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# The formula for aluminum sulfate is Al_2(SO_4)_3. What is the molar mass of aluminum sulfate to the nearest gram?

Jun 12, 2017

${\text{342 g mol}}^{- 1}$

#### Explanation:

The idea here is that the molar mass of aluminium sulfate, "Al"_ 2("SO"_4)_color(blue)(3), will be equal to the sum of the molar masses of all the atoms that make up a mole of this compound.

As you can see, $1$ mole of aluminium sulfate contains

• two moles of aluminium, $2 \times \text{Al}$
• three moles of sulfur, $\textcolor{b l u e}{3} \times \text{S}$
• twelve moles of oxygen, $\textcolor{b l u e}{3} \times 4 \times \text{O}$

Now, grab a Periodic Table and look for the molar masses of aluminium, sulfur, and oxygen

• M_ ("M Al") = "26.981538 g mol"^(-1)
• M_ ("M S") = "32.065 g mol"^(-1)
• M_ ("M O") = "15.9994 g mol"^(-1)

In order to find the molar mass of aluminium sulfate, you will need to do

$2 \times {M}_{\text{M Al") + color(blue)(3) xx M_ ("M S") + 12 xx M_ ("M O}}$

You will thus have

$2 \times {\text{26.981538 g mol"^(-1) + color(blue)(3) xx "32.065 g mol"^(-1) + 12 xx "15.9994 g mol}}^{- 1}$

which will get you

M_ ("M Al"_ 2("SO"_ 4)_ 3) = "342.150876 g mol"^(-1)

Rounded to the nearest gram, the answer will be

M_ ("M Al"_ 2("SO"_ 4)_ 3) = color(darkgreen)(ul(color(black)("342 g mol"^(-1))))

format: plain


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CosmoMC: Octave plots from the output files

Use of Healpix, camb, CLASS, cosmomc, compilers, etc.
Rajeev Kumar Jain
Posts: 6
Joined: July 01 2005
Affiliation: Department of Theoretical Physics, University of Geneva

CosmoMC: Octave plots from the output files

Hi,
Can anybody tell me whether I can use OCTAVE to plot the output files that I got by running COSMOMC as I don't have MATLAB.
If yes, what do I need to do?
How do I make 2d and 3d plots using OCTAVE?

Sarah Bridle
Posts: 144
Joined: September 24 2004
Affiliation: University College London (UCL)
Contact:

CosmoMC: Octave plots from the output files

It would be great to have some nice non-matlab tools for analysing sets of MCMC samples.
I don't know octave well enough to help you properly, but I recall from my brief use of it, you should be able to get something scientifically useful from octave for this problem.
Can you read the samples files in?
I don't think you can make 3d plots in octave, but can definitely do 2d.
Can you make the 1d plots fine? (So I can understand what you want help with.)

Joe Zuntz
Posts: 9
Joined: October 22 2004
Affiliation: UCL

CosmoMC: Octave plots from the output files

It would be great to have some nice non-matlab tools for analysing sets of MCMC samples.
I have a version of GetDist that produces simple plotting codes for PyLab, the python plotting code with a similar syntax to matlab. The code and instructions are at http://astro.imperial.ac.uk/~jaz/code

Boud Roukema
Posts: 87
Joined: February 24 2005
Affiliation: Institute of Astronomy, Nicolaus Copernicus University
Contact:

Re: CosmoMC: Octave plots from the output files

Sarah Bridle wrote: I don't think you can make 3d plots in octave, but can definitely do 2d.
Can you make the 1d plots fine? (So I can understand what you want help with.)
Starting from www.octave.org and clicking on "docs" gets quickly to: Three dimensional plotting in octave
octave wrote: 17.3 Three-Dimensional Plotting

The Matlab-style three-dimensional plotting commands are:

— Function File: mesh (x, y, z)

Plot a mesh given matrices x, and y from meshdom and a matrix z corresponding to the x and y coordinates of the mesh. If x and y are vectors, then a typical vertex is (x(j), y(i), z(i,j)). Thus, columns of z correspond to different x values and rows of z correspond to different y values.

— Function File: [xx, yy, zz] = meshgrid (x, y, z)
— Function File: [xx, yy] = meshgrid (x, y)
— Function File: [xx, yy] = meshgrid (x)

Given vectors of x, y, and z coordinates, return three dimensional arrays corresponding to the x, y, and z coordinates of a mesh. Given only x and y, return matrices corresponding to the x and y coordinates of a mesh. The rows of xx are copies of x, and the columns of yy are copies of y. If y is omitted, then it is assumed to be the same as x.

— Function File: meshdom (x, y)

Given vectors of x and y coordinates, return two matrices corresponding to the x and y coordinates of the mesh.

Note: this function is provided for compatibility with older versions of Matlab. You should use meshgrid instead.

So the answer is: yes, you can make 3d plots in octave, using old matlab style commands. However, it is recommended in octave to use meshgrid rather than mesh.

Sarah Bridle
Posts: 144
Joined: September 24 2004
Affiliation: University College London (UCL)
Contact:

CosmoMC: Octave plots from the output files

I guess it depends what is meant by "3d".
Boud, I agree that the meshgrid plots you describe exist in 3 dimensions e.g. could plot x, y and likelihood like this.
In matlab you can also do plots with x,y,z *and* likelihood. Whether you call this 4d or 3d I don't know. I think you can't do this in octave, but don't know if this is a problem for many people.

Boud Roukema
Posts: 87
Joined: February 24 2005
Affiliation: Institute of Astronomy, Nicolaus Copernicus University
Contact:

Re: CosmoMC: Octave plots from the output files

Sarah Bridle wrote:I guess it depends what is meant by "3d".
Boud, I agree that the meshgrid plots you describe exist in 3 dimensions e.g. could plot x, y and likelihood like this.
In matlab you can also do plots with x,y,z *and* likelihood. Whether you call this 4d or 3d I don't know. I think you can't do this in octave, but don't know if this is a problem for many people.
Hi Sarah, could you give the name of the function in matlab and maybe an example plot? i'm not going to buy a copy of matlab, so I can't search for it myself. i'm curious what these plots would look like (projected on a 2D screen, of course ;).

Sarah Bridle
Posts: 144
Joined: September 24 2004
Affiliation: University College London (UCL)
Contact:

CosmoMC: Octave plots from the output files

Sure. e.g. the bottom two plots of
http://www.sarahbridle.net/cosmologui/s ... logui.html
Also there are movies there. But indeed no 3d glasses..
Great if you can find some way of doing this in octave (a colored scatter plot like the 3rd plot from the bottom of the above page would also be almost as useful).

Jochen Weller
Posts: 45
Joined: September 24 2004
Affiliation: Ludwig-Maximilians-University Munich
Contact:

CosmoMC: Octave plots from the output files

Another nice program to use for this would be R

http://www.r-project.org/

Boud Roukema
Posts: 87
Joined: February 24 2005
Affiliation: Institute of Astronomy, Nicolaus Copernicus University
Contact:

Re: CosmoMC: Octave plots from the output files

Sarah Bridle wrote:Sure. e.g. the bottom two plots of
http://www.sarahbridle.net/cosmologui/s ... logui.html
OK, in that case I would call that a 3D plot with one 2D contour, though it does represent 4D information in the same sense that a traditional 2D plot with 1D contours represents 3D info.

I didn't find any obvious general octave routine for plotting one or more 2D contours in a 3D plot - probably time to write one and contribute it back to the community under the GPL so that nobody can take it and lock it up. :)

BTW, the package name for anybody wishing to install R in debian is:

Code: Select all

r-base
. Try

Code: Select all

aptitude install r-base
. :)

Hopefully sometime i'll have the time to start learning it...


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Browse Questions

# The charge deposited on $4 \mu F$ capacitor the circuit is

$(C) 24 \times 10^{–6} C$


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# Computing unkown entries of a matrix given a desired eigenvalue

Let $$M$$ be an irreducible non-negative square matrix with spectral radius $$\lambda$$. Due to the Perron-Frobenius theorem, we know that $$\lambda > 0$$ and $$\lambda$$ is an eigenvalue of $$M$$. Also, all other eigenvalues of $$M$$ are strictly smaller than $$\lambda$$.

Now consider such a matrix with an unknown entry, e.g. $$M = \begin{pmatrix} 0 & 0 & 40 & 60 & 80\\ p & 0 & 0 & 0 & 0\\ 0 & 0.35 & 0 & 0 & 0\\ 0 & 0 & 0.16 & 0 & 0\\ 0 & 0 & 0 & 0.08 & 0 \end{pmatrix}.$$ Question: How can I determine $$p$$ such that $$\lambda = 1$$ (in general)? I'm not sure how simple finding an exact solution is, I would already be happy with a numerical solution. I thought about guessing some values for $$p$$ such that $$\lambda \approx 1$$ and using interpolation, but am not sure which interpolation would be appropriate, since I don't know how $$p$$ and $$\lambda$$ should relate to each other.

