Primera versión terminada

codigo/zariski_vankampen.py

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+r"""
+Zariski-Van Kampen method implementation
+
+This file contains functions to compute the fundamental group of
+the complement of a curve in the complex affine or projective plane,
+using Zariski-Van Kampen approach. It depends on the package ``sirocco``.
+
+The current implementation allows to compute a presentation of the
+fundamental group of curves over the rationals or number fields with
+a fixed embedding on `\QQbar`.
+
+Instead of computing a representation of the braid monodromy, we
+choose several base points and a system of paths joining them that
+generate all the necessary loops around the points of the discriminant.
+The group is generated by the free groups over these points, and
+braids over this paths gives relations between these generators.
+This big group presentation is simplified at the end.
+
+.. TODO::
+
+    Implement the complete braid monodromy approach.
+
+AUTHORS:
+
+- Miguel Marco (2015-09-30): Initial version
+
+EXAMPLES::
+
+    sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
+    sage: R.<x,y> = QQ[]
+    sage: f = y^3 + x^3 -1
+    sage: fundamental_group(f) # optional - sirocco
+    Finitely presented group < x0 |  >
+"""
+# ****************************************************************************
+#       Copyright (C) 2015 Miguel Marco <mmarco@unizar.es>
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+from __future__ import division, absolute_import
+
+from sage.groups.braid import BraidGroup
+from sage.groups.perm_gps.permgroup_named import SymmetricGroup
+from sage.rings.rational_field import QQ
+from sage.rings.qqbar import QQbar
+from sage.rings.all import CC
+from sage.parallel.decorate import parallel
+from sage.misc.flatten import flatten
+from sage.groups.free_group import FreeGroup
+from sage.misc.misc_c import prod
+from sage.rings.complex_field import ComplexField
+from sage.rings.complex_interval_field import ComplexIntervalField
+from sage.combinat.permutation import Permutation
+from sage.functions.generalized import sign
+
+
+def braid_from_piecewise(strands):
+    r"""
+    Compute the braid corresponding to the piecewise linear curves strands.
+
+    INPUT:
+
+    - ``strands`` -- a list of lists of tuples ``(t, c)``, where ``t``
+      is a number between 0 and 1, and ``c`` is a complex number
+
+    OUTPUT:
+
+    The braid formed by the piecewise linear strands.
+
+    EXAMPLES::
+
+        sage: from sage.schemes.curves.zariski_vankampen import braid_from_piecewise # optional - sirocco
+        sage: paths = [[(0, I), (0.2, -1 - 0.5*I), (0.8, -1), (1, -I)],
+        ....:          [(0, -1), (0.5, -I), (1, 1)],
+        ....:          [(0, 1), (0.5, 1 + I), (1, I)]]
+        sage: braid_from_piecewise(paths) # optional - sirocco
+        s0*s1
+    """
+    L = strands
+    i = min(val[1][0] for val in L)
+    totalpoints = [[[a[0][1].real(), a[0][1].imag()]] for a in L]
+    indices = [1 for a in range(len(L))]
+    cuts = [0,i]
+    while i < 1:
+        for j, val in enumerate(L):
+            if val[indices[j]][0] > i:
+                xaux = val[indices[j] - 1][1]
+                yaux = val[indices[j]][1]
+                aaux = val[indices[j] - 1][0]
+                baux = val[indices[j]][0]
+                interpola = xaux + (yaux - xaux) * (i - aaux) / (baux - aaux)
+                totalpoints[j].append([interpola.real(), interpola.imag()])
+            else:
+                totalpoints[j].append([val[indices[j]][1].real(),
+                                       val[indices[j]][1].imag()])
+                indices[j] = indices[j] + 1
+        i = min(val[indices[k]][0] for k, val in enumerate(L))
+        cuts.append(i)
+    for j, val in enumerate(L):
+        totalpoints[j].append([val[-1][1].real(), val[-1][1].imag()])
+
+
+    braid = []
+    G = SymmetricGroup(len(totalpoints))
+
+    def sgn(x, y):
+        if x < y:
+            return 1
+        if x > y:
+            return -1
+        return 0
+    for i in range(len(totalpoints[0]) - 1):
+        l1 = [totalpoints[j][i] for j in range(len(L))]
+        l2 = [totalpoints[j][i + 1] for j in range(len(L))]
+        M = [[l1[s], l2[s]] for s in range(len(l1))]
+        M.sort()
+        l1 = [a[0] for a in M]
+        l2 = [a[1] for a in M]
+        cruces = []
+        for j in range(len(l2)):
+            for k in range(j):
+                if l2[j] < l2[k]:
