for random γ, γ′ ∈ C.
• Irreducible decomposition: a partition of thewitness set wY,L stable under this action.
• Linear trace test: the average of thepoints in a witness set of an irreduciblecomponent behaves linearly.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Example with an embedded component
ExampleI = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1).
y + 1 = 0
x = 0
������������u(0, 0)
��solution set of
{x · x(y + 1) = 0;y · x(y + 1) = 0.
Numerical irreducible decomposition*sees two 1-dimensional components ...
... but does not discover the embedded point.
*NID reference: Sommese, Verschelde, Wampler “Numerical decompositionof the solution sets...” (2001)
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Example with an embedded component
ExampleI = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1).
y + 1 = 0
x = 0������������
u(0, 0)
��solution set of
{x · x(y + 1) = 0;y · x(y + 1) = 0.
Numerical irreducible decomposition*sees two 1-dimensional components ...
... but does not discover the embedded point.
*NID reference: Sommese, Verschelde, Wampler “Numerical decompositionof the solution sets...” (2001)
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Example with an embedded component
ExampleI = (f1, f2), where f1 = x2(y + 1) and f2 = xy(y + 1).
y + 1 = 0
x = 0������������u(0, 0) ��
solution set of{x · x(y + 1) = 0;y · x(y + 1) = 0.
Numerical irreducible decomposition*sees two 1-dimensional components ...
... but does not discover the embedded point.
*NID reference: Sommese, Verschelde, Wampler “Numerical decompositionof the solution sets...” (2001)
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflation matrix, system, and ideal
Example (Deflation ideal of order d = 2)original system f(x, y) → deflation matrix A(2)
I (x, y)
→→ deflation system D(2)f(x, y,a) →→ deflation ideal I(2)
∂x ∂y ∂2x ∂x∂y ∂2yf1 ∗ ∗ ∗ ∗ ∗f2 ∗ ∗ ∗ ∗ ∗xf1 ∗ ∗ 6x(y + 1) ∗ ∗xf2 ∗ ∗ ∗ ∗ ∗yf1 ∗ ∗ ∗ ∗ ∗yf2 ∗ ∗ ∗ ∗ ∗
axayaxx
axyayy
=: D(2)f(x, y,a)
6
∂2(xf1)∂x2
6
C[x, y,a]6
Deflation idealI(2) = 〈f , D(2)f〉 ⊂ C[x, y,a]
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflation matrix, system, and ideal
Example (Deflation ideal of order d = 2)original system f(x, y) → deflation matrix A(2)
I (x, y) →→ deflation system D(2)f(x, y,a)
→→ deflation ideal I(2)
∂x ∂y ∂2x ∂x∂y ∂2yf1 ∗ ∗ ∗ ∗ ∗f2 ∗ ∗ ∗ ∗ ∗xf1 ∗ ∗ 6x(y + 1) ∗ ∗xf2 ∗ ∗ ∗ ∗ ∗yf1 ∗ ∗ ∗ ∗ ∗yf2 ∗ ∗ ∗ ∗ ∗
ax
ay
axx
axy
ayy
=: D(2)f(x, y,a)6
∂2(xf1)∂x2
6
C[x, y,a]6
Deflation idealI(2) = 〈f , D(2)f〉 ⊂ C[x, y,a]
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflation matrix, system, and ideal
Example (Deflation ideal of order d = 2)original system f(x, y) → deflation matrix A(2)
I (x, y) →→ deflation system D(2)f(x, y,a) →→ deflation ideal I(2)
∂x ∂y ∂2x ∂x∂y ∂2yf1 ∗ ∗ ∗ ∗ ∗f2 ∗ ∗ ∗ ∗ ∗xf1 ∗ ∗ 6x(y + 1) ∗ ∗xf2 ∗ ∗ ∗ ∗ ∗yf1 ∗ ∗ ∗ ∗ ∗yf2 ∗ ∗ ∗ ∗ ∗
ax
ay
axx
axy
ayy
=: D(2)f(x, y,a)6
∂2(xf1)∂x2
6
C[x, y,a]6
Deflation idealI(2) = 〈f , D(2)f〉 ⊂ C[x, y,a]
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Deflated variety
• For the given variety X = V (I) = {x | f(x) = 0 for all f ∈ I}define deflated variety of order d:X(d) = V (I(d)) ⊂ CB(n,d), where B(n, d) = n− 1 +
(n+d+1
d
).
• Projection: πd : CB(n,d) → Cn, πd(x,a) 7→ x; πdX(d) = X.
• A component Y ⊂ X is called visible at order d if Y = πdZ for anisolated component Z ⊂ X(d).
s(0, 0) Theorem (L.)Every component is visible at some order d.
ExampleFor the running example d = 1 is sufficient.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Example: I = (f1, f2), f1 = x2(y + 1), f2 = xy(y + 1)
• Isolated components of X = V (I):
V (y + 1), V (x).