Motivation: Leslie matrices are often used to study changes of a population. The entries must be estimated and the case $$\lambda = 1$$ is the equilibrium state of the population.

It's easy to determine all values of $$p$$ for which $$M$$ has any fixed eigenvalue $$\lambda$$: simply compute the determinant of $$M-\lambda I$$, which is a linear function of $$p$$, and find the value of $$p$$ that makes this determinant equal to $$0$$. In particular, the unique value of $$p$$ for which $$1$$ is an eigenvalue of $$M$$ is $$p=625/11074 \approx 0.0564385$$, and one can then check by computer that $$1$$ is indeed the eigenvalue of largest modulus.


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# Tag Info

Let $X$ be the algebraic variety over $\mathbb{R}$ obtained from the projective plain $\mathbb{P}^2$ by blowing up one point $P \in \mathbb{P}^2(\mathbb{R})$. Since $X$ is a smooth projective algebraic variety the space of complex points $X(\mathbb{C})$ admits a Kahler structure and hence the underlying $4$-dimensional real manifold admits a symplectic ...


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# 2D slice of a 6D Calabi-Yau manifold , and other?

1. Apr 13, 2014

### Spinnor

"2D slice of a 6D Calabi-Yau manifold", and other?

Mathematically what does it mean to take a "2D slice of a 6D Calabi-Yau manifold"?

Part of quote taken from the top of,

http://en.wikipedia.org/wiki/Calabi–Yau_manifold

Is there a finite number of slices of a 6D Calabi-Yau manifold that in principle could define that Calabi-Yau manifold?

How are 6D Calabi-Yau manifolds classified?

Does it make sense to take some 6D Calabi-Yau manifold and distort its shape a little and still keep it the "same" topologically?

Can we distort one 6D Calabi-Yau manifold into another?

Does it make sense to say that there is a minimum number of "flat" complex dimensions a typical 6D Calabi-Yau manifold can be embedded?

Is there a 6D Calabi-Yau manifold that could be considered most simple?

What cool things "happen" when we allow for complex manifolds that don't happen for real manifolds?

In string theory do the vibrating strings interact with the Calabi-Yau manifold to change the Calabi-Yau manifold even if a little bit?

If some questions don't make sense, sorry, hopefully they can be modified so that they do make sense.

Thanks for any help!

Last edited: Apr 13, 2014
2. Apr 13, 2014

### Staff: Mentor

3. Apr 13, 2014

### Ben Niehoff

Most of your questions are very vague and open-ended, and you should do some reading. However, this one is simple to answer:

No. In real geometry it is easy to show that every real manifold is isometrically embeddable in some $\mathbb{R}^n$. However, the analogous situation with $\mathbb{C}^n$ is not true in complex geometry. The catch is that complex embeddings have to be holomorphic, and this severely restricts the kinds of embeddings you can have.

It is true, however, that every compact complex manifold can be embedded in some $\mathbb{CP}^n$ (i.e. complex projective space).

4. Apr 13, 2014

### Spinnor

That is neat, thanks! Do we know what the smallest n is in CP^n that would handle all 6D Calabi-Yau manifolds?

Last edited: Apr 13, 2014
5. Apr 13, 2014

### Spinnor

Nice demo! Amazon lets me take a pretty good look at the book,

https://www.amazon.com/String-Theor...958&sr=1-1&keywords=string+theory+for+dummies

Thanks!

Last edited by a moderator: May 6, 2017
6. Apr 14, 2014

### Therodre

Actually no, to be projective you need at least to be Kähler, and some compact complex surfaces are not even Kähler, like Hopf surface.
There are cohomological obstructions to being projective, for instance, all your H^p for p odd must be of even rank, this is not the case for Hopf surfaces, whose H^1 is easily seen to be Z.

7. Apr 15, 2014

### Ben Niehoff

You're right. I was thinking Calabi-Yau and I wrote "complex".

8. Apr 21, 2014

### mathwonk

although there exist non projective complex manifolds, if a compact complex manifold of dimension n is projective, then it embeds holomorphically in CP^(2n+1), by the usual method of projecting it down from wherever you were able to embed it. I.e.a projection map from a point off the manifold is always holomorphic, and is an embedding if it does not lie on a secant of the embedded manifold.

Since the bundle of secants has dimension 2n+1, it does not fill up CP^k when k > 2n+1, so you can always choose a point to project from to a lower dimension. In particular all complex 3 dimensional projective Calabi Yau's embed in CP^7. So the quintic threefold in CP^4 is rather special, and in some ways is one of the most simple, or at least one of the most familiar such animals. The other examples given in the wikipedia article of degree k+2 in CP^(k+1), seem also rather special, being hyper surfaces.

I cannot find many reference to 2 dimensional slices of a 6d Calabi Yau in your link. Complkex manifolds do often vary in families, in which the complex structure but not the topology varies. In fact in any smooth family of complex compact manifolds the topology is constant, so the question of "deforming" one manifold into another is one of determining how many different families of such manifolds exist, and whether the different families are somehow connected, all deep questions.

As an example, all quoin tic three folds in CP^4 form one family, obtained by varying the quintic polynomial, so all of these are deformations of one another and all are thus homeomorphic. (In fact diffeomorphic, cf "theorem of Ehresmann".)

Taking a complex 1 diml slice of an embedded cx 3diml manifold is done by intersecting it with a general linear space of complex codimension 2.

Last edited: Apr 21, 2014


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Do not in general produce triangular matrices are precisely those that stabilize the standard flag ( a ) takes than... Sections 2.4.4 and A.1 numerical analysis and lower triangular if all the entries above the main diagonal zero.


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## stanford algebraic geometry seminar 2021

The Stanford Algebraic Geometry Seminar meets online, usually Fridays at noon pacific time.

Register in advance for the seminar: https://stanford.zoom.us/meeting/register/tJEvcOuprz8vHtbL2_TTgZzr-_UhGvnr1EGv

Then attend the seminar:  https://stanford.zoom.us/w/95272114542 (pwd 362880)

## stanford algebraic geometry seminar 2020

Click on the title to see the abstract (when available).

Posted in seminars | Comments Off on stanford algebraic geometry seminar 2020

## abstracts for 2019-20 seminars

(The seminar webpage is here.)

Isabel Vogt (Stanford)

September 27, 2019

Stability of normal bundles of space curves

The normal bundle controls the deformation theory of a curve embedded in projective space. In this talk we study the stability of normal bundles of curves in P^3 using degeneration. This is joint work with Izzet Coskun and Eric Larson.

Jordan Ellenberg (Wisconsin)

October 4, 2019

Heights of rational points on (some) stacks

The study of rational points on algebraic stacks over global fields is in many respects very similar to the familiar world of Diophantine geometry of schemes. But one key element of that world is missing; a theory of heights. I will propose such a theory and explain how it recovers many already-in-use notions of complexity for points on stacks, while also generating new ones. I talked about this project in the Stanford number theory seminar in spring 2018; I will explain the basic ideas again and then talk about some examples we understand better now than we did then, such as points on the moduli stack of abelian varieties, and discuss some questions that remain open for future work.

Stefan Schreieder (Munich)

October 11, 2019

Stably irrational hypersurfaces of small slopes

We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension n>2 and degree at least log_2(n)+2 is not stably rational. This improves earlier results of Koll\’ar and Totaro, who proved the same result under a linear bound on the degree.

Jim Bryan (University of British Columbia)

October 25, 2019, 2:30 pm

K3 surfaces with symplectic group actions, enumerative geometry, and modular forms

The Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the “G-fixed Hilbert schemes” parametrizing G-invariant collections of points on X are related to modular forms of level |G| and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as chi_y genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.

Nathan Pflueger (Amherst)

October 25, 2019, 4 pm

Relative Richardson Varieties

A Richardson variety is an intersection of two Schubert varieties defined by transverse flags in a vector space. Richardson varieties have many nice geometric properties; for example, a theorem of Knutson, Woo, and Yong shows that their singularities are completely determined by those of Schubert varieties. I will discuss a generalization of this theorem to a relative context, where the two transverse flags are replaced by a moving pair of flags in a vector bundle that become non-transverse at some points. I will also discuss a theorem describing the cohomology of the resulting relative Richardson variety. I will describe an application to Brill-Noether theory, and some related conjectures. This is joint work with Melody Chan.

November 8, 2019

The intersection of the Torelli locus with the non-ordinary locus in PEL-type Shimura varieties

Around 1980, mathematicians developed several techniques to study the Newton polygon stratification of the moduli space of principally polarized abelian varieties in positive characteristic p.  In 2004, Faber and Van der Geer used these techniques to prove that the Torelli locus of Jacobians of smooth curves intersects every p-rank stratum.  In 2013, Viehmann and Wedhorn proved that every Newton polygon satisfying the Kottwitz conditions occurs on Shimura varieties of PEL-type.  In most cases, it is still not known whether the Torelli locus intersects these Newton polygon strata.  We provide a positive answer for the mu-ordinary and non-mu ordinary strata in infinitely many cases.  As an application, we produce infinitely many new examples of unusual Newton polygons which occur for Jacobians of smooth curves.  This is joint work with Li, Mantovan, and Tang.