+                    t = (l1[j][0] - l1[k][0])/(l2[k][0] - l1[k][0] + l1[j][0] - l2[j][0])
+                    s = sgn(l1[k][1]*(1 - t) + t*l2[k][1], l1[j][1]*(1 - t) + t*l2[j][1])
+                    cruces.append([t, k, j, s])
+        if cruces:
+            cruces.sort()
+            P = G(Permutation([]))
+            while cruces:
+                # we select the crosses in the same t
+                crucesl = [c for c in cruces if c[0] == cruces[0][0]]
+                altura = cuts[i]*(1-cruces[0][0])+cuts[i+1]*cruces[0][0]
+                crossesl = [(P(c[2] + 1) - P(c[1] + 1), c[1], c[2], c[3])
+                            for c in crucesl]
+                cruces = cruces[len(crucesl):]
+                while crossesl:
+                    crossesl.sort()
+                    c = crossesl.pop(0)
+                    braid.append((altura, c[3]*min(map(P, [c[1] + 1, c[2] + 1]))))
+                    P = G(Permutation([(c[1] + 1, c[2] + 1)])) * P
+                    crossesl = [(P(cr[2]+1) - P(cr[1]+1), cr[1], cr[2], cr[3])
+                                for cr in crossesl]
+
+    return braid
+
+
+def discrim(f):
+    r"""
+    Return the points in the discriminant of ``f``.
+
+    The result is the set of values of the first variable for which
+    two roots in the second variable coincide.
+
+    INPUT:
+
+    - ``f`` -- a polynomial in two variables with coefficients in a
+      number field with a fixed embedding in `\QQbar`
+
+    OUTPUT:
+
+    A list with the values of the discriminant in `\QQbar`.
+
+    EXAMPLES::
+
+        sage: from sage.schemes.curves.zariski_vankampen import discrim # optional - sirocco
+        sage: R.<x,y> = QQ[]
+        sage: f = (y^3 + x^3 - 1) * (x + y)
+        sage: discrim(f) # optional - sirocco
+        [1,
+        -0.500000000000000? - 0.866025403784439?*I,
+        -0.500000000000000? + 0.866025403784439?*I]
+    """
+    x, y = f.variables()
+    F = f.base_ring()
+    poly = F[x](f.discriminant(y).resultant(f, y)).radical()
+    return poly.roots(QQbar, multiplicities=False)
+
+
+def segments(points):
+    """
+    Return the bounded segments of the Voronoi diagram of the given points.
+
+    INPUT:
+
+    - ``points`` -- a list of complex points
+
+    OUTPUT:
+
+    A list of pairs ``(p1, p2)``, where ``p1`` and ``p2`` are the
+    endpoints of the segments in the Voronoi diagram.
+
+    EXAMPLES::
+
+        sage: from sage.schemes.curves.zariski_vankampen import discrim, segments # optional - sirocco
+        sage: R.<x,y> = QQ[]
+        sage: f = y^3 + x^3 - 1
+        sage: disc = discrim(f) # optional - sirocco
+        sage: segments(disc) # optional - sirocco # abs tol 1e-15
+        [(-2.84740787203333 - 2.84740787203333*I,
+        -2.14285714285714 + 1.11022302462516e-16*I),
+        (-2.84740787203333 + 2.84740787203333*I,
+        -2.14285714285714 + 1.11022302462516e-16*I),
+        (2.50000000000000 + 2.50000000000000*I,
+        1.26513881334184 + 2.19128470333546*I),
+        (2.50000000000000 + 2.50000000000000*I,
+        2.50000000000000 - 2.50000000000000*I),
+        (1.26513881334184 + 2.19128470333546*I, 0.000000000000000),
+        (0.000000000000000, 1.26513881334184 - 2.19128470333546*I),
+        (2.50000000000000 - 2.50000000000000*I,
+        1.26513881334184 - 2.19128470333546*I),
+        (-2.84740787203333 + 2.84740787203333*I,
+        1.26513881334184 + 2.19128470333546*I),
+        (-2.14285714285714 + 1.11022302462516e-16*I, 0.000000000000000),
+        (-2.84740787203333 - 2.84740787203333*I,
+        1.26513881334184 - 2.19128470333546*I)]
+    """
+    from numpy import array, vstack
+    from scipy.spatial import Voronoi
+    discpoints = array([(CC(a).real(), CC(a).imag()) for a in points])
+    added_points = 3 * abs(discpoints).max() + 1.0
+    configuration = vstack([discpoints, array([[added_points, 0],
+                                               [-added_points, 0],
+                                               [0, added_points],
+                                               [0, -added_points]])])
+    V = Voronoi(configuration)
+    res = []
+    for rv in V.ridge_vertices:
+        if -1 not in rv:
+            p1 = CC(list(V.vertices[rv[0]]))
+            p2 = CC(list(V.vertices[rv[1]]))
+            res.append((p1, p2))
+    return res
+
+
+def followstrand(f, x0, x1, y0a, prec=53):
+    r"""
+    Return a piecewise linear approximation of the homotopy continuation
+    of the root ``y0a`` from ``x0`` to ``x1``.
