User richard eager - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:51:53Z http://mathoverflow.net/feeds/user/874 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129926/cohomology-of-twisted-holomorphic-forms-on-fano-threefolds Cohomology of twisted holomorphic forms on Fano threefolds Richard Eager 2013-05-07T04:41:02Z 2013-05-08T15:52:53Z <p>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?</p> <p>Added: There are few scattered results in the literature. For example:</p> <p>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].</p> http://mathoverflow.net/questions/120324/mock-modular-forms-and-indefinite-quadratic-forms Mock modular forms and (indefinite) quadratic forms Richard Eager 2013-01-30T14:12:03Z 2013-02-01T15:28:16Z <p>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 $$c(n,m,l) = \begin{cases} (-1)^{s+l} &amp; \mbox{if } 4n - m^2 + l^2 = 2s(s+1) \end{cases}$$ $$\begin{cases} 0 &amp; otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;<br> \end{cases}$$ 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?</p> <p>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.</p> http://mathoverflow.net/questions/118157/a-question-on-the-laurent-phenomenon/118368#118368 Answer by Richard Eager for A question on the Laurent phenomenon Richard Eager 2013-01-08T15:36:30Z 2013-01-08T15:36:30Z <p>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.</p> http://mathoverflow.net/questions/10146/good-books-on-problem-solving-math-olympiad/109154#109154 Answer by Richard Eager for Good books on problem solving / math olympiad Richard Eager 2012-10-08T15:06:51Z 2012-10-08T15:06:51Z <p>My favorite olympiad books were</p> <p>"Winning Solutions" by Edward Lozansky and Cecil Rousseau</p> <p>"Mathematical Miniatures" by Svetoslav Savchev and Titu Andreescu</p> <p>"Geometry Unbound" by Kiran Kedlaya [online notes]</p> <p>"Geometry Revisited" H. S. M. Coxeter and Samuel L. Greitzer</p> <p>Notes by Po-Shen Loh <a href="http://www.math.cmu.edu/~ploh/olympiad.shtml" rel="nofollow">http://www.math.cmu.edu/~ploh/olympiad.shtml</a></p> http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/101736#101736 Answer by Richard Eager for Examples of theorems with proofs that have dramatically improved over time Richard Eager 2012-07-09T04:04:08Z 2012-07-09T04:04:08Z <p>Witten's proof of the positive energy theorem using spinors drastically simplified the original proof by Schoen and Yau.</p> http://mathoverflow.net/questions/47770/string-theory-computation-for-math-undergrad-audience/47786#47786 Answer by Richard Eager for String theory "computation" for math undergrad audience Richard Eager 2010-11-30T11:50:57Z 2010-11-30T11:50:57Z <p>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.</p> <p>The proof can be explained without any reference to physics, but the crucial difference in statistics (Boson/Fermion) employed in the proof becomes obscured.</p> http://mathoverflow.net/questions/14212/tiling-a-rectangle-with-a-hint-of-magic/14215#14215 Answer by Richard Eager for Tiling A Rectangle With A Hint of Magic Richard Eager 2010-02-05T02:24:36Z 2010-06-15T01:41:53Z <p>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.</p> <p>[Edited after Reid's comment. Here $\mathbb{Z}(\mathbb{R}/\mathbb{Z})$ is the free $\mathbb{Z}$ module with basis $\mathbb{R}/\mathbb{Z}$]</p> http://mathoverflow.net/questions/20263/software-for-computing-multi-graded-hilbert-series Software for computing multi-graded Hilbert series Richard Eager 2010-04-03T20:43:32Z 2010-05-02T17:51:15Z <p>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 </p> <p>$\frac{1 - abcd}{(1-ac)(1-ad)(1-bc)(1-bd)}$</p> <p>Is there a software package that can compute multigraded Hilbert series? Can it be computed using Macaulay2?</p> <p>Alternatively, is there software that can compute the multigraded Hilbert series of a toric variety, specified by its fan?</p> <p>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 $S_{C^{*}}$ $\sum_{m \in S_{C^{*}}} q^m = \frac{(1 - q_1)}{ (1 - q_2)(1 - q_3)(1 - q_1 q_2^{-1}) (1 - q_1 q_3^{-1}) }$</p> <p>References:</p> <p>"Linear diophantine equations and local cohomology," R.P. Stanley, 1982.</p> <p>"Combinatorial commutative algebra," E. Miller and B. Sturmfels, 2005.</p> <p>"Sasaki-Einstein manifolds and volume minimisation," Martelli, Sparks, Yau, 2006.</p> http://mathoverflow.net/questions/20478/matrix-factorization-categories-for-ade-singularities/20481#20481 Answer by Richard Eager for Matrix factorization categories for ADE singularities Richard Eager 2010-04-06T08:01:17Z 2010-04-06T08:01:17Z <p>See: "Matrix Factorizations and Representations of Quivers II: type ADE case" (math/0511155) by Kajiura, Saito, and Takahashi for a recent account.