Eric Riedl (Notre Dame)

November 15, 2019

Linear subvarieties of hypersurfaces and unirationality

The de Jong-Debarre Conjecture predicts that the space of lines on any smooth hypersurface of degree d <= n in P^n has dimension 2n-d-3. We prove this conjecture for n > 2d, improving on the previously-known exponential bounds. We prove an analogous result for k-planes, and use this generalization to prove that an arbitrary smooth hypersurface is unirational if n > 2^{d!}. This is joint work with Roya Beheshti.

November 22, 2019

Metric Algebraic Geometry

Metric algebraic geometry is a term proposed for the study of properties of real algebraic varieties that depend on a distance metric. The distance metric can be the Euclidean metric in the ambient space or a metric intrinsic to the variety. In this talk, we introduce metric algebraic geometry through discussion of Voronoi cells, bottlenecks, offset hypersurfaces, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.

Carl Lian (Columbia)

January 10, 2020

Enumerating pencils with moving ramification on curves

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.

Geoffrey Smith (Harvard)

January 24, 2020, 2:30pm

Covering gonalities in positive characteristic

The covering gonality of an irreducible projective variety over the complex numbers is the minimum gonality of a curve through a general point on the variety. This definition has two reasonable generalizations to positive characteristic, the covering gonality and the separable covering gonality. Of the two, separable covering gonalities are much easier to bound, and I’ll give an easy lower bound for smooth hypersurfaces essentially due to Bastianelli-de Poi-Ein-Lazarsfeld-Ullery. I’ll then give an analogous bound for the covering gonality of very general hypersurfaces, using a Chow-theoretic argument that extends work of Riedl-Woolf.

Jackson Morrow (Emory)

January 24, 2020, 4pm

Non-Archimedean entire curves in varieties

The classical conjectures of Green—Griffiths—Lang—Vojta predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, and groupless. In his thesis, Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the “correct” version, as it does not mirror complex hyperbolicity. In recent work, Javanpeykar and Vezzani introduced a new non-Archimedean notion of hyperbolicity, which fixed this issue and also stated a non-Archimedean version of the Green-Griffiths-Lang-Vojta conjecture.
In this talk, I will discuss algebraic, complex, and non-Archimedean notions of hyperbolicity and a proof of the non-Archimedean Green—Griffiths—Lang–Vojta conjecture for closed subvarieties of semi-abelian varieties and projective surfaces admitting a dominant morphism to an elliptic curve.

Izzet Coskun (UIC)

January 31, 2020, 4pm

The stabilization of the cohomology of moduli spaces of sheaves on surfaces

The Betti numbers of the Hilbert scheme of points on a smooth, irreducible projective surface have been computed by Gottsche. These numbers stabilize as the number of points tends to infinity. In contrast, the Betti numbers of moduli spaces of semistable sheaves on a surface are not known in general. In joint work with Matthew Woolf, we conjecture these also stabilize and that the stable numbers do not depend on the rank. We verify the conjecture for large classes of surfaces. I will discuss our conjecture and provide the evidence for it.

Jack Huizenga (Penn State)

February 7, 2020, 4pm

Moduli of sheaves on Hirzebruch surfaces

Let X be a Hirzebruch surface.  Moduli spaces of semistable sheaves on X with fixed numerical invariants are always irreducible by a theorem of Walter.  On the other hand, many other basic properties of sheaves on Hirzebruch surfaces are unknown.  I will discuss two different problems on this topic.  First, what is the cohomology of a general sheaf on X with fixed numerical invariants?  Second, when is the moduli space
actually nonempty? The latter question should have an answer reminiscient of the Drezet-Le Potier classification of semistable sheaves on the projective plane; in particular, there is a fractal-like hypersurface in the space of numerical invariants which bounds the invariants of semistable sheaves.  This is joint work with Izzet Coskun.

Sarah Frei (Rice)

February 14, 2020, 4pm

Derived equivalence and rational points

It is natural to ask which properties of a smooth projective variety are recovered by its derived category. In this talk, I will consider the question: is the existence of a rational point preserved under derived equivalence? In recent joint work with Nicolas Addington, Ben Antieau, and Katrina Honigs, we show that over Q, the answer is no. We give two counterexamples: an abelian variety and a torsor over it, and a pair of hyperkaehler fourfolds.

Alex Perry (IAS)

February 21, 2020, 2:30pm

The integral Hodge conjecture for 2-dimensional Calabi-Yau categories

I will formulate a version of the integral Hodge conjecture for categories, discuss its proof for categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and explain how this implies new cases of the usual integral Hodge conjecture for varieties.

Daniel Litt (UGA)

February 21, 2020, 4pm

The section conjecture at the boundary of moduli space

Grothendieck’s section conjecture predicts that over arithmetically interesting fields (e.g. number fields), rational points on a smooth projective curve X of genus at least two can be detected via the arithmetic of the etale fundamental group of X. We construct infinitely many curves of each genus satisfying the section conjecture in interesting ways, building on work of Stix, Harari, and Szamuely. The main input to our result is an analysis of the degeneration of certain torsion cohomology classes on the moduli space of curves at various boundary components. This is (preliminary) joint work with Padmavathi Srinivasan, Wanlin Li, and Nick Salter.

Alex Smith (Harvard)

February 28, 2020, 2:30pm

8-class ranks of imaginary quadratic fields and 4-Selmer groups of elliptic curves

We prove that the two-primary subgroups of the class groups of imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra-Gerth heuristic. In this talk, we will detail our method for proving the 8-class rank portion of this theorem and will compare our approach to one that uses the governing fields predicted by Cohn and Lagraias. We will also connect this work to related questions on the 4-Selmer groups of elliptic curves in quadratic twist families.

Anand Patel (Oklahoma State)

February 28, 2020, 4pm

Projection and Ramification

When a projective variety is linearly projected onto a projective space of the same dimension, a ramification divisor appears. In joint work with Anand Deopurkar and Eduard Duryev, we study basic questions about the map which sends a projection to its ramification divisor. I will present proven results, open problems, and if time permits, some curious numerology.

Dinesh Thakur (University of Rochester)

March 13, 2020, 4pm

Multizeta and motives in function fields

The study of multizeta values in intimately connected with current developments in non-abelian and homotopical directions in number theory.  We explain some background with multizeta values and their interpretation as periods in terms of higher-dimensional generalizations of Drinfeld modules (called mixed t-motives), as well as conjectures on relations and non-relations among various multizeta values.  We also discuss the implications for huge Galois representations and  lifting of multizeta values to functions on (pro)algebraic groups.  Finally, we describe new results for higher-genus curves as well as several open questions.

## stanford algebraic geometry seminar 2019-20

Fridays 4-5:30 pm in 383-N (with exceptions)

DateSpeakerTitle
September 27 12:30pm in 384-IIsabel Vogt (Stanford)Stability of normal bundles of space curves
October 4Jordan Ellenberg (Wisconsin)Heights of rational points on (some) stacks
October 11Stefan Schreieder (Munich)Stably irrational hypersurfaces of small slopes
October 18no seminar
October 25 2:30pm in 384-IJim Bryan (UBC)K3 surfaces with symplectic group actions, enumerative geometry, and modular forms
October 25 4pm in 383-NNathan Pflueger (Amherst)Relative Richardson Varieties
November 1no seminar (WAGS at Utah)
November 8Rachel Pries (Colorado State)The intersection of the Torelli locus with the non-ordinary locus in PEL-type Shimura varieties
November 15Eric Riedl (Notre Dame)Linear subvarieties of hypersurfaces and unirationality
November 22Maddie Weinstein (Berkeley)Metric Algebraic Geometry
November 29no seminar (thanksgiving break)
December 6maybe no seminar
January 10Carl Lian (Columbia)Enumerating pencils with moving ramification on curves
January 17no seminar (JMM)
January 24 2:30pm in 383-NGeoff Smith (Harvard)Covering gonalities in positive characteristic
January 24 4pm in 383-NJackson Morrow (Emory)Non-Archimedean entire curves in varieties
January 31Izzet Coskun (UIC)The stabilization of the cohomology of moduli spaces of sheaves on surfaces
February 7Jack Huizenga (Penn State)Moduli of sheaves on Hirzebruch surfaces
February 14Sarah Frei (Rice)Derived equivalence and rational points
February 21 2:30pm in 383-NAlex Perry (IAS)The integral Hodge conjecture for 2-dimensional Calabi-Yau categories
February 21 4pm in 383-NDaniel Litt (UGA)The section conjecture at the boundary of moduli space
February 28 2:30pm in 383-NAlex Smith (Harvard University)8-class ranks of imaginary quadratic fields and 4-Selmer groups of elliptic curves
February 28 4pm in 383-NAnand Patel (Oklahoma State)Projection and Ramification
May 1Rohini Ramadas (Brown) and Rob Silversmith (Northeastern)

## abstracts for 2018-19 seminars

(The seminar webpage is here.)