+
+    INPUT:
+
+    - ``f`` -- a polynomial in two variables
+    - ``x0`` -- a complex value, where the homotopy starts
+    - ``x1`` -- a complex value, where the homotopy ends
+    - ``y0a`` -- an approximate solution of the polynomial `F(y) = f(x_0, y)`
+    - ``prec`` -- the precision to use
+
+    OUTPUT:
+
+    A list of values `(t, y_{tr}, y_{ti})` such that:
+
+    - ``t`` is a real number between zero and one
+    - `f(t \cdot x_1 + (1-t) \cdot x_0, y_{tr} + I \cdot y_{ti})`
+      is zero (or a good enough approximation)
+    - the piecewise linear path determined by the points has a tubular
+      neighborhood  where the actual homotopy continuation path lies, and
+      no other root intersects it.
+
+    EXAMPLES::
+
+        sage: from sage.schemes.curves.zariski_vankampen import followstrand # optional - sirocco
+        sage: R.<x,y> = QQ[]
+        sage: f = x^2 + y^3
+        sage: x0 = CC(1, 0)
+        sage: x1 = CC(1, 0.5)
+        sage: followstrand(f, x0, x1, -1.0) # optional - sirocco # abs tol 1e-15
+        [(0.0, -1.0, 0.0),
+         (0.7500000000000001, -1.015090921153253, -0.24752813818386948),
+         (1.0, -1.026166099551513, -0.32768940253604323)]
+    """
+    CIF = ComplexIntervalField(prec)
+    CC = ComplexField(prec)
+    G = f.change_ring(QQbar).change_ring(CIF)
+    (x, y) = G.variables()
+    g = G.subs({x: (1 - x) * CIF(x0) + x * CIF(x1)})
+    coefs = []
+    deg = g.total_degree()
+    for d in range(deg + 1):
+        for i in range(d + 1):
+            c = CIF(g.coefficient({x: d - i, y: i}))
+            cr = c.real()
+            ci = c.imag()
+            coefs += list(cr.endpoints())
+            coefs += list(ci.endpoints())
+    yr = CC(y0a).real()
+    yi = CC(y0a).imag()
+    from sage.libs.sirocco import contpath, contpath_mp
+    try:
+        if prec == 53:
+            points = contpath(deg, coefs, yr, yi)
+        else:
+            points = contpath_mp(deg, coefs, yr, yi, prec)
+        return points
+    except Exception:
+        return followstrand(f, x0, x1, y0a, 2 * prec)
+
+
+@parallel
+def braid_in_segment(f, x0, x1):
+    """
+    Return the braid formed by the `y` roots of ``f`` when `x` moves
+    from ``x0`` to ``x1``.
+
+    INPUT:
+
+    - ``f`` -- a polynomial in two variables
+    - ``x0`` -- a complex number
+    - ``x1`` -- a complex number
+
+    OUTPUT:
+
+    A braid.
