Hello. Excuse me. My name is Alicia Walker. I am the Interim Chair and Professor of Medieval Art History at Bryn Mawr. I'm so pleased to be here today to introduce my student, but I would first just like to thank the co-organizing institutions Temple and especially Penn for this wonderful event. I'm so happy to be here and to hear all these wonderful papers. So I'm pleased to introduce Shannon Steiner, who is in her final stages of the PhD at Bryn Mawr College. Shannon completed her BA down the road at Temple University and her MA at the University of Texas Austin before coming to Bryn Mawr. Her work is focused on issues of materiality in the devotional objects of early and middle Byzantine society and she has researched and published on tokens from the pilgrimage site of the St. Simeon and on gold glass medallions embedded in the walls of early Christian catacombs in Rome. Today she will present aspects of her dissertation research in which she is exploring the technical and ideological aspects of Byzantine enamel and how the understanding of enamel is a technical scientific art informed its meaning in Byzantine religious and political culture. Since her first year doctoral studies, Shannon has held the prestigious Saratay Graduate Fellowship at Bryn Mawr and in addition she has pursued her dissertation research with the support of a two-year Crest Foundation institutional fellowship at the Central Institute for Kunstgeschichte in Munich, a Mary Jeharis Center for Byzantine Art and Culture Dissertation Grant and the Summer Institute in Technical Art History participation in that program which is sponsored by the Conservation Center of the IFA. She currently holds a junior fellowship in Byzantine studies at Dunbart-Nokes in Washington, D.C. where she has her nose in the books and her fingers on the keyboard as she completes her dissertation research and writing but where she is also engaged in hands-on work with a group of contemporary metalsmiths to explore the technical aspects of Byzantine enamel work through the reconstruction of medieval artisanal practices and the production of original enamel objects using these historical methods. Her talk today is titled, Reified Knowledge, Byzantine Enamel as a Staticized Technology. I mean just a second I lost my microphone when I stood up. A little bit awkward but we'll work through it. All right, I'm giving up. Thank you Professor Walker for that introduction and I'd also like to extend my thanks to the Barnes and to Temple and University of Pennsylvania in addition to Bryn Mawr for organizing this and especially to Bryn Mawr for their support. So renowned for its sumptuous beauty and radiant brilliance, cloissanet enameling is a technique distinctive to Byzantine art production and its use was kept restricted to the highest echelons of Byzantine society. Byzantine goldsmiths were matchless in their ability to fuse vividly colored glass to golden plaques divided into cells by delicate strips of pulled wire. The French term cloissanet refers to these cells or cloissons which render the complex designs of this medium within a shimmering tracery. Soon after enamel production began in earnest around the mid 9th century goldsmiths rapidly developed an unprecedented range of glass colors often varying its transparency and translucency from one cell to the next. Wire was thinned to a minute width that allowed for pictorial compositions unparalleled in their degree of detail which you can see here which is itself a detail of a much larger object. This is a very small section of it. Each of these innovations required deep familiarity with the properties of natural substances such as gold, metal oxides and salts and even fire and heat. Enameling also demanded proficiency in manipulating those materials into impressive displays of artistic ingenuity. In my paper today I consider how Byzantine enamel could bear unique meaning through its display of the rarefied knowledge required to produce cloissanet and the skill evident in its execution. It is important to note that while scholars often describe enamel as a material in its own right, it is more accurately defined as an artistic process. Following the work of historians of early modern art and science, most notably Pamela H. Smith, I consider processes of making to be crucial occasions for the creation and demonstration of knowledge. Knowledge of materials, of iconography and a self aware knowledge on the part of the artist of his own ability to overcome the challenges presented by both, something we would call skill. Within this understanding, I posit that Byzantine enameling was aestheticized technology. Indeed, the Greek roots of the word technology embody this understanding. Technique meaning art or skill and logos meaning knowledge bespeak the processes of display and making that I argue are realized in the visual and material properties of Byzantine cloissanet enamel. Technology and process where the concepts through which enamel was defined in Byzantium, the terms for enamel in Byzantine literature, himmephton and area himmephtah derive from the verb heal to flow poor or melt. They can be translated literally as melted things or melted work. However, the stem of the words and their etymological origins link them to himia and hemesis, the medieval Greek terms for alchemy. An alternative translation of Byzantine enamel vocabulary might then be things by himmephtis or more simply alchemical things and alchemical work. The relationship between enameling and alchemy in Byzantium was more than just etymological. Byzantine texts on the subject of alchemy survive and they contain recipes and instructions for enameling that scholars have noted, but that have not been discussed in any great detail. In the 19th century, French chemist Marceline Bertelot compiled, edited and translated the collection des enseignes alchemist Grec, known