June Huh (Institute for Advanced Study)

October 26, 2018

Distinguished Lecture 2 of 3: Kazhdan-Lusztig theory for matroids

There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration), based on their combinatorial cohomology theories. After giving an overview of the similarity, I will report on a cohomological approach to some conjectures in enumerative combinatorics. Joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang.

David Jensen (Kentucky)

November 2, 2018, 3:15-4:15 pm

Linear Systems on General Curves of Fixed Gonality

The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the Brill-Noether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Sam Payne (UT Austin and MSRI)

November 2, 2018, 4:30-5:30 pm

Tropical methods for the Strong Maximal Rank Conjecture

I will present joint work with Dave Jensen using tropical methods on a chain of loops to prove new cases of the Strong Maximal Rank Conjecture of Aprodu and Farkas that are relevant to computing the Kodaira dimensions of the moduli spaces M_22 and M_23.  As time permits, I will also discuss relations to an analogous approach via limit linear series on chains of genus 1 curves, developed in the work of Liu, Osserman, Teixidor, and Zhang.

Margaret Bilu (Courant)

April 19, 2019, 2:30-3:30 pm

Motivic Euler products and motivic height zeta functions

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin’s conjecture and on its motivic analog: the latter predicts the behavior of moduli spaces of curves of large degree on some algebraic varieties. It may be formulated in terms of the generating series of the classes of these moduli spaces in the Grothendieck ring, called the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition and how this can be used to give a proof of the motivic version of Manin’s conjecture for equivariant compactifications of vector groups.

Ronno Das (University of Chicago)

April 19, 2019, 4-5 pm

Points and lines on cubic surfaces

The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a ‘universal family’ of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I’ll also explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the average smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

Chenyang Xu

Thursday, May 23, 2019

Department Colloquium: Compact moduli spaces of varieties

Moduli space is ubiquitous in modern algebraic geometry. In this talk, I will discuss the recent progress on the construction of compact moduli spaces parametrising varieties whose Chern classes are positive. Ideas from differential geometry play an essential role.

## stanford algebraic geometry seminar 2018-19

Fridays 4-5 pm in 383-N (with exceptions)

 October 26 Distinguished Lecture 2 of 2 June Huh (Institute for Advanced Study) Kazhdan-Lusztig theory for matroids November 2, 3:15-4:15 pm David Jensen (Kentucky) Linear Systems on General Curves of Fixed Gonality November 2, 4:30-5:30 pm Sam Payne (UT Austin and MSRI) Tropical methods for the Strong Maximal Rank Conjecture November 30 Jack Hall (Arizona) GAGA theorems January 25 Arnav Tripathy (Harvard) A geometric model for complex analytic equivariant elliptic cohomology February 1 Eugene Gorsky (UC Davis) Soergel bimodules and Hilbert schemes February 15 Padmavathi Srinivasan (Georgia Tech) An arithmetic count of lines meeting four lines in P^3 Tuesday March 12 Colloquium Christopher Hacon (Utah) On the classification of algebraic varieties March 15 Dan Bragg (UC Berkeley and MSRI) Supersingular twistor spaces April 12, 2:30-3:30 pm, 384-I Akhil Mathew (Chicago and MSRI) A gentle approach to the de Rham-Witt complex April 12, 4-5 pm, 383-N Laure Flapan (Northeastern and MSRI) Chow motives, L-functions, and powers of algebraic Hecke characters April 13-14, Berkeley Western Algebraic Geometry Symposium (WAGS) April 19, 2:30-3:30 pm Margaret Bilu (NYU) Motivic Euler products and motivic height zeta functions April 19, 4-5 pm Ronno Das (Chicago) Points and lines on cubic surfaces Thursday April 25 Colloquium Xinwen Zhu (Caltech) April 26 Kristin DeVleming (UCSD and MSRI) Comparing compactifications of the moduli space of plane curves May 3 Juliette Bruce (UW Madison) Semi-Ample Asymptotic Syzygies May 10 Dan Erman (UW Madison) Limits of polynomials rings Thursday May 23 Colloquium Chenyang Xu (MIT) Compact moduli spaces of varieties May 24, 2:30-3:30 pm Anne-Sophie Kaloghiros (MSRI and Brunel University) Threefold Calabi-Yau pairs May 24, 4:00-5:00 pm Milena Hering (MSRI and Edinburgh) Stability of Toric Tangent bundles June 7 Sam Payne (UT Austin) Top weight cohomology of M_{g,n}

## abstracts for 2017-18 seminars

(The seminar webpage is here.)

Sam Grushevsky (Stony Brook University)

September 29, 2017

Geometry of compactified moduli of cubic threefolds

Starting from considering the GIT compactification of the moduli of cubic threefolds, the “wonderful” compactification, which is smooth with normal crossing boundary, is constructed by an explicit sequence of blowups. We show that there exists a family of intermediate jacobians over the wonderful compactification. We compute the cohomology of the wonderful compactification by comparing it to the symplectic resolution. Based on joint works with Casalaina-Martin, Hulek, Laza

Felix Janda (University of Michigan)

October 6, 2017

Genus two curves on quintic threefolds

Virtual (Gromov-Witten) counts of maps from algebraic curves to quintic 3-folds in projective space have been of significant interest for mathematicians and physicists since the early 90s. While there are (very inefficient) algorithms for computing any specific Gromov-Witten invariant, explicit formulae are only known in genus zero and one. On the other hand, physicists have explicit conjectural formulas up to genus 51.

I will discuss a new approach to the Gromov-Witten theory of the quintic (using logarithmic geometry) which yields an explicit formula in genus two that agrees with the physicists’ conjecture.

This is based on joint works in progress with Q. Chen, S. Guo and Y. Ruan.

Remy van Dobben de Bruyn (Columbia)

October 20, 2017

Dominating varieties by liftable ones

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.

Jason Lo (Cal State Northridge)

October 27, 2017

The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations

Slope stability is a type of stability for coherent sheaves on smooth projective varieties. On a variety where the derived category of coherent sheaves admits a non-trivial autoequivalence, it is natural to ask how slope stability `transforms’ to a different stability under the autoequivalence. This question also has implications for understanding the symmetries within various counting invariants. In this talk, we will give an answer to the above question for elliptic surfaces and threefolds under a Fourier-Mukai transform.

Daniel Litt (Columbia University)

December 1, 2017

Galois actions on fundamental groups

Let X be a variety over a field k, and let x be a k-rational point of X. Then the absolute Galois group of k acts on the etale fundamental group of X. If k is an arithmetically interesting field (i.e. a number field, a p-adic field, or a finite field), then this action reveals a great deal about the geometry of X; if X is a variety with an interesting fundamental group, this action reveals a great deal about the arithmetic of k.

This talk will discuss (1) joint work with Alexander Betts about the structure of Galois actions on fundamental groups, (2) how to describe invariants of these actions in terms of more geometric invariants of X, and (3) applications of this work to classical algebraic geometry, and, if time permits, arithmetic.

Pablo Solis (Stanford)

January 19, 2018

Hunting Vector Bundles on $\mathbf{P}^1 \times \mathbf{P}^1$

Motivated by Boij-Soderberg theory, Eisenbud and Schreyer conjectured there should be vector bundles on $\mathbf{P}^1 \times \mathbf{P}^1$ with natural cohomology and prescribed Euler characteristic. I’ll give some background on Boij-Soderberg theory, explain what natural cohomology means and prove the conjecture in “most” cases.

Izzet Coskun (University of Illinois at Chicago)

January 26, 2018

The geometry of moduli spaces of sheaves on surfaces

In this talk, I will discuss recent results concerning the Brill-Noether Theory of higher rank bundles on rational surfaces and stable cohomology of moduli spaces of sheaves. In joint work with Jack Huizenga, we characterize when the cohomology of a general stable sheaf on a Hirzebruch surface is determined by its Euler characteristic. We use these results to classify moduli spaces where the general bundle is globally generated. If time permits, I will discuss joint work with Matthew Woolf on the stable cohomology of moduli spaces on rational surfaces.

Katrina Honigs (Utah)

February 2, 2018

Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic

There are many results characterizing when derived categories of two complex surfaces are equivalent, including theorems of Bridgeland and Maciocia showing that derived equivalent Enriques or bielliptic surfaces must be isomorphic. The proofs of these theorems strongly use Torelli theorems and lattice-theoretic methods which are not available in positive characteristic. In this talk I will discuss how to prove these results over algebraically closed fields of positive characteristic (excluding some low characteristic cases). This work is joint with M. Lieblich and S. Tirabassi.

