Archives for April 2024

Introduction The central objects of study in topology are spaces called manifolds, which look flat when you zoom in on them. The surface of a sphere, for instance, is a two-dimensional manifold. Topologists understand such two-dimensional manifolds very well. And they have developed tools that let them make sense of three-dimensional manifolds and those with five or more dimensions. But in four dimensions, “everything goes a bit crazy,” said Sam Hughes, a postdoctoral researcher at the University of Oxford. Tools stop working; exotic behavior emerges. As Tom Mrowka of the Massachusetts Institute of Technology explained, “There’s just enough room to have interesting phenomena, but not so much room that they fall apart.” In the early 1990s, Mrowka and Peter Kronheimer of Harvard University were studying how two-dimensional surfaces can be embedded within four-dimensional manifolds. They developed new techniques to characterize these surfaces, allowing them to gain crucial insights into the otherwise inaccessible structure of four-dimensional manifolds. Their findings suggested that the members of a broad class of surfaces all slice through their parent manifold in a relatively simple way, leaving a fundamental property unchanged. But nobody could prove this was always true. In February, together with Daniel Ruberman of Brandeis University, Hughes constructed a sequence of counterexamples — “crazy” two-dimensional surfaces that dissect their parent manifolds in ways that mathematicians had believed to be impossible. The counterexamples show that four-dimensional manifolds are even more remarkably diverse than mathematicians in earlier decades had realized. “It’s really a beautiful paper,” Mrowka said. “I just keep looking at it. There’s lots of delicious little things there.” Making a List Making a List
Late last year, Ruberman helped organize a conference that created a new list of the most significant open problems in low-dimensional topology. In preparing for it, he looked at a previous list of important unsolved topological problems from 1997. It included a question that Kronheimer had posed based on his work with Mrowka. “It was in there, and I think it was a little bit forgotten,” Ruberman said. Now he thought he could answer it. To understand the question, it helps to first consider two key ideas: simply connected manifolds, and the fundamental group. Simply connected manifolds are spaces without any holes passing through them. In one dimension, an infinite line is simply connected, but a circle isn’t. In two dimensions, an infinite plane and the surface of a sphere are simply connected, but the surface of a doughnut is not. Mathematicians make this distinction rigorous by placing loops on a manifold and considering how they can be deformed. If any loop can be shrunk to a point, then a manifold is simply connected. On a plane or the surface of a sphere, for instance, this is possible — think about pulling a string taut. But if that string goes around a circle, it can’t shrink. Similarly, on the surface of a doughnut, loops that go either around or through the central hole can’t be deformed into a single point. The doughnut itself gets in the way. Mathematicians classify spaces that are not simply connected by computing their “fundamental group,” an object whose structure reflects how loops shrink. Manifolds that are simply connected have a “trivial” fundamental group with just one element. But manifolds with holes in them have more complicated fundamental groups. Introduction Four-dimensional manifolds that are simply connected can still be plenty strange. To understand them, mathematicians ponder what can happen to the two-dimensional surfaces embedded in them. By analogy, think about laying a loop of string flat on a piece of paper. There’s not much you can do with it. But lift it up into three-dimensional space, and you can tie it into complicated knots. The ways in which you can manipulate the string — a one-dimensional manifold — clarify the nature of the space it is embedded in. Similarly, in the more complicated world of four dimensions, two-dimensional surfaces are “kind of key to the whole business, in many different ways,” Ruberman said. “Surfaces tell you much more about a four-dimensional manifold than you have any right to expect.” Surfaces let you distinguish between manifolds: If a surface can live inside one manifold but not another, you know the manifolds are different. And surfaces can be used to build new manifolds out of old ones. Surfaces also have corresponding fundamental groups. And so do their complements — the part of a manifold that is left over when you take the surface away. Remove the equator from two-dimensional manifolds like the surface of a sphere or doughnut, for instance, and you get two disconnected hemispheres. But the doughnut’s surface remains in one piece if you remove a vertical ring instead of a horizontal one. Similarly, depending on how you cut a surface out of a four-dimensional manifold, you can get different kinds of complements. Introduction Back in the 1990s, Mrowka and Kronheimer investigated what happens when you excise a two-dimensional surface from a four-dimensional manifold. If the manifold itself is simply connected, what conditions must surfaces meet to guarantee that their complements must also be simply connected? Kronheimer and Mrowka knew that some kinds of surfaces could have complements that weren’t simply connected. But their work seemed to indicate that another broad class of surfaces must always have simply connected complements. For nearly three decades, nobody could find an example of a surface in that class whose complement was not simply connected. But in the fall of 2023, after coming across the problem, Ruberman thought he could. Instead of starting with a four-dimensional manifold and cutting out a surface, he began with a two-dimensional surface that had the necessary properties and built a manifold around it. First, he fattened the surface into a four-dimensional blob. This four-dimensional blob had a three-dimensional boundary, just as a three-dimensional object like a ball has a two-dimensional boundary. Ruberman wanted to attach a carefully chosen four-dimensional manifold to the other side of the boundary, which would serve as the surface’s complement. If the gambit worked, then this manifold would have a complicated fundamental group, yet the fundamental group of everything taken together would be trivial. The newly constructed four-dimensional manifold would therefore be simply connected. But to be able to glue everything together in the right way, he had to show that the fundamental group of the new addition satisfied all sorts of properties. “I had no idea how to do that,” Ruberman said. Then in January, Hughes — a group theorist — gave a talk at Brandeis. Ruberman was in the audience. He recognized that Hughes might have the missing piece he was looking for. The two met up the following day, and within a few hours, they’d worked out the main ideas they needed. What Ruberman was missing “is something group theorists have been computing for 70, 80 years at this point,” Hughes said. “We’ve been at this forever.” By the end of the week, they had a completed proof. “I knew some things, and he knew some things, and between the two of us, we knew enough to just do it,” Ruberman said. Because of the way group theory gets used in the proof, “it’s a little bit unusual,” said Maggie Miller of the University of Texas, Austin. “It’s written a little bit different than most four-dimensional topologists would be comfortable with.” The result is yet another example of how complicated four-dimensional topology can get. “There are more interesting embeddings of surfaces than we thought,” Hughes said. This makes it more difficult to classify manifolds, and harder to prove other kinds of results about them. Nevertheless, in March, İnanç Baykur of the University of Massachusetts, Amherst, who organized last year’s list-making conference with Ruberman, announced the solution to another problem involving simply connected four-dimensional manifolds from the 1997 list. It seems the topologists are cleaning house.