• Additional equations A(1)I a = 0:[
2x(y + 1) x2
y(y + 1) x(2y + 1)
] [axay
]= 0.
• Isolated components of X(1) = V (I(1)):
V (y + 1, ay), V (x, y), V (x, ax), V (x, y + 1)
• The first three project onto the components of X, the last one (a“pseudo-component”) projects onto a singular point.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Witness sets
For an irreducible subvariety Y ⊂ X = V (I), where I ⊂ R is an ideal,a
generalized
witness set consists of• a generic “slicing” plane L with dimL = codimY
(d)
• witness points w = Y
(d)
∩ L
and their projections via πd
• generators of I
(d)
Definition: Y (d) is an isolated irreducible component of the deflated varietyX(d) mapping onto Y under projection πd.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Witness sets
For an irreducible subvariety Y ⊂ X = V (I), where I ⊂ R is an ideal,a generalized witness set consists of• a generic “slicing” plane L with dimL = codimY (d)
• witness points w = Y (d) ∩ L and their projections via πd• generators of I(d)
Definition: Y (d) is an isolated irreducible component of the deflated varietyX(d) mapping onto Y under projection πd.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
The algorithm and NPD concept
• Idea of the algorithm: Use numerical irreducible decompositionof a deflated variety to find (generalized) witness setsrepresenting components.
• Definition: Numerical primary decomposition is a collection ofsuch witness sets, one per component.
• Deficiencies:• There is an apriori bound on the order d of needed to make all
components visible, however it is not practical.• Pseudocomponents (projections of components of X(d) that are
not components of X) are hard to eliminate.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Local global?
Local knowledge:• Dx[I] describes the local ring (R/I)x = Rx/Ix.• A generic point on a component Y is a smooth point of Y that
does not belong to any component not containing Y properly.• A generic point x ∈ Y together with the algorithm for computingD
(d)x [I] describe Y .
Global knowledge:• Given a NPD (as a collection of witness sets), mark one generic
point on each component.• The set of marked points describe the ideal I.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Path switching
Applications in mathematics wherepath certification is desirable:• Numerical irreducible
decomposition algorithms• Galois group computation based
on monodromy• Problems where the root count is
impossible by other methods
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Newton’s method and approximate zerosGiven f ∈ C[x], consider the Newton operator associated to f ,
N(f)(x) = x−Df(x)−1f(x),
where Df(x) is the n× n derivative (Jacobian) matrix of f at x ∈ Cn.
• Definition: x ∈ Cn is an approximate zero of f with associatedzero η ∈ Cn if
‖N(f)l(x)− η‖ ≤ ‖x− η‖22l−1
, l ≥ 0.
• γ-theorem(Smale): Let x ∈ Cn, η ∈ f−1(0), and
‖x− η‖ ≤ 3−√
7
2γ(f, x), where γ(f, x) = sup
k≥2
∥∥∥∥Df(x)−1Dkf(x)
k!
∥∥∥∥1
k−1
.
Then x is an approximate zero of f associated to η.• Hauenstein, Sottile: alphaCertify, software for certification of
regular solutions.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Smale’s α theory
• α-theorem: Let
β(f, x) = ‖x−N(f)(x)‖ = ‖Df(x)−1f(x)‖ .
Then α(f, x) = β(f, x)γ(f, x) < 0.15767 certifies that x is anapproximate zero of f .
• "robust" theorem: Let x ∈ Cn with α(f, x) < 0.03. If
‖x− y‖ < 1
20γ(f, x),
then y is an approximate zero associated to the same zero as x.
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Newton’s operator attraction basins
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Certified regions
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Robust regions
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Linear and segment homotopy• Let H(d) be the space of systems of homogeneous polynomials
of fixed degrees (d) = (d1, . . . , dn) (with the Bombieri-Weylnorm).
• Consider f, g ∈ S = {f ∈ H(d) : ‖f‖ = 1} ⊂ H(d).• Using α-theory we design certified homotopy tracking (CHT)
algorithm that tracks a linear homotopy on S (assuming BSSmodel of computation).
• The “robust” α-theory leads to the robust CHT algorithm (Beltrán,L.):
g
f Take input f, g with coefficients in Q[i].Use the segment homotopy:
t→ ht = (1− t)g + tf, t = [0, 1].
All computations use exact linearalgebra over Q[i].
Homotopy continuation Singular isolated solutions Positive dimension Certified homotopy tracking
Future
• General methods• (Numerical) local ring structure• (Numerical) primary decomposition• Real solutions, real homotopy continuation
• Certification• Generalizations to higher order methods• Certification of singular isolated solutions• Certification of (irreducible/primary) decomposition
• Upcoming events• SI(AG)2: SIAM activity group in algebraic geometry• IMA PI summer program for graduate students on
Algebraic Geometry for Applications
June 18th – July 6th at Georgia Tech
top related