</p> <p>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)</p> http://mathoverflow.net/questions/3721/programming-languages-based-on-category-theory/3726#3726 Answer by Richard Eager for Programming Languages based on Category Theory Richard Eager 2009-11-02T01:58:51Z 2010-04-02T10:57:32Z <p>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. </p> <p>One of the simplest ways to get a monad is from a pair of adjoint functors.</p> <p>For more on monads see:</p> <p>(1) Embûches tissues blog: <a href="http://embuchestissues.wordpress.com/2009/02/25/monads-in-mathematics-1-examples/" rel="nofollow">Monads in Mathematics 1: examples</a> </p> <p>(2) A series of <a href="http://www.youtube.com/watch?v=9fohXBj2UEI" rel="nofollow">lectures on youtube</a> by TheCatsters:</p> <p>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.</p> <p>Here is one of several introductions to <a href="http://concretenonsense.wordpress.com/2009/10/02/a-free-association-on-basic-adjoints/#more-713" rel="nofollow">pairs of adjoint functors</a> from the Concrete Nonsense blog.</p> http://mathoverflow.net/questions/10512/theories-of-noncommutative-geometry/14469#14469 Answer by Richard Eager for Theories of Noncommutative Geometry Richard Eager 2010-02-07T05:22:23Z 2010-02-07T05:22:23Z <p>There is a beautiful three-part series of lectures by Jonathan Block that introduces both the Connes and Kontsevich schools of non-commutative geometry:</p> <p><a href="http://www.math.upenn.edu/~tpantev/rtg09bc/lecnotes/block-ncg.pdf" rel="nofollow">http://www.math.upenn.edu/~tpantev/rtg09bc/lecnotes/block-ncg.pdf</a></p> <p>One of the main motivating examples throughout the lectures is Lusztig's proof proof of the Novikov conjecture.</p> http://mathoverflow.net/questions/13220/quillens-morphism-inverting-functors Quillen's Morphism Inverting Functors Richard Eager 2010-01-28T03:41:40Z 2010-01-29T23:12:30Z <p>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.</p> <p>Proposition 1: The category of covering spaces of BC is canonically isomorphic to the category of morphism-inverting functors $F: C\rightarrow Sets$.</p> <p>[For $C$ a small category, its classifying space $BC$ is the geometric realization of its nerve, $NC$]</p> <p>This proposition plays an essential role in Quillen's Theorem 1 showing that his Q-construction agrees with Grothendieck's construction for $K_0$.</p> <p>Theorem 1: $\pi_1(B(QC))$ is canonically isomorphic to the Grothendieck group $K_0(M)$</p> <p>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?</p> http://mathoverflow.net/questions/12284/topologically-distinct-calabi-yau-threefolds/12290#12290 Answer by Richard Eager for Topologically distinct Calabi-Yau threefolds Richard Eager 2010-01-19T07:55:09Z 2010-01-19T07:55:09Z <p>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).</p> <p>There are some ideas from type IIA-heterotic duality suggesting that this is extremal, but isn't even a physics proof.</p> <p>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.</p> <p>(1) A. Degeratu, K. Wendland: Friendly giant meets pointlike instantons? On a new conjecture by John McKay <a href="http://www.opus-bayern.de/uni-augsburg/volltexte/2007/700/pdf/mpreprint_07_037.pdf" rel="nofollow">http://www.opus-bayern.de/uni-augsburg/volltexte/2007/700/pdf/mpreprint_07_037.pdf</a></p> <p>(2) Kachru and Vafa, hep-th/9505105</p> <p>For 4-folds see: (3) <a href="http://arxiv.org/abs/hep-th/9701023v2" rel="nofollow">http://arxiv.org/abs/hep-th/9701023v2</a></p> http://mathoverflow.net/questions/129926/cohomology-of-twisted-holomorphic-forms-on-fano-threefolds/130001#130001 Comment by Richard Eager Richard Eager 2013-05-08T00:40:54Z 2013-05-08T00:40:54Z The Hodge groups are for $k=0$. Can anything be said about $k &gt; 0$? For the quadric, the groups $H^1(X, \Omega_X^2(k))$ vanish for all $k &gt; 1.$ http://mathoverflow.net/questions/120324/mock-modular-forms-and-indefinite-quadratic-forms Comment by Richard Eager Richard Eager 2013-01-31T09:50:02Z 2013-01-31T09:50:02Z @Jeff Harvey - Thanks! Typo fixed and (some) details added. http://mathoverflow.net/questions/26848/looking-for-an-interesting-problem-riddle-involving-triple-integrals/26875#26875 Comment by Richard Eager Richard Eager 2012-10-19T15:56:32Z 2012-10-19T15:56:32Z Only a single integral is needed if you use spherical symmetry.