Bill Maher doesn’t always hit the nail on the head, but when he does, it’s usually funny. During a recent episode of “Club Random with Bill Maher,” Candace Owens couldn’t help but laugh at Maher’s description of “what this country is.”“This country reminds me of like a medieval European country wher…

This post may contain affiliate links. For more information, read my disclosure policy. Blueberry Banana Bread is an easy banana bread recipe full of fresh blueberries and topped with a crunch cinnamon sugar topping. Cinnamon sugar is always a good thing, don’t you think? It’s one of the best blueberry breads EVER! Table of Contents Banana Blueberry Bread Recipe This Blueberry Banana Bread recipe is the softest and moistest bread and is also incredibly easy to make! It’s full of fresh berries (or frozen) and tons of banana flavor. I created this from my favorite banana bread recipe – my moms – and it’s EPIC. I also put cinnamon sugar on the crust of this recipe, which is the best topping for any bread recipe. The blueberry and cinnamon sugar will always be a favorite pairing of mine that you need to try! If you have any overripe ugly brown bananas that need to be used…this is your recipe! Ingredients Needed Bananas : Be sure to use ripe bananas – the more spotted the better! Bananas get sweeter as they get spotted and brown – so they’re perfect for baking. Blueberries : I prefer fresh berries in this recipe, but you can also use thawed frozen blueberries. Vinegar : Mixes with the milk to create sour milk, which helps the bread rise – the acid activates the baking soda. Milk : You can use any kind of milk or nondairy milk. How to make The Best Banana Bread Preheat oven to 375°F and prepare your pan. Whisk together the all purpose flour with salt and baking soda in a large bowl. Set aside flour mixture. Mix milk and vinegar; set aside. Cream butter and sugar until fluffy, then mix in mashed bananas, eggs, and vanilla, then add in dry ingredients and milk mixture. Fold in 2 cups blueberries. Pour into pans, sprinkle with cinnamon sugar, and bake until a cake tester or toothpick comes out clean – the loaf will be golden brown. Expert Tips Be sure to read the FULL recipe and tips before starting! If you are using frozen blueberries , be sure to thaw them first and drain them well. I like to sugar my pan – this is like greasing and flouring (where you add 1-2 tablespoons of flour to a crisco or butter greased pan) but instead of flour you use granulated sugar. This is optional but adds amazing flavor. You can use a stand mixer or hand mixer to make this recipe. This will also make 3 mini loaves – you can use 3 mini loaf pans in place of one large pan. Store banana bread wrapped in plastic wrap on the counter for up to 3 days or freeze it for up to 3 months. FAQs Can you freeze this bread? Yes! You can freeze this bread recipe in an airtight container for up to 3 months. Why is my banana bread not done in the middle? Quick breads take a long time to bake, and that center crack in the middle is the last to finish. If your bread isn’t done all the way, the center might stay gooey, or the bread will fall as it’s cooling. If you notice the bread getting too brown, tent it with foil or reduce the oven temperature to 350°F. Bread is done when a toothpick comes out clean from the center of the crack/very center of the bread. Blueberry Banana Bread Recipe with Cinnamon Sugar Topping Equipment Ingredients Bread: 2 cups all purpose flour 1 teaspoon baking soda ¼ teaspoon salt 7 tablespoons (105ml) milk (see note) 1 teaspoon white vinegar (see note) 1 teaspoon vanilla extract ¾ cup (150g)granulated sugar ½ cup (113g) unsalted butter , softened 2 medium bananas (brown spotted, overripe) 2 large eggs 2 cups fresh or frozen blueberries (be sure to thaw and drain frozen berries) Topping: ¼ cup (50g) granulated sugar 1 teaspoon ground cinnamon Instructions Preheat oven to 375°F. Grease a 9x5x3” loaf pan with butter and coat it with sugar. (You do this like you would grease and flour a pan. Grease it first, then add about 2 tablespoons sugar to the pan and move the pan side to side until the bottom and sides are coated with sugar. Do NOT substitute cooking spray for the butter. You can skip the sugaring and just use cooking spray, if you wish.) Whisk flour, baking soda, and salt in a medium or large bowl. Set aside. Place milk, vinegar, and vanilla extract in a measuring cup and set aside. Cream butter and ¾ cup sugar with a hand mixer or stand mixer. Mash bananas with a fork and add to the butter mixture along with the eggs. Mix until chunky. Add the dry ingredients into the banana mixture along with the milk mixture. Mix until just combined (batter will be slightly lumpy). Fold in blueberries. Pour batter into prepared pan. Combine 1/4 cup sugar and cinnamon in a small bowl. Sprinkle evenly over the top of the batter. Bake for 45-50 minutes until a toothpick comes out clean from the center of the bread. The edges will be a dark brown and there will be a nice crack down the center. Cool completely before removing loaf from pan, but you can cut slices from the pan after it’s cooled for about 15-20 minutes. Recipe Video Recipe Notes My original recipe (and the video) shows blending the bananas, eggs, baking soda, milk, and vinegar using a blender. You can definitely do this, but I’ve started making my banana breads by just mashing the banana by hand and mixing everything in because it means less dishes. The top of the bread will get deep golden, but if you notice the bread browning too fast, you can tent the pan with foil or reduce oven temperature to 350°F. This happens in various ovens, depending on how the oven cooks. You can use fresh or frozen berries, but make sure frozen berries are thawed and drained. Basically this recipe calls for “sour milk” which I’ve separated as vinegar and milk. It’s essentially buttermilk – the acid in the vinegar activates the baking soda for a moist bread. You can use 7 tablespoons of buttermilk if you have it and omit the milk and vinegar. Don’t have vinegar? Use lemon juice instead. You can also use ANY kind of milk – dairy or nondairy) Pan sizes: a 9×5 pan is the best to use, although an 8×4 will work, but you’ll want to watch it to make sure it doesn’t get too brown on top before it’s done in the middle. You can also make this in 3 mini loaf pans or as 12 muffins. Recipe Nutrition Other Banana Bread Recipes Last Updated on March 10, 2024