Junliang Shen (ETH)

February 9, 2018

K3 categories, cubic 4-folds, and the Beauville-Voisin conjecture

We discuss recent progress on the connection between 0-cycles of holomorphic symplectic varieties and structures of K3 categories. We propose that there exists a sheaf/cycle correspondence for any K3 category, which controls the geometry of algebraically coisotropic subvarieties of certain holomorphic symplectic varieties. Two concrete cases will be illustrated in details:
(1) the derived category of a K3 surface (joint with Qizheng Yin and Xiaolei Zhao),
(2) Kuznetsov category of a cubic 4-fold (joint with Qizheng Yin).
If time permits, we will also discuss the connection to rational curves in cubic 4-folds.

Michael Viscardi (Berkeley)

February 16, 2018

Quantum cohomology and 3D mirror symmetry

Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a semisimple group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These symplectic resolutions are expected to be symplectic duals of Nakajima quiver varieties, so that our result is an analogue of (part of) the work of Maulik and Okounkov in the symplectic dual setting.

Roya Beheshti (Washington U. St. Louis)

February 23, 2018

Moduli spaces of rational curves on hypersurfaces

I will talk about the geometry of moduli spaces of rational curves on Fano hypersurfaces and discuss some results concerning their dimension and birational geometry.

Yongbin Ruan (Michigan)

March 9, 2018

The structure of higher genus Gromov-Witten invariants of quintic 3-fold

The computation of higher genus Gromov-Witten invariants of quintic 3–fold (or compact Calabi-Yau manifold in general) has been a focal point of research of geometric and physics for more than twenty years. A series of deep conjectures have been proposed via mirror symmetry for the specific solutions as well as structures of its generating functions. Building on our initial success for a proof of genus two conjecture formula of BCOV, we present a proof of two conjectures regarding the structure of the theory. The first one is Yamaguchi-Yau’s conjecture that its generating function is a polynomial of five generators and the other one is the famous holomorphic anomaly equation which governs the dependence on four out of five generators. This is a joint work with Shuai Guo and Felix Janda.

Andrei Calderaru (Wisconsin)

April 13, 2018

Computing a categorical Gromov-Witten invariant

In his 2005 paper “The Gromov-Witten potential associated to a TCFT” Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.

In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.

Daniel Halpern-Leistner (Cornell)

April 20, 2018

What is wall-crossing?

I will discuss recent progress in understanding geometric invariant theory from an “intrinsic” perspective. This leads to a conceptually clean meta-principle for how to study the birational geometry of moduli spaces as well as a universal wall-crossing formula for the integrals of tautological K-theory classes on these moduli spaces. I will apply this perspective to the example of Bridgeland semistable complexes on an algebraic surface. The result is a relatively straightforward construction of K-theoretic Donaldson invariants, along with wall-crossing formulas for these invariants which are new-ish (conjecturally equivalent to Mochizuki’s cohomological wall crossing formulas in the context where both are defined).

Sean Howe (Stanford)

April 27, 2018

Motivic random variables and random matrices

As first shown by Katz-Sarnak, the zero spacing of L-functions of smooth plane curves over finite fields approximate the infinite random matrix statistics observed experimentally for the zero spacing of the Riemann-Zeta function (arbitrarily well by first taking the size of the finite field to infinity, and then the degree of the curve to infinity). The key geometric inputs are a computation of the image of the monodromy representation and Deligne’s purity theorem, which ensures that only the zeroth cohomology group of irreducible local systems will contribute asymptotically to the statistics. In this talk, we explain how higher order terms (i.e. the lower weight part of cohomology) can be computed starting from a simple heuristic for the number of points on a random smooth plane curve.

Isabel Vogt (MIT)

May 4, 2018, 3-4 pm

Interpolation problems for curves in projective space

In this talk we will discuss the following question: When does there exist a curve of degree $d$ and genus $g$ passing through $n$ general points in $\mathbb{P}^r$?

Eric Larson (MIT)

May 4, 2018, 4:30-5:30 pm

The Maximal Rank Conjecture

We find the Hilbert function of a general curve of genus $g$, embedded in $\mathbb{P}^r$ via a general linear series of degree $d$. Note that Isabel Vogt’s talk earlier this afternoon is a pre-requisite for this talk.

Dori Bejleri (Brown)

May 18, 2018, 3-4 pm

Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations

A degree one del Pezzo surface is the blowup of $\mathbb{P}^2$ at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we can construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs $(X,D)$ is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of a space of weighted stable elliptic surface pairs and explain how it relates to some previous compactifications of the space of degree one del Pezzo surfaces. This is joint work with Kenny Ascher.

Francois Greer (Stony Brook)

May 18, 2018, 4:30-5:30 pm

Elliptic fibrations in the presence of singularities

The Gromov-Witten generating series for an elliptic fibration is expected to have modular properties by mirror symmetry. When the homology class in the base is irreducible and the total space is smooth, we obtain a classical modular form for the full modular group. If the base class is reducible, we expect the series to be quasi-modular. If the fibration does not admit a section, then the modular form has higher level. Both of these relaxations are related to the presence of singularities in the geometry.

Soren Galatius (Stanford)

May 25, 2018, 3-4 pm, Room TBA

$M_g$, $M_g^{trop}$, GRT, and Kontsevich’s complex of graphs

I will report on recent joint work with Melody Chan and Sam Payne on the cohomology of $M_g$ in degree $4g-6$. It is known that the rational cohomology vanishes above this degree. We prove that the rational cohomology in this degree is non-trivial for all $g \geq 7$ and that its dimension grows faster than $1.324^g +$ constant, making it asymptotically larger than the entire tautological ring and disproving a recent conjecture of Church-Farb-Putman and an older conjecture of Kontsevich. Our proof relates the weight filtration on compactly supported cohomology of $M_g$ with the moduli space of tropical curves and with the cohomology of Kontsevich’s graph complex. We then use a theorem of Willwacher to construct an injection of the Grothendieck-Teichmüller Lie algebra into $H^{2g}_c(M_g)$.

Tathagata Basak (Iowa State University)

May 25, 2018, 4:30-5:30 pm

A complex ball quotient and the monster

We shall talk about an arithmetic lattice M in $PU(13,1)$ acting on the the unit ball B in thirteen dimensional complex vector space. Let X be the space obtained by removing the hypersurfaces in B that have nontrivial stabilizer in M and then quotienting the rest by M. The fundamental group G of the ball quotient X is a complex hyperbolic analog of the braid group. We shall state a conjecture that relates this fundamental group G and the monster simple group and describe our results (joint with D. Allcock) towards this conjecture.

The discrete group M is related to the Leech lattice and has generators and relations analogous to Weyl groups. Time permitting, we shall give a second example in $PU(9,1)$ related to the Barnes-Wall lattice for which there is a similar story.

## stanford algebraic geometry seminar 2017-18

Fridays 4-5 pm in 383-N (with exceptions)

 September 29 Samuel Grushevsky (Stony Brook University) Geometry of compactified moduli of cubic threefolds October 6 Felix Janda (University of Michigan) Genus two curves on quintic threefolds October 13 no seminar (WAGS weekend) Weekend of October 14-15 Western Algebraic Geometry Symposium, at UCLA October 20 Remy van Dobben de Bruyn (Columbia University) Dominating varieties by liftable ones October 27 Jason Lo (Cal State Northridge) The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations November 10 November 17 (probably no seminar, Ravi away) November 24 no seminar (Thanksgiving break) December 1 Daniel Litt (Columbia University) Galois actions on fundamental groups January 19 Pablo Solis (Stanford) Hunting Vector Bundles on $\mathbf{P}^1 \times \mathbf{P}^1$ January 26 Izzet Coskun (UIC) The geometry of moduli spaces of sheaves on surfaces February 2 Katrina Honigs (Utah) Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic February 9 Junliang Shen (ETH) K3 categories, cubic 4-folds, and the Beauville-Voisin conjecture February 16 Michael Viscardi (Berkeley) Quantum cohomology and 3D mirror symmetry February 23 Roya Beheshti (Washington U. St. Louis) Moduli spaces of rational curves on hypersurfaces March 9 Yongbin Ruan (Michigan) The structure of higher genus Gromov-Witten invariants of quintic 3-fold April 13 Andrei Caldararu (Wisconsin) Computing a categorical Gromov-Witten invariant April 20 Daniel Halpern-Leistner (Cornell) What is wall-crossing? April 27 Sean Howe Motivic random variables and random matrices May 4 (3-4 pm, in 380-W) Isabel Vogt (MIT) Interpolation problems for curves in projective space May 4 (4:30-5:30 pm, in 383-N) Eric Larson (MIT) The Maximal Rank Conjecture May 11 Sheldon Katz (UIUC) (joint with physics) May 18 (3-4 pm) Dori Bejleri (Brown) Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations May 18 (4:30-5:30 pm) Francois Greer (Stony Brook) Elliptic fibrations in the presence of singularities May 25 (3-4 pm, location TBA) Soren Galatius (Stanford) $M_g$, $M_g^{trop}$, the Grothendieck-Teichmuller group, and Kontsevich’s complex of graphs May 25 (4:30-5:30 pm) Tathagata Basak (Iowa State) A complex ball quotient and the monster
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## abstracts for 2016-17 seminars

(The seminar webpage is here.)