today as the Greek alchemical corpus. The manuscripts range and date from the 11th to 15th centuries. Each is a compilation or miscellany rather than a single authored composition. And the texts contain within them date through the second through 13th centuries. Bibliotheca Marchiana's Greicus to 99 and Paris Greicus 2327, shown here with their really fantastic depictions of alchemical imagery and apparatus are the two best known Byzantine alchemical manuscripts. Now, defining exactly what the practice of Byzantine alchemy entailed is difficult. The education and social demographic of Byzantine alchemical authors also remains mysterious, but the practical technical nature of many of these texts suggests that the authors observed working artisans and may have even collaborated outright with them in experimental endeavors. The subject of chrysopia or gold making abounds within the alchemical corpus, but in late antiquity in Byzantium, alchemical practice was not restricted to transmuting based metal into gold. Interspersed throughout are a wide range of recipes and commentaries for processes involving some kind of physical transformation, such as making silver, fabricating gemstones and pearls, making purple dye, baking bread and creating explosives. Most importantly, from my study of Byzantine enamel, much of the alchemical corpus examines how to artificially reproduce physical qualities and behaviors of natural matter. Now, artifice is the key term here. Artifice and artificiality today carry strong connotations of inauthenticity and even deception. Something artificial can't be genuine, and what is artificial is often considered antithetical to nature. In the Byzantine alchemical corpus, we see none of these modern judgments. Instead, what is artificial is that which is made through human action and human intention, rather than spontaneously occurring in nature. Far from being at odds with nature, Byzantine alchemical writings often signal that man-made replications of natural phenomena were proof of fully comprehending and even mastering nature. The enamel recipe that I will speak about today is especially evocative of this belief in the power of artifice, a belief that had a long history in the Byzantine world stretching as far back as the first few centuries CE. One of the earliest proto-alchemical treatises to survive from late antiquity is the fourth century Stockholm papyrus, a compilation of instructions on making gold, silver, purple dye, and gems. One of the most compelling features of the Stockholm papyrus is the author's unabashed celebration of the artificial. In the course of listing ingredients and methods for coloring glass and quartz into convincing imitations of rubious sapphires and emeralds, the author of the papyrus often states that the final product will surpass the natural kind.  Yet glass enhancements and imitation gems appear with such regularity and sophisticated late Roman and Byzantine goldsmiths work that they seem unlikely to have gone unnoticed. Colored glass could be adhered to valuable rock crystal to enhance the visibility of carving and intensify the natural crystals reflectivity as seen in a pendant now in Dunbarton Oaks. This practice is known in goldsmithing as creating doublets and is an artificial intervention made for the purpose of enhancing a stone's natural visual qualities and preventing scratches and breakage. Artificial glass emeralds are incorporated alongside genuine garnets and sapphires on an intricate 10th century earring or diadem ornament now in Athens. And a recent analysis of a 7th century jewel re-hoard using Roman spectroscopy revealed gemstones previously thought to be emeralds and lapis lazuli to in fact be glass imitations. So the necklace is genuine emerald but the the earrings up here, those are glass. It is possible that the makers and owners of these expensive luxury jewels were duped by some dishonest artisan but a more intriguing possibility is that artificial replications of precious natural materials had a value and power of their own. This notion lies at the heart of a hitherto unidentified recipe for enameling in the Greek alchemical corpus and these are the two manuscript pages of the beginning of this treatise. Imitation gems are the central focus of this treatise elaborately titled The Deep Dying of Stones, Emeralds, Rubies and Sapphires, according to the book taken from the inter-sanctuary of the temple and that's significant but I don't have time to talk about it today. The treatise contains certain Arabic loanwords and other philological signatures that place its date somewhere between the 8th and 10th centuries, a moment when the volume of Byzantine enamel production intensified. The text is a long anthology of recipes for artificial gems attributed to semi-mythological alchemists such as the Egyptian priest Ostanis or a Jewish female alchemist named Maria. The text is arduous and somewhat fragmented in its current state of addition but its format is very regular. Each recipe for a stone is preceded by a theoretical explanation of a stone's natural properties. In most cases, color is emphasized as a distinguishing feature and then this is then followed by a list of ingredients that can approximate that effect when applied to either glass or crystal. In other words, the recipes are offered as an explanation of how to reproduce the natural phenomena described above it. The instructions for enameling are preceded by a theoretical investigation not of color but of luminescence. Concerning the making of emerald, just as Ostanis, the keeper of all things, Arcane, thinks, the ingredients used are copper rest and the bile of all sorts of animals and things like that. For sapphires, the ingredients used are the hyacinth plant and woad root boiled together. For rubies, the ingredients used are alkanet and dragon's blood. That's a resin, not actual