Jamie Lee Curtis Pays Tribute to ‘My Friend’ Richard Lewis: ‘Deep and So Freaking Funny’ Anything But Love co-star
Jamie Lee Curtis remembered Richard Lewis, her co-star on the late-Eighties sitcom Anything But Love , and credited the stand-up legend as “the reason I am sober” in a tribute following the Curb Your Enthusiasm actor’s death. Related “I’ve just read that my friend Richard Lewis has died,” Curtis wrote on social media. “I remember exactly where I was when I saw a billboard of him about a stand up special on Sunset Boulevard when we were casting the ABC pilot Anything But Love and asked the casting people to bring him in to audition to play my best friend/maybe boyfriend, Marty Gold.” Curtis continued, “I thought he was handsome. He made me laugh, which is the one thing that a strong, capable woman, can’t really do for herself. He got the part when I snort laughed when he mispronounced the word Bundt cake. He blew everyone else away.” Anything But Love first aired on ABC in March 1989 with an abbreviated first season, after which most of the sitcom — except for Curtis and Lewis, who developed an on-screen chemistry — was overhauled. “They came back to me and said that the chemistry with Richard was so great and could we revamp the original pilot which is the show we ended up making for a couple years,” Curtis added; Anything But Love ran for four seasons, ceasing production in 1992.

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Outside of Curb , Anything But Love , and films like Robin Hood: Men in Tights and Leaving Las Vegas , Lewis was a beloved comedian. “He was also a stand-up comic and hated the live audience, where I, who had never done a play, loved it,” Curtis wrote, noting how Lewis “used to hide his lines everywhere on the set, on props, door frames, on my face in a close up and was always carrying a clipboard with his lines on them. It turns out he was a wonderful actor. Deep and so freaking funny.” Curtis added that Lewis’ last text to her was lobbying for her to convince Disney/ABC to put a second volume of Anything But Love episodes on DVD.

Curtis also credited Lewis, who was open about his own struggles with addiction, as “the reason I am sober.” “He helped me. I am forever grateful for him for that act of grace alone,” Curtis said. “I’m weeping as I write this. Strange way of saying thank you to a sweet and funny man. Rest in laughter, Richard.”