Michael Kemeny (Stanford)
October 7, 2016

Syzygies, scrolls and Hurwitz spaces

A famous conjecture of Mark Green predicts a close relationship between the geometry of a curve and the algebraic properties of its coordinate ring. Namely, the Clifford index of the curve should equal the length of the linear part of the resolution of its coordinate ring under the canonical embedding. A conjecture of Schreyer goes beyond this by specifying that the last piece of the linear part should moreover tell you whether or not the curve has a unique pencil of minimal degree. We will discuss a proof of Schreyer’s conjecture for general curves of prescribed gonality, obtained jointly with Gavril Farkas. Two of the key actors in this story are the scroll associated to a pencil and the geometry of Hurwitz space.  (poster)

Christian Schnell (Stony Brook University)
October 21, 2016

Pushforwards of pluricanonical bundles and morphisms to complex abelian varieties

In the past few years, people working on the analytic side of algebraic geometry have obtained two important new results: a version of the Ohsawa-Takegoshi extension theorem with sharp estimates (Blocki, Guan-Zhou), and the existence of canonical singular hermitian metrics on pushforwards of relative pluricanonical bundles (Berndtsson, Paun, Takayama, and others). In this talk, I will explore some consequences of their work for the study of morphisms to complex abelian varieties, including the recent proof of Iitaka’s conjecture over abelian varieties (Cao-Paun).  (The talk will be understandable without any background in analysis.)

Wenhao Ou (UCLA)
October 28, 2016

Fano varieties where all pseudoeffective divisors are also numerically effective

We recall that a divisor in a smooth projective variety is said to be numerically effective (or nef) if it meets each curve with non negative intersection number, and is called pseudoeffective if it is the limit of effective Q-divisor classes. Both of these properties are ways in which a divisor can be in some sense “positive”. A nef divisor is always pseudoeffective, but the converse is not true in general. A Fano varity is a special variety whose anti-canonical divisor is ample. From the Cone Theorem, it turns out that the geometry of a Fano variety is closely related to its nef divisors. In this talk, we will consider Fano varieties such that all pseudoeffective divisors are nef. Wiśniewski shows that the Picard number of such a variety is at most equal to its dimension. Druel classifies these varieties when these two numbers are equal. We classify the case when the Picard number is equal to the dimension minus 1.

November 4, 2016

Isospectrality of compact locally symmetric spaces and weak commensurability of arithmetic groups

Quotients of symmetric spaces of semi-simple Lie groups by torsion-free arithmetic subgroups are particularly interesting Riemannian manifolds which can be studied by using diverse techniques coming from the theories of Lie Groups, Lie Algebras, Algebraic Groups and Automorphic Forms. In my talk, I will discuss a well-known problem which was formulated by Mark Kac as “Can one hear the shape of a drum?”, and its solution, for arithmetic quotients of symmetric spaces, obtained in a joint paper (in Publ Math IHES, vol 109) with Andrei Rapinchuk. For its solution, we introduced a notion of “weak commensurability” of arithmetic, and more general Zariski-dense, subgroups and derive very strong consequences of weak commensurability.

Donghai Pan (Stanford)
November 11, 2016

Galois cyclic covers of the projective line and pencils of Fermat hypersurfaces

Classically, there are two objects that are particularly interesting to algebraic geometers: hyperelliptic curves and quadrics. The connection between these two seemingly unrelated objects was first revealed by M. Reid, which roughly says that there’s a correspondence between hyperelliptic curves and pencil of quadrics. I’ll give a brief review of Reid’s work and then describe a higher degree generalization of the correspondence.

Giulia Sacca (Stony Brook)
December 2, 2016

Intermediate Jacobians and hyperkahler manifolds

In recent years, there have been an increasing number of connections between cubic fourfolds and hyperkahler manifolds. The first instance of this was noticed by Beauville-Donagi, who showed that the Fano varieties of lines on a cubic fourfolds X is holomorphic symplectic. The aim of the talk is to describe another instance of this phenomenon, which is carried out in joint work with Laza and Voisin. The resulting hyperkahler manifold is fibered in intermediate Jacobians and is deformation equivalent to O’Grady’s ten-dimensional example. I will also present a more recent proof of this result, which is obtained in joint work with Laza, Kollár, and Voisin.

Hannah Larson (Harvard)
January 13, 2017, 4-4:45 pm

Lines on hypersurfaces with certain normal bundles

Let $X$ be a smooth hypersurface. The Fano scheme of lines $F(X)$ is the parameter space of all lines $L \subset X$. Given such a line, the normal bundle of $L$ in $X$ controls the deformation theory of $L$ in $X$, and thus provides local information about $F(X)$ near $L$. Being a vector bundle on ${\mathbf{P}}^1$, the normal bundle of $L$ in $X$ always splits as a direct sum of line bundles. In this talk, we consider natural subschemes of $F(X)$ parameterizing lines $L$ whose normal bundle in $X$ has a certain splitting type.

Gavril Farkas (Humboldt University)
January 13, 2017, 5-6 pm

K3 surfaces of genus 14 via cubic fourfolds

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera $g<21$, with the exception of the case $g=14$. Using Hassett's identification between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. The proof relies on a degenerate version of Mukai's structure theorem for K3 surfaces of genus 8. This is joint work with Verra.

Ashvin Swaminathan (Harvard)
Thursday, January 19, 2017, 12:15-1 pm in 384-I

Inflection Points of Linear Systems on Families of Curves

It is a classic theorem in enumerative geometry that a general plane curve of degree $d$ has exactly $3d(d-2)$ flex points (these are inflection points at which the tangent line has contact of order $3$). Given this result, there are two natural generalizations to consider: (1) what can we say about inflection points of higher contact order, and (2) what happens when we look at such inflection points in families of curves acquiring a singularity? In this talk, I will discuss joint work with Anand Patel, in which we develop a method for answering these more general questions. Moreover, I will describe how to apply our method to tackle three interesting problems: (1) counting hyperflexes in a general pencil of plane curves, (2) describing the analytic-local behavior of the divisor of flexes in a family of plane curves acquiring a nodal singularity, and (3) computing the divisors of Weierstrass points of arbitrary order on the moduli space of curves.

Jake Levinson (Michigan)
January 20, 2017, 3:45-4:45 pm

Boij-Söderberg Theory for Grassmannians

Boij-Söderberg theory is a structure theory for syzygies of graded modules: a near-classification of the possible Betti tables of such modules (these tables record the degrees of generators in a minimal free resolution). One of the surprises of the theory was the discovery of a “dual” classification of sheaf cohomology tables on projective space.

I’ll tell part of this story, then describe some recent extensions of it to the setting of Grassmannians. Here, the algebraic side concerns modules over a polynomial ring in $kn$ variables, thought of as the entries of a $k \times n$ matrix. The goal is to classify “$GL(k)$-equivariant Betti tables”, recording the syzygies of equivariant modules, and to relate them to sheaf cohomology tables on the Grassmannian $Gr(k,n)$. This work is joint with Nic Ford and Steven Sam.

Ben Bakker (Georgia)
January 20, 2017, 5-6 pm

A global Torelli theorem for singular symplectic varieties

Holomorphic symplectic manifolds are the higher-dimensional analogs of K3 surfaces and their local and global deformation theories enjoy many of the same nice properties.  By work of Namikawa, some aspects of the story generalize to singular symplectic varieties, but the lack of a well-defined period map means the moduli theory is ill-defined.  In joint work with C. Lehn, we consider locally trivial deformations—deformations along which the singularities don’t change — and show that in this context most of the results from the smooth case extend.  In particular, we prove a version of the global Torelli theorem and derive some applications to the geometry of birational contractions of moduli spaces of vector bundles on K3 surfaces.

Srikanth Iyengar (Utah)
January 27, 2017

A local Serre duality for modular representations of finite groups (and group schemes)

This talk will be about the representations of a finite group (or a finite group scheme) G defined over a field k of positive characteristic. In recent work, Dave Benson, Henning Krause, and Julia Pevtsova, and I discovered that stable module category of finite dimensional representations of G has local Serre duality. My plan is to explain what this means and also present some of the ideas, mostly from commutative algebra, that go into its proof.