+
+    EXAMPLES::
+
+        sage: from sage.schemes.curves.zariski_vankampen import braid_in_segment # optional - sirocco
+        sage: R.<x,y> = QQ[]
+        sage: f = x^2 + y^3
+        sage: x0 = CC(1,0)
+        sage: x1 = CC(1, 0.5)
+        sage: braid_in_segment(f, x0, x1) # optional - sirocco
+        s1
+
+    TESTS:
+
+    Check that :trac:`26503` is fixed::
+
+        sage: wp = QQ['t']([1, 1, 1]).roots(QQbar)[0][0]
+        sage: Kw.<wp> = NumberField(wp.minpoly(), embedding=wp)
+        sage: R.<x, y> = Kw[]
+        sage: z = -wp - 1
+        sage: f = y*(y + z)*x*(x - 1)*(x - y)*(x + z*y - 1)*(x + z*y + wp)
+        sage: from sage.schemes.curves import zariski_vankampen as zvk
+        sage: g = f.subs({x: x + 2*y})
+        sage: p1 = QQbar(sqrt(-1/3))
+        sage: p2 = QQbar(1/2+sqrt(-1/3)/2)
+        sage: B = zvk.braid_in_segment(g,CC(p1),CC(p2)) # optional - sirocco
+        sage: B.left_normal_form()  # optional - sirocco
+        (1, s5)
+    """
+    CC = ComplexField(64)
+    (x, y) = f.variables()
+    I = QQbar.gen()
+    X0 = QQ(x0.real()) + I * QQ(x0.imag())
+    X1 = QQ(x1.real()) + I * QQ(x1.imag())
+    F0 = QQbar[y](f(X0, y))
+    y0s = F0.roots(multiplicities=False)
+    strands = [followstrand(f, x0, x1, CC(a)) for a in y0s]
+    complexstrands = [[(a[0], CC(a[1], a[2])) for a in b] for b in strands]
+    centralbraid = braid_from_piecewise(complexstrands)
+
+    return centralbraid
+
+
+def fundamental_group(f, simplified=True, projective=False):
+    r"""
+    Return a presentation of the fundamental group of the complement of
+    the algebraic set defined by the polynomial ``f``.
+
+    INPUT:
+
+    - ``f`` -- a polynomial in two variables, with coefficients in either
+      the rationals or a number field with a fixed embedding in `\QQbar`
+
+    - ``simplified`` -- boolean (default: ``True``); if set to ``True`` the
+      presentation will be simplified (see below)
+
+    - ``projective`` -- boolean (default: ``False``); if set to ``True``,
+      the fundamental group of the complement of the projective completion
+      of the curve will be computed, otherwise, the fundamental group of
+      the complement in the affine plane will be computed
+
+    If ``simplified`` is ``False``, the returned presentation has as
+    many generators as degree of the polynomial times the points in the
+    base used to create the segments that surround the discriminant. In
+    this case, the generators are granted to be meridians of the curve.
+
+    OUTPUT:
+
+    A presentation of the fundamental group of the complement of the
+    curve defined by ``f``.
+
+    EXAMPLES::
+
+        sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
+        sage: R.<x,y> = QQ[]
+        sage: f = x^2 + y^3
+        sage: fundamental_group(f) # optional - sirocco
+        Finitely presented group < ... >
+        sage: fundamental_group(f, simplified=False) # optional - sirocco
+        Finitely presented group < ... >
+
+    ::
+
+        sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
+        sage: R.<x,y> = QQ[]
+        sage: f = y^3 + x^3
+        sage: fundamental_group(f) # optional - sirocco
+        Finitely presented group < ... >
+
+    It is also possible to have coefficients in a number field with a
+    fixed embedding in `\QQbar`::
+
+        sage: from sage.schemes.curves.zariski_vankampen import fundamental_group # optional - sirocco
+        sage: zeta = QQbar['x']('x^2+x+1').roots(multiplicities=False)[0]
+        sage: zeta
+        -0.50000000000000000? - 0.866025403784439?*I
+        sage: F = NumberField(zeta.minpoly(), 'zeta', embedding=zeta)
+        sage: F.inject_variables()
+        Defining zeta
+        sage: R.<x,y> = F[]
+        sage: f = y^3 + x^3 +zeta *x + 1
+        sage: fundamental_group(f) # optional - sirocco
+        Finitely presented group < x0 |  >
+    """
+    (x, y) = f.variables()
+    F = f.base_ring()
+    g = f.radical()
+    d = g.degree(y)
+    while not g.coefficient(y**d) in F or (projective and g.total_degree() > d):
+        g = g.subs({x: x + y})
+        d = g.degree(y)
+    disc = discrim(g)
+    segs = segments(disc)
+    vertices = list(set(flatten(segs)))
+    Faux = FreeGroup(d)
+    F = FreeGroup(d * len(vertices))
+    rels = []
+    if projective:
+        rels.append(prod(F.gen(i) for i in range(d)))
+    braidscomputed = braid_in_segment([(g, seg[0], seg[1]) for seg in segs])
+    for braidcomputed in braidscomputed:
+        seg = (braidcomputed[0][0][1], braidcomputed[0][0][2])
+        b = braidcomputed[1]
+        i = vertices.index(seg[0])
+        j = vertices.index(seg[1])
+        for k in range(d):
+            el1 = Faux([k + 1]) * b.inverse()
+            el2 = k + 1
+            w1 = F([sign(a) * d * i + a for a in el1.Tietze()])
+            w2 = F([d * j + el2])
+            rels.append(w1 / w2)
+    G = F / rels
+    if simplified:
+        return G.simplified()
+    else:
+        return G

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seminario.css

@@ -83,7 +83,7 @@ div.output_wrapper {
     border-radius: 25px;
     color: white;
     text-align: center;
-    background: #223d71;
+    background: #3333b3;
 }
 
 .reveal h1.title {
@@ -110,7 +110,7 @@ div.output_wrapper {
 
 .reveal h2 {
     margin-top: 0 !important;
-    background: #223d71;
+    background: #3333b3;
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     margin-right: 0;
     margin-bottom: 4%;
@@ -155,7 +155,7 @@ div.output_wrapper {
     margin-left: auto;
     margin-right: auto;
     display: block;
-    max-height: 350px;
+    max-height: 400px;
 }
 
 .reveal li {
@@ -212,3 +212,54 @@ div.output_wrapper {
 .reveal th {
     font-size: 25px
 }
+
+
+
+
+.reveal div.thm {
+
+    margin-right: 5%;
+    margin-left:5%;
+    margin-top: 2%;
+    padding-left: 2%;
+    padding-right: 2%;
+    padding-top: 0%;
+    padding-bottom: 2%;
+    border-radius:15px;
+    background: #eaeaf3;
+    box-shadow: 5px 5px 10px grey;
+}
+
+.reveal div.thm h3 {
+    padding-top: 1%;
+    padding-bottom: 1%;
+    padding-right: 1%;
+    padding-left: 3%;
+/*     width: 100.8% !important; */
+    margin-right: -2.4%;
+    margin-left: -2.4%;
+    margin-bottom: 0;
+    border-radius: 15px 15px 0px 0px;
+/*     font-size: 1.9em; */
+    background: #3333b3;
+    color: white;
+    text-align: left;
+}
+
+.reveal div.thm p {
+
+    margin-top: 1%;
+    margin-bottom: 1%;
+    padding-top: 1%;
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+}
+
+.reveal div.thm ul {
+    display: contents;
+}
+
+.reveal div.thm ul li {
+    padding-left: 3%;
+    padding-top: 0.5%;
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+}

seminario.ipynb

@@ -117,11 +117,15 @@
    "cell_type": "markdown",
    "metadata": {
     "slideshow": {
-     "slide_type": "fragment"
+     "slide_type": "slide"
     }
    },
    "source": [
-    "**Ejemplo**: $y^2= 1-x^2$"
+    "## Ejemplo \n",
+    "\n",
+    "$y^2= x^2-1$\n",
+    "\n",
+    "![](hiperbola.png)"
    ]
   },
   {
@@ -132,7 +136,720 @@
     }
    },
    "source": [
-    "La proyección sobre el eje $x$ es $2:1$ excepto cuando $1-x^2=0$"
+    "La proyección sobre el eje $x$ es $2:1$ excepto en $x=\\pm 1$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Ejemplo\n",
+    "\n",
+    "![](supriemannelev-0.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Ejemplo\n",
+    "\n",
+    "<video data-autoplay src=\"supriemannelev.mp4\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Topología del encaje\n",
+    "\n",
+    "$C\\subseteq \\mathbb{C}^2$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Codimensión 2: situación análoga a los nudos"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Uno de los invariantes principales es el grupo fundamental del complementario $\\pi_1(\\mathbb{C}^2\\setminus C)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "![](monodtrenzas1.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "![](monodtrenzas2.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "![](monodtrenzas3.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas4.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas5.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas6.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas7.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas8.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Lema</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "    Esta construcción determina un morfismo de grupos $\\pi_1(\\mathbb{C}\\setminus \\Delta)\\longrightarrow B_d$.\n",
+    "    </p>\n",
+    "</div>\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Definición</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "    Este morfismo recibe el nombre de <i>monodromía de trenzas</i> de la curva.\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Se puede representar como una lista de trenzas correspondientes a una base geométrica de $\\pi_1(\\mathbb{C}\\setminus \\Delta)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas9.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<video data-autoplay src=\"monodtrenzas10.webm\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Monodromía de trenzas\n",
