User richard eager - MathOverflow

Cohomology of twisted holomorphic forms on Fano threefolds

2013-05-07T04:41:02Z

Given a Fano threefold $X$, its index $ind(X)$ is the largest integer $r$ such that there exists a divisor $H$ such that $rH \cong -K_X$. Let $\mathcal{L}$ be the associated (ample) line bundle and define the twisted forms $\Omega^q(k) = \Omega^q \otimes \mathcal{L}^k$. When do the twisted cohomology groups $H^p(X, \Omega^q(k))$ vanish? The Kodaira-Nakano vanishing theorem states that the cohomology groups vanish for $p+q > 3.$ For simple examples such as $\mathbb{CP}^3$ and the quadric hypersurface, many more of the twisted cohomology groups vanish. What results are known? Can anything stronger be said if $X$ has a Kahler-Einstein metric?

Added: There are few scattered results in the literature. For example:

The only two threefolds with non-vanishing $H^0(X, \Omega^1_X (1))$ are the Mukai–Umemura threefold $V_{22}$ and $V_{18}$ [math.AG/0310390].

Mock modular forms and (indefinite) quadratic forms

2013-01-30T14:12:03Z

Define the function $$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$ where $c(n,m,l)$ is defined by $$ c(n,m,l) = \begin{cases} (-1)^{s+l} & \mbox{if } 4n - m^2 + l^2 = 2s(s+1) \end{cases} $$ $$ \begin{cases} 0 & otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;
\end{cases} $$ for some integer $s$ and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$ $f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that $$f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$ Is there an elementary proof of the above conjecture? Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?

Update: $f(q,z,y)$ is a product of Jacobi theta functions and $\mu(q;z,y),$ where $\mu(q;z,y)$ is a Lerch sum studied by Zweger in his thesis. Zweger's thesis also relates mock modular forms to indefinite quadratic forms of signature $(1,n),$ so perhaps it isn't too unsurprising that $f(q,z,y)$ takes a "nice" form.

Answer by Richard Eager for A question on the Laurent phenomenon

2013-01-08T15:36:30Z

Many recurrence relations such as the Somos-4 sequence can be embedded into the octahedron recurrence with some periodic identifications. It would be interesting to see if the simultaneous recursions in 109955 can also be embedded into the octahedron recurrence.

Answer by Richard Eager for Good books on problem solving / math olympiad

2012-10-08T15:06:51Z

My favorite olympiad books were

"Winning Solutions" by Edward Lozansky and Cecil Rousseau

"Mathematical Miniatures" by Svetoslav Savchev and Titu Andreescu

"Geometry Unbound" by Kiran Kedlaya [online notes]

"Geometry Revisited" H. S. M. Coxeter and Samuel L. Greitzer

Notes by Po-Shen Loh http://www.math.cmu.edu/~ploh/olympiad.shtml

Answer by Richard Eager for Examples of theorems with proofs that have dramatically improved over time

2012-07-09T04:04:08Z

Witten's proof of the positive energy theorem using spinors drastically simplified the original proof by Schoen and Yau.

Answer by Richard Eager for String theory "computation" for math undergrad audience

2010-11-30T11:50:57Z

Ginsparg's Applied Conformal Field Theory (hep-th/9108028 section 7.6) has a nice proof of the Jacobi Triple-Product formula and Euler's pentagonal number theorem. The equalities can be interpreted as the equivalence between the partition function of a free chiral boson and the partition function of two chiral fermions on a torus. This is an example of bosonization and plays an important role in string theory.

The proof can be explained without any reference to physics, but the crucial difference in statistics (Boson/Fermion) employed in the proof becomes obscured.

Answer by Richard Eager for Tiling A Rectangle With A Hint of Magic

2010-02-05T02:24:36Z

There is an exercise in "Modern Graph Theory" by Bollobas section II.4 pg 63 that is essentially the same argument, but eliminates the trigonometric functions. For each rectangle $U = [x_1, x_2] \times [y_1, y_2]$ let $\psi(U) = (x_2 - x_1) \otimes (y_2 - y_1)$ in $\mathbb{Z}(\mathbb{R}/\mathbb{Z}) \otimes \mathbb{Z}(\mathbb{R}/\mathbb{Z})$ (viewed as a $\mathbb{Z}$ module). Then $\sum_{U} \psi(U) = 0$ so the original rectangle must have an integer side.

[Edited after Reid's comment. Here $\mathbb{Z}(\mathbb{R}/\mathbb{Z})$ is the free $\mathbb{Z}$ module with basis $\mathbb{R}/\mathbb{Z}$]

Software for computing multi-graded Hilbert series

2010-04-03T20:43:32Z

The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights (1,1,-1,-1) is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series

$\frac{1 - abcd}{(1-ac)(1-ad)(1-bc)(1-bd)}$

Is there a software package that can compute multigraded Hilbert series? Can it be computed using Macaulay2?