Steven Sam (Wisconsin)
February 3, 2017

Secant varieties of Veronese embeddings

Given a projective variety X over a field of characteristic 0, and a positive integer r, we study the rth secant variety of Veronese re-embeddings of X. In particular, I’ll explain recent work which shows that the degrees of the minimal equations (and more generally, syzygies) defining these secant varieties can be bounded in terms of X and r independent of the Veronese embedding. This is based on http://arxiv.org/abs/1510.04904 and http://arxiv.org/abs/1608.01722.

February 10, 2017

Double Ramification Cycles and Tautological Relations

Tautological relations are certain equations in the Chow ring of the moduli space of curves. I will discuss a family of such relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line. These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations. This is joint work with various subsets of S. Grushevsky, F. Janda, X. Wang, and D. Zakharov.

Ionut Ciocan-Fontanine (Minnesota)
February 17, 2017

Wall-crossing in quasimap theory

Quasimap theory is concerned with curve counting on certain GIT quotients. In fact, one has a family of curve counting theories, with Gromov-Witten included, depending on the linearization in the GIT problem. I will present a wall-crossing formula, in all genera and at the level of virtual classes, as the size of the linearization changes. Some numerical consequences will be discussed as well. The talk will focus primarily on the recently established case of complete intersections in projective space (in this case, stable quasimaps coincide with stable quotients). This is joint work with Bumsig Kim.

Dusty Ross (SFSU)
February 24, 2017

Genus-One Landau-Ginzburg/Calabi-Yau Correspondence

First suggested by physicists in the late 1980’s, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence, which is determined from genus-zero data alone. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.

Dhruv Ranganathan (MIT)
April 14, 2017

A Brill-Noether theorem for curves of a fixed gonality

The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill–Noether theorem, which determines the dimensions of the Brill–Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus”. The proof blends recent advances in tropical linear series theory, Berkovich geometry, and ideas from logarithmic Gromov-Witten theory. This is joint work with Dave Jensen.

Erik Carlsson (UC Davis)
April 21, 2017

Geometry behind the shuffle conjecture

The original “shuffle conjecture” of Haglund, Haiman, Loehr, Ulyanov, and Remmel predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A, in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the ideas that went into my recent proof of this conjecture with Anton Mellit, namely the construction of a new algebra which has many elements in common with DAHA’s, and which has been expected to have a geometric construction. I will then describe a current project with Eugene Gorsky and Mellit, in which we have discovered the desired action of this algebra on the torus-equivariant K-theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals of finite codimension in C[x,y].

Arnav Tripathy (Harvard)
May 12, 2017

Motivic Donaldson-Thomas invariants of K3 times an elliptic curve

I’ll describe a new chapter in the enumerative geometry of the K3 surface and its product with an elliptic curve in a long line of extensions starting from the classic Yau-Zaslow formula for counts of rational nodal curves. In particular, I’ll describe a string-theoretic prediction for the threefold’s motivic Donaldson-Thomas invariants given the Hodge-elliptic genus of the K3, a new quantity interpolating between the Hodge polynomial and the elliptic genus.

John Lesieutre (UIC)
May 19, 2017

A projective variety with discrete, non-finitely generated automorphism group

I will outline the construction of a projective variety over $\mathbf{C}$ for which the group of automorphisms is discrete, but not finitely generated. I’ll also discuss a related example of a variety over $\mathbf{R}$ with infinitely many non-isomorphic $\mathbf{C}/\mathbf{R}$-forms.

Yuchen Liu (Princeton)
May 26, 2017

Construction of hyperbolic cyclic covers

A complex algebraic variety is Brody hyperbolic if there are no non-constant holomorphic maps from the complex plane to the variety. As shown by Duval, Shiffman, Zaidenberg and many other people, degeneration methods are very useful in constructing Brody hyperbolic varieties. Using degeneration to the normal cone, we construct lots of examples of Brody hyperbolic cyclic covers when their branch loci have sufficiently large degree. If time permits, I will also discuss a different approach to construct hyperbolic double covers of $\mathbf{P}^2$ and Hirzebruch surfaces with the smallest possible degree of branch loci.


text:
# Contents

## Idea

A derived $\infty$-Lie algebroid is an ∞-Lie algebroid whose underlying space is a derived space , for instance a derived smooth manifold.

More precisely, let $T$ be any abelian Lawvere theory and $T \hookrightarrow C \hookrightarrow T Alg^{op}$ a small full subcategory of geometric test objects formally dual to T-algebras. Then by the theory of function algebras on ∞-stacks over $C$ one identifies $\infty LieAlgd$ with the subcategory

$(T Alg^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} [C^{op}, sSet]$

of the (∞,1)-category of (∞,1)-sheaves on $C$ modeled by cosimplicial T-algebras. Under the Dold-Kan correspondence applied to the underlying cosimplicial ordinary algbra, this identifies with non-negatively graded dg-algebras: the Chevalley-Eilenberg algebras of the corresponding dual $\infty$-Lie algebroids.

But in full derived geometry this setup is further generalized: instead of considering ∞-stacks on just a category of duals of $T$-algebras, one considers derived ∞-stacks over an (∞,1)-site of simplicial algebras. Let $T \hookrightarrow C^{\Delta^{op}} \hookrightarrow T Alg^{\Delta^{op}}$ be accordingly a small subcategory of simplicial $T$-algebra, then the above adjunction generalizes to

$((T Alg^{\Delta^{op}})^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} [C^{op}, sSet] \,,$

where now on the left we have cosimplicial simplicial algebras . Under the Dold-Kan correspondence these now identify with unbounded dg-algebras $CE(\mathfrak{a})$. We have that

• the categorical degree $k$ coming from k-morphisms of the $\infty$-sheaves contributes to positive degrees in $CE(\mathfrak{a})$;

• the derived degree $l$ coming from $l$-morphisms in the $\infty$-function algebras ontribute to negative degree.

This category $(T Alg^{\Delta^{op} \times \Delta})^{op}$ we identify with that of derived $\infty$-Lie algebroids.

## Examples

### BV-BRST complex

In the literature the most familiar example of a derived $\infty$-Lie algebroids – even if not under this name – are the BV-BRST complexes. These are action Lie algebroids for actions of ∞-Lie algebras on derived smooth manifolds.


text:
The area (in sq. units) of the region

Question:

The area (in sq. units) of the region

\begin{aligned} A=&\{(x, y) \in R \times R \mid 0 \leq x \leq 3,0 \leq y \leq 4\\ &\left.y \leq x^{2}+3 x\right\} \text { is : } \end{aligned}

1. $\frac{53}{6}$

2. $\frac{59}{6}$

3. 8

4. $\frac{26}{3}$

Correct Option: , 2

Solution:


text:
# Must we test whether e.g. $A=B$ and $A=C$ implies $B=C$ by experiment?

Chaper 10, conservation of momentum in "The Feynman Lectures on Physics" in the chapter entitled, the authors write that

Suppose we know from the foregoing experiment that two pieces of matter, $A$ and $B$ (of copper and aluminum), have equal masses, and we compare a third body, say a piece of gold, with the copper in the same manner as above, making sure that its mass is equal to the mass of the copper. If we now make the experiment between the aluminum and the gold, there is nothing in logic that says these masses must be equal; however, the experiment shows that they actually are. So now, by experiment, we have found a new law. A statement of this law might be: If two masses are each equal to a third mass (as determined by equal velocities in this experiment), then they are equal to each other. (This statement does not follow at all from a similar statement used as a postulate regarding mathematical quantities.)

I assume that by "a similar statement used as a postulate regarding mathematical quantities" the authors refer to the transitive axiom of algebra, that is, if $A=B$ and $A=C$, then $B=C$.

It seems to me that, using Feynman's definition of mass and the transitive axiom of algebra, one must conclude that it is the case that $B=C$ even without making an experiment. Why do Feynman et al claim that

there is nothing in logic that says these masses must be equal; however, the experiment shows that they actually are

• The transitive axiom of algebra relates symbols written down on a piece of paper. Feynman is talking about the outcomes of physical measurements. I think the broad point he is making is that one should not assume a priori that the abstract mathematics of algebra describes relationships between measurable quantities, but that it turns out experimentally that in this case (and in many others) it does. – Mark Mitchison Jul 9 '15 at 13:52
• I think you should read the whole section again. I believe Feynman is talking about you can always add more and more "laws" that work in special cases, but you have no reason to believe they will work in other cases. In a sense he is just saying be careful before making claims that something "should always happen" verse "happened in this case" – Peter Anderson Jul 9 '15 at 13:59
• what are the appropriate tags for this question? to be honest, this question is really about philosophy/epistemology of science. what knowledge can we justify etc – innisfree Jul 9 '15 at 14:02
• I'd say that assuming the masses were equal was very safe knowledge, even without the experiment. – innisfree Jul 9 '15 at 14:04
• A slightly different example, but in Aristotle's day everyone simply knew that if two unequal masses were dropped from the same height, the greater mass would fall faster than the lesser mass. There was no need to do the experiment because it was obvious. Then Galileo made the same observation as Feynman - There is nothing in logic that says the heavier mass must fall faster. So he did the experiment and found the unexpected result that they fell at the same rate. I believe that's Feynman's point - you cannot assume something just because it appears obvious - you must do the experiment. – Mark Jul 9 '15 at 17:10

Who can with certainty fathom the mind of a Feynman? :-)
However, it sounds like he is saying that there is no reason to decide with certainty that Gold, Copper and Aluminum do not have properties which affect how they behave when accelerated by a force.

