28 Jul

This salad is perfect for any summer gathering, party or lunch!

The combination of fruit, nuts and dark leafy greens makes this salad not only delicious, but high up on a healthy scale.

It also only took me 5 minutes to put it together. Easy and delicious, my favorite combination 🙂

Ingredients:

• Apple Pecan Dressing
• Green Apple Slices
• Strawberries
• Cucumbers
• Sliced Almonds
• Dark Leafy Spring Lettuce
• Spinach Leaves

I found that this salad is better shaken. When you just pour the dressing over the lettuce, it tends to clump up on various ingredients and doesn’t reach to others. Trust me, this incredible dressing is so tasty, you’ll want it to reach every piece!

Make sure not to over-dress your salad. The delicate sweetness of the Apple Pecan dressing is delicious, but it’s easy to over do it. Start small, and add as needed.

Enjoy!

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28 Jul

Ingredients:

• Fresh sliced red grapes
• Strawberries
• Cucumbers
• Feta Cheese
• Sliced Almonds
• Spinach Leaves
• Red Cabbage
• Black Pepper
• Apple Cider Vinaigrette Dressing

I’ve got to admit this is one of my favorite salads yet!

Perfect for a hot day, light and refreshing

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## Talk by John Huerta – The Functor of Points approach to Supermanifolds

28 Jul

John Huerta visited here for about a week earlier this month, and gave a couple of talks. The one I want to write about here was a guest lecture in the topics course Susama Agarwala and I were teaching this past semester. The course was about topics in category theory of interest to geometry, and in the case of this lecture, “geometry” means supergeometry. It follows the approach I mentioned in the previous post about looking at sheaves as a kind of generalized space. The talk was an introduction to a program of seeing supermanifolds as a kind of sheaf on the site of “super-points”. This approach was first proposed by Albert Schwartz, though see, for instance, this review by Christophe Sachse for more about this approach, and this paper (comparing the situation for real and complex (super)manifolds) for more recent work.

It’s amazing how many geometrical techniques can be…

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## Computing convolutions of measures

28 Jul

If $latex {f: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ and $latex {g: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ are two absolutely integrable functions on a Euclidean space $latex {{\bf R}^n}&fg=000000$, then the convolution $latex {f*g: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ of the two functions is defined by the formula

$latex \displaystyle f*g(x) := \int_{{\bf R}^n} f(y) g(x-y)\ dy = \int_{{\bf R}^n} f(x-z) g(z)\ dz.&fg=000000$

A simple application of the Fubini-Tonelli theorem shows that the convolution $latex {f*g}&fg=000000$ is well-defined almost everywhere, and yields another absolutely integrable function. In the case that $latex {f=1_F}&fg=000000$, $latex {g=1_G}&fg=000000$ are indicator functions, the convolution simplifies to

$latex \displaystyle 1_F*1_G(x) = m( F \cap (x-G) ) = m( (x-F) \cap G ) \ \ \ \ \ (1)&fg=000000$

where $latex {m}&fg=000000$ denotes Lebesgue measure. One can also define convolution on more general locally compact groups than $latex {{\bf R}^n}&fg=000000$, but we will restrict attention to the Euclidean…

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## Watermelon Cucumber Pineapple protein shakes

28 Jul

One of my favorite parts of summer is sweet sweet sweet watermelon!
I love putting it in smoothies, shakes and salads.
I used my new protein powder that I purchased yesterday, and made more of a shake than a smoothie.

Simple Ingredients:

• Freshly cut sweet watermelon
• Sliced cucumber
• Pineapple
• Guava juice
• Watermelon juice
• Ice
• 3 scoops protein powder

Blend, add a cute garnish and enjoy!

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## BSD for a Large Class of Elliptic Curves

28 Jul

I’m giving up on the p-divisible group posts for awhile. I would have to be too technical and tedious to write anything interesting about enlarging the base. It is pretty fascinating stuff, but not blog material at the moment.

I’ve been playing around with counting fibration structures on K3 surfaces, and I just noticed something I probably should have been aware of for a long time. This is totally well-known, but I’ll give a slightly anachronistic presentation so that we can use results from 2013 to prove the Birch and Swinnerton-Dyer conjecture!! … Well, only in a case that has been known since 1973 when it was published by Artin and Swinnerton-Dyer.

Let’s recall the Tate conjecture for surfaces. Let $latex {k}&fg=000000$ be a finite field and $latex {X/k}&fg=000000$ a smooth, projective surface. We’ve written this down many times now, but the long exact sequence associate to the Kummer sequence

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