+    "\n",
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Teorema</b> (Zariski-Van Kampen)\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "    El grupo fundamental del complementario $\\pi_1(\\mathbb{C}^2\\setminus C)$ es el cociente del grupo libre $F_d$ por la acción de la monodromía de trenzas.\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Teorema</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "    La monodromía de trenzas determina completamente la topología del par $(\\mathbb{C}^2,C)$.\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Pares de Zariski\n",
+    "\n",
+    "La monodromía de trenzas es un objeto algebraico (morfismo de grupos) definido a partir de otro objeto algebraico (ideal en el anillo de polinomios)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Definición</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "        Dos curvas algebraicas $C_1$ y $C_2$ forman un <i>par de Zariski aritmético</i> si\n",
+    "    <ul>\n",
+    "        <li class=\"fragment\">Los polinomios que las definen $f_1,f_2$ son conjugados de Galois.</li>\n",
+    "        <li class=\"fragment\">Los grupos fundamentales $\\pi_1(\\mathbb{C}^2\\setminus C_1)$ y $\\pi_1(\\mathbb{C}^2\\setminus C_1)$ son no isomorfos.</li>\n",
+    "    </ul>\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Pares de Zariski\n",
+    "\n",
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Teorema</b> (ACG_)\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "        Existen pares de Zariski aritméticos\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Esto implica que la monodromía de trenzas (o el grupo fundamental) no se puede calcular de manera puramente algebraica."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Se necesitan cálculos **numéricos**, pero **certificados**."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo de la monodromía de trenzas\n",
+    "\n",
+    "Primer paso: elegir los caminos\n",
+    "\n",
+    "![](voronoi0.jpg)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo de la monodromía de trenzas\n",
+    "\n",
+    "Primer paso: elegir los caminos\n",
+    "\n",
+    "![](voronoi1.jpg)\n",
+    "\n",
+    "El *Diagrama de Voronoi* nos da segmentos que rodean los puntos del discriminante, acercándose lo menos posible."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo de la trenza sobre un segmento\n",
+    "\n",
+    "Mediante cambio de variable, podemos normalizar a $[0,1]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "![](trenzasola.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo de la trenza sobre un segmento\n",
+    "\n",
+    "Mediante cambio de variable, podemos normalizar a $[0,1]$\n",
+    "\n",
+    "![](trenzapoligonal.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo de la trenza sobre un segmento\n",
+    "\n",
+    "Mediante cambio de variable, podemos normalizar a $[0,1]$\n",
+    "\n",
+    "![](trenzaentorno.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Aritmética intervalar\n",
+    "\n",
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Definición</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "        Un <i>intervalo complejo</i> es un conjunto $$[r_0,r_1]+[i_0,i_1]\\cdot i:=\\{a+b\\cdot i \\mid a \\in [r_0,r_1], b\\in [i_0,i_1]\\}\\subseteq \\mathbb{C}$$\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Definición</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "        Dado $Y\\subseteq \\mathbb{C}$, su <i>clausura intervalar</i> $[Y]$ es el menor intervalo que lo contiene.\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Se puede operar con intervalos tomando la clausura intervalar de las operaciones punto a punto."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Los intervalos con estas operaciones **no** forman un anillo (pero para ciertos efectos, se puede operar como si lo fuera)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Criterio del intervalo de Newton\n",
+    "\n",
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Teorema</b>\n",
+    "    </h3>\n",
+    "    <p>\n",
+    "        Sea $Y$ un intervalo complejo, $y0\\in Y$. Sea $f: Y\\longrightarrow \\mathbb{C}$ una función holomorfa, tal que $0\\in [f'(Y)]$. Si\n",
+    "        $$\n",