An example that has some correspondence is carrying out the same experiment in air.
Mount 1 kg spheres of each metal on massless, frictionless, area less "carts" (made of unobtainium (or nonexisteum) ) such that they can be slid across a surface in a manner which is independent of the carts' properties. Now place any two spheres in contact on their carts and apply an explosion based force as before. The initial velocities will be the same as before and equal but some mysterious new force will apply that causes the spheres to act differently. Gold-Al and Gold-Cu will have different results for the non Al material. A few nano-seconds thought will allow us to see that in both cases the Al sphere behaves the same way and that there is some property of each material that affects how they behave once proceeding at a given velocity. We will have discovered air resistance and drag and the different densities will cause the different masses to have non-equal frontal areas so they will have different deceleration forces from drag.

While this is so obvious that it seems trivial, there may have been other properties of Au.Al/Cu that affected how they interacted in the initial experiment. Until we perform the experiment we cannot know what curve ball nature has or hasn't got in store.

This leads to his extremely valuable observation

   "From this example we can see how quickly we start to infer things if we are careless."

If we were unaware of air drag then expecting equal behaviour for all 3 materials after the initial explosion would leave us puzzled. That we are not puzzled in the free space case is because our expectation happened to ,match reality. But, we can never be sure that this is the case.

Real (out of this) world example:

For some while it seemed that there were 1/3 as many Neutrinos coming from the sun as calculations predicted. In due course it was posited and then experimentally "proven" that the Neutrinos 'change colour' as they amble on their way at (or just above or just below :-) ) light speed, and that only the 1/ which are the 'right' colour as they pass through our detectors get detected.

Solar neutrino problem

• Measurements of solar neutrino types were not consistent with models of the Sun's interior.

• Former Standard Model: Neutrinos should have been massless according to the then-accepted theory; this means that the type of neutrino would be fixed when it was produced. The Sun should emit only electron neutrinos as they are produced by H–H fusion. Observation

• Only one third to one half of predicted number of electron neutrinos were detected; neutrino oscillation explains the difference but requires neutrinos to have mass.

• Resolution: Neutrinos have mass and so can change type.

• Neutrino oscillation is a quantum mechanical phenomenon whereby a neutrino created with a specific lepton flavor (electron, muon or tau) can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates through space.

• First predicted by Bruno Pontecorvo in 1957, neutrino oscillation has since been observed by a multitude of experiments in several different contexts; it also turned out to be the resolution to the long-standing solar neutrino problem.)

• I don't really think this tackles the crux (as I see it) of the matter - do we have to test logical results experimentally? or is that $B=C$ safe, reliable knowledge already? – innisfree Jul 9 '15 at 14:06
• @innisfree Can you explain why you do not think it explains the issue? It seems to do to me. In maths we define certain properties and as maths is a model we= can make the model fit whatever assumptions we wish before we "turn it on". | Nature is not like that. We are unable to be certain of ANYTHING that we do not measure, and even then , we may be fooled by laws that do not apply as we expect. Au/Cu/Al might have had interactions between atoms or ...? It is only by having measured the results that we know it is not so. | Dark matter & energy are excellent candidates. Do they exist per se ... – Russell McMahon Jul 9 '15 at 14:15
• @Innisfree ... or are they attributes of mass or "gravity" (if the two can be distinguished) which we have yet to come to grips with. | Solve that and you can start on QM and gravity :-). || Rutherford said there was "Physics and stamp collecting". In reality - it's ALL stamp collecting. Physics is just descriptions of what we observe framed in maths speak. That's perfectly fine - but when something breaks (eg dark xxx) we have to re-examine some of the stamps :-). – Russell McMahon Jul 9 '15 at 14:20
• interesting ideas, not sure I agree though. surely an important aspect of science is making inferences from data, once you've collected it (i.e. it isn't all stamp/data collecting, but inferences from stamp/data collections). – innisfree Jul 9 '15 at 14:31

I think the content of his statement is that "the assignment of mass to an object as defined using the conservation of momentum experiment is a transitive property". Not all relations are transitive properties (for instance, "Wolves eat Deer, Deer eat grass, but wolves don't eat grass"). For instance, if we had defined "the mass of an object" as a weight $mg$, the transitive property might not be satisfied unless the experiment was performed at the same location on Earth.

So in my mind, the story from Feynman is a statement about how mass is well-defined using the conservation of momentum. When we say "the mass of the car is 500 kg", we hope that can mean everything that the statement "$m_{car}=500$ kg" does, and when we perform the experiment we have some evidence that it is at least transitive, as it should be.

We know how real numbers (in the mathematical sense) behave. There is no a priori reason however to assume that masses of objects behave as real numbers. The proposed experiment can be seen as a check to see if masses actually can be modeled by (a subset of the) real numbers.


text:
# Labour Productivity Growth and Reallocation of Labour

Using a shift-share methodology, labour productivity growth can be arise from productvity growth within a sector (within effect) or reallocation of labour to sectors with higher productivity (reallocation effect).

### 1)

However we know that Productivity = Real Value-added / Employment.

So won't an increase in employment in a sector result in a fall in the sector's productivity and hence overall productivity?

employment intensity is the direct inverse of labour productivity

So does this mean the government should not employ policies to reallocate labour to higher productivity sectors?

It will be good to refer me to some paper that address these questions

You make a hand waiving claim. In these scenarios making concise mathematic statements helps improving your arguments.

Denote employment as $L$, define real value added as $F(L) - wL$, and productivity as real value / employment:

$$A = \frac{F(L) - w(L)}{L}$$

The labor share should be some constant $\beta$ of total value added. Without going deep into mathematics, straight away assume $w(L) = \beta\alpha F(L)$. Hence, we have that

$$A = \frac{F(L) (1 - \beta\alpha)}{L}$$

Now, what happens if $L$ increases? Let $L_2 > L_1$

$$A_2 - A_1 = (1-\alpha\beta)(\frac{F(L_2)}{L_2} - \frac{F(L_1)}{L_1})$$

Assume $F$ to be concave. Then we have that the change of productivity, $A_2 - A_1$ is negative - as you say.

However, $A$ here is average productivity. Total productivity is $AL$.

$$A_2L_2 - A_1L_1 = (1-\alpha\beta)(F(L_2) - F(L_1))$$.

Independent of the curvature of $F$, for any strictly increasing production function, this will be positive.

tl;dr

• Don't confuse average productivity with total productivity.
• Whenever you find something weird, use algebra to make your statements precise

Again, you are not being precise enough. What is the government's objective? If government is only about maximizing average productivity, the solution is to let $L\to 0$, as then $F'(L) \to \infty$ (and for Cobb-Douglas, $F'(L)$ is proportional to average productivity).


text:
Correct! To get ice's melting point in Kelvin, we just add $$273$$ to its melting point in Celsius ($$0 ^{\circ}\mathrm{C}$$) So, we get $$0 + 273 = 273 \, \mathrm{K}$$
So, like we discussed, a solid turning into a liquid is called melting. Obviously, there are other changes of matter as well. You can see them in the animation. For example, if a liquid turns into a gas, it is called Vaporisation.

Every time a change of state occurs via heating, latent heat is absorbed. Every time a change of state occurs via cooling, latent heat is released.
During condensation, is latent heat absorbed or released?
• Latent Heat is released
• Latent Heat is absorbed
• Latent Heat is neither released nor absorbed


text:
## College Physics (4th Edition)

Published by McGraw-Hill Education

# Chapter 22 - Problems - Page 863: 19

#### Answer

(a) Humans are able to perceive approximately 1 octave of visible light. (b) The microwave region is approximately 13.3 octaves wide.

#### Work Step by Step

(a) The range of frequencies for visible light is about $4\times 10^{14}~Hz$ to $7.5\times 10^{14}~Hz$. We can find the ratio: $\frac{7.5\times 10^{14}~Hz}{4\times 10^{14}~Hz} = 1.9$ Since the ratio is 1.9:1, humans are able to perceive approximately 1 octave of visible light. (b) The range of frequencies for microwaves is about $10^{8}~Hz$ to $10^{12}~Hz$. We can find the ratio: $\frac{10^{12}~Hz}{10^{8}~Hz} = 10,000$ We can find the number of octaves $N$: $2^N = 10,000$ $N~ln(2)= ln(10,000)$ $N= \frac{ln(10,000)}{ln(2)}$ $N = 13.3$ The microwave region is approximately 13.3 octaves wide.

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