Alternatively, is there software that can compute the multigraded Hilbert series of a toric variety, specified by its fan?

For this example $v_1 = (0,0,1), v_2 = (1,0,1), v_3 = (1,1,1), v_4 = (0,1,1)$ specify the vertices of the toric fan. The multigraded Hiblert series is given by the index which counts points in the dual cone $S_{C^{*}}$ $\sum_{m \in S_{C^{*}}} q^m = \frac{(1 - q_1)}{ (1 - q_2)(1 - q_3)(1 - q_1 q_2^{-1}) (1 - q_1 q_3^{-1}) }$

References:

"Linear diophantine equations and local cohomology," R.P. Stanley, 1982.

"Combinatorial commutative algebra," E. Miller and B. Sturmfels, 2005.

"Sasaki-Einstein manifolds and volume minimisation," Martelli, Sparks, Yau, 2006.

Answer by Richard Eager for Matrix factorization categories for ADE singularities

2010-04-06T08:01:17Z

See: "Matrix Factorizations and Representations of Quivers II: type ADE case" (math/0511155) by Kajiura, Saito, and Takahashi for a recent account.

Older references include: "Construction geometrique de la correspondance de McKay" Gonzalez-Sprinberg,and Verdier (1983) Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings (1990)

Answer by Richard Eager for Programming Languages based on Category Theory

2009-11-02T01:58:51Z

Haskell is a purely functional language. However side-effects are (almost by definition) difficult to incorporate into a functional language. This is an important problem since I/O is a very important side-effect for most computer programs. Haskell's method of incorporating side-effects is to use monads.

One of the simplest ways to get a monad is from a pair of adjoint functors.

For more on monads see:

(1) Embûches tissues blog: Monads in Mathematics 1: examples

(2) A series of lectures on youtube by TheCatsters:

Pairs of adjoint functors are fairly common and I've found they provide a useful way for seeing part of the "big-picture" in many different branches of mathematics.

Here is one of several introductions to pairs of adjoint functors from the Concrete Nonsense blog.

Answer by Richard Eager for Theories of Noncommutative Geometry

2010-02-07T05:22:23Z

There is a beautiful three-part series of lectures by Jonathan Block that introduces both the Connes and Kontsevich schools of non-commutative geometry:

http://www.math.upenn.edu/~tpantev/rtg09bc/lecnotes/block-ncg.pdf

One of the main motivating examples throughout the lectures is Lusztig's proof proof of the Novikov conjecture.

Quillen's Morphism Inverting Functors

2010-01-28T03:41:40Z

In "Higher algebraic K-theory I" Quillen defines a morphism inverting functor to be a functor from a category C to the category Sets which maps "arrows" in C to isomorphisms in Sets.

Proposition 1: The category of covering spaces of BC is canonically isomorphic to the category of morphism-inverting functors $F: C\rightarrow Sets$.

[For $C$ a small category, its classifying space $BC$ is the geometric realization of its nerve, $NC$]

This proposition plays an essential role in Quillen's Theorem 1 showing that his Q-construction agrees with Grothendieck's construction for $K_0$.

Theorem 1: $\pi_1(B(QC))$ is canonically isomorphic to the Grothendieck group $K_0(M)$

Questions: Have morphism-inverting functors played an important role in other contexts? Is there a more modern incarnation of morphism-inverting functors related to the fundamental groupoid of an infinity-category?

Answer by Richard Eager for Topologically distinct Calabi-Yau threefolds

2010-01-19T07:55:09Z

The smallest known Euler characteristic of a Calabi-Yau threefold is -960. The threefold is a hypersurface in the weighted projective space P_(1,1,12,28,42).

There are some ideas from type IIA-heterotic duality suggesting that this is extremal, but isn't even a physics proof.

Note that 42 occurs as the largest possible denominator in writing 1 as the sum of Egyptian fractions 1 = 1/2+1/3+1/7+1/42.

(1) A. Degeratu, K. Wendland: Friendly giant meets pointlike instantons? On a new conjecture by John McKay http://www.opus-bayern.de/uni-augsburg/volltexte/2007/700/pdf/mpreprint_07_037.pdf

(2) Kachru and Vafa, hep-th/9505105

For 4-folds see: (3) http://arxiv.org/abs/hep-th/9701023v2

Comment by Richard Eager

2013-05-08T00:40:54Z

The Hodge groups are for $k=0$. Can anything be said about $k > 0$? For the quadric, the groups $H^1(X, \Omega_X^2(k))$ vanish for all $k > 1.$

Comment by Richard Eager

2013-01-31T09:50:02Z

@Jeff Harvey - Thanks! Typo fixed and (some) details added.

Comment by Richard Eager

2012-10-19T15:56:32Z

Only a single integral is needed if you use spherical symmetry.