+    "        N(f,y_0,Y):=y_0-\\frac{f(y_0)}{[f'(Y)]}\\subseteq Y\n",
+    "        $$\n",
+    "        entonces $f$ tiene un único cero en $Y$. Además, este cero está en $N(f,y_0,Y)$.\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Podemos usar este teorema para asegurar que tenemos un entorno alrededor de una raíz aproximada, que contiene a la raíz auténtica, pero sin tocar a las demás raíces."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Notar que, ya que estamos operando con intervalos, podemos usar intervalos en los propios coeficientes."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Ejemplo\n",
+    "\n",
+    "$$f = [0.9,1.1]\\cdot y^2-1-x$$\n",
+    "$$y_0=1$$\n",
+    "$$Y=[0.5,1.5]$$\n",
+    "$$x_0=0$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$N(f,y_0,Y)=1-\\frac{[0.9,1.1]-1}{2\\cdot[0.9,1.1][0.5,1.5]}=$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$=1-\\frac{[-0.1,0.1]}{[0.9,3.3]}\\approx [0.888,1.111]$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Ejemplo: extender $x$ a un intervalo\n",
+    "\n",
+    "$$f = [0.9,1.1]\\cdot y^2-1-x$$\n",
+    "$$y_0=1$$\n",
+    "$$Y=[0.5,1.5]$$\n",
+    "$$x=[0,0.25]$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$N(f,y_0,Y)=1-\\frac{[0.9,1.1]-1-[0,0.25]}{2\\cdot[0.9,1.1][0.5,1.5]}=$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$$=1-\\frac{[-0.35,0.1]}{[0.9,3.3]}\\approx [0.888,1.13888]$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo numérico certificado de la trenza\n",
+    "\n",
+    "![](cubos1.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo numérico certificado de la trenza\n",
+    "\n",
+    "![](cubos2.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo numérico certificado de la trenza\n",
+    "\n",
+    "![](cubos3.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo numérico certificado de la trenza\n",
+    "\n",
+    "![](cubos4.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo numérico certificado de la trenza\n",
+    "\n",
+    "![](cubos5.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Cálculo numérico certificado de la trenza\n",
+    "\n",
+    "![](cubos6.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## SIROCCO: demostración"
    ]
   },
   {
@@ -140,7 +857,184 @@
    "execution_count": null,
    "metadata": {},
    "outputs": [],
-   "source": []
+   "source": [
+    "A.<x,y> = AffineSpace(QQ,2)\n",
+    "C = A.curve(y^2-x^3-x^2)\n",
+    "C.plot((x,-3,3),(y,-3,3))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "show(C.fundamental_group())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## SIROCCO: demostración"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "C2 = A.curve((x^2+y^2)^2+18*(x^2+y^2) - 27 -8*(x^3-3*x*y^2))\n",
+    "C2.plot((x,-3,3),(y,-3,3))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "show(C2.fundamental_group())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Diagramas de Rudolf\n",
+    "\n",
+    "Volviendo a las superficies de Riemann:\n",
+    "\n",
+    "![](diagramarudolfhiperbola2.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Diagramas de Rudolf\n",
+    "\n",
+    "Volviendo a las superficies de Riemann:\n",
+    "\n",
+    "![](diagramarudolfhiperbola1.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Diagramas de Rudolf\n",
+    "\n",
+    "<div class=\"thm\">\n",
+    "    <h3>\n",
+    "        <b>Definición</b> \n",
+    "    </h3>\n",
+    "    <p>\n",
+    "        Sea $f\\in\\mathbb{C}[x,y]$, su <i>diagrama de Rudolf</i> es el conjunto\n",
+    "        $$\\{x\\in\\mathbb{C} \\mid \\exists y_0\\neq y_1, Re(y_0)=Re(y_1), f(x,y_0)=f(x,y_1)=0\\}$$\n",
+    "    </p>\n",
+    "</div>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "- Como conjunto de $\\mathbb{R}^2\\cong \\mathbb{C}$, es un conjunto semialgebraico"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "- En general, son difíciles de calcular"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Diagramas de Rudolf\n",
+    "\n",
+    "Diagrama de Rudolf de $(4z^3+12z+3-w^2)^2-8w-48z+52$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "<img src=\"diagramarudolforevkov.svg\" style=\"max-height:450px !important; height:450px;\"/>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Diagramas de Rudolf\n",
+    "\n",
+    "![](laberinto0.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Diagramas de Rudolf\n",
+    "\n",
+    "![](laberinto1.svg)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Muchas gracias"
+   ]
   }
  ],
  "metadata": {