dragon's blood. It's a bit clearer in these things. For the rubies that is called both night shining and sea dyed, the ingredients used are the bile of sea creatures, fish or whales, on account of their luminescing at night and their lustrous gleaming as their entrails, their scales and their bones clearly shine at night. This passage summarizes the ingredients of the recipes that came before it, before launching into an evocative description of a naturally fluorescing stone and the bioluminescence of sea creatures. While the author focuses on the replication of color in ordinary emeralds, sapphires and rubies, his concern with this mysterious night shining ruby is its autonomous generation of light. He continues. But from where do the stones receive their fiery brightness? Neither bile nor copper rest have the ability to grant this property, being green by nature. What do we say? Did such an important process elude Maria? No. She wrote on fabricating rubies. She also described it above. Here the artisan explains to us that the most precious stone appropriate to make and die is that which at night emits rays of light in the matter of the sun and those being in possession of it have the ability to read and write and do anything nearly as well as they do in the day. For each ruby has the property of shining at night according to its particular size and clarity, whether it is natural or artificial. But the property of shining at night belongs uniquely to the night shining ruby. Ruby, excuse me. So how to make sense of this description of a supernatural shining stone? What's this business with fish bile and what has it got to do with enamel? First, it's clear that the author of this text was not an artisan and was instead collecting knowledge of the causes of interesting natural phenomena and the techniques for reproducing them. Second, there is a pattern of material sympathy in the ingredients. So, for example, by the author's logic, the fact that some sea creatures are bioluminescent proves that the stone's luminosity is natural, not abnormal, and that the addition of fluids from those animals to a process would help to synthesize that effect in the final product. And this passage is not a complete flight of fancy. Given some clues in other parts of the passage where the ruby that shines at night is also called a sapphire, I believe that the author is describing fluorite. When seen in daylight, fluorite has a varied blue, green, purple hue. After hours of UV absorption, it glows a bright watery green. While no Byzantine objects made of fluorite survive, the stone was well known in classical antiquity and was popular for its use in luxury objects such as the first or second century Crawford and Barber Cups, found in a tomb in Anatolia where the stone is plentiful. However, I do not believe that Byzantine enamel was understood as artificial fluorite in a literal sense. It's not the stone that fascinates the author of the treatise, but rather the fluorite's natural quality of emitting its own light. This is a wondrous yet natural phenomenon that the stone shares with bioluminescent sea creatures, which the author identifies as particularly valuable. The recipe for enamel that follows prioritizes the reproduction of bright light over all else. It reads, take three liters of lead oxide, one liter of clear glass, two hexagia of tin and about a poos of finely ground sulfur. Put them in a clean small cup and heat them over charcoal until it becomes green glass. If the heating is extended, it becomes golden. If extended even longer, white like crystal. This recipe describes how to color the glass that an artist would grind and mix with water to fill the cells of an enameled work. The dominant additive to the glass is lead oxide, an ingredient that not only improves the feasibility of glass, but also dramatically increases its transparency and refractive index, allowing light to bend rather than just reflect and thereby enhancing effects of luminosity. With this in mind, a thoughtful arrangement of glass in relation to the direction of the reflective metal around it can capitalize on very little light in order to artificially simulate the autonomous glowing of a natural stone. This is seen especially in the brilliant green glass so prevalent in Byzantine enamels, exemplified by a mid-ninth century reliquary of the true cross. The glow of Byzantine enamel is difficult to photograph, not least of all because it's difficult now to see them in low lighting, but I attempted to generate this effect in the image on the right through very dim, indirect light. Under these conditions, the placement of the wire that formed inscriptions caused the letters to dissolve into unexpectedly, almost disproportionately bright light behind the heads of the saints that populate the reliquary's lid and sides. This technical strategy persists in the most accomplished of Byzantine cloisonnay, for example in the 11 extant medallions from an icon frame. X-ray imaging has revealed the presence of wire hidden beneath the surface of the glass fill in all of the medallions, seen here on the medallion of Saint Demetrius in the saints halo, hair, cheeks, and hands. In addition to securing the glass flux around them, the wires in the halos achieve the same refraction of light seen in the true cross reliquary when the halos are filled with green glass. But in this instance, the light follows the graceful loops and swirls of the wire between minuscule crosses and tiny circles. In both the reliquary and the medallions, the enameling goldsmiths demonstrated a consciousness of the kinds of natural qualities the Byzantines valued knowledge of and deployed their own skill and knowledge of materials to bring those qualities into existence. Here, enamel does not merely imitate a stone, but rather exhibits Byzantine understanding and control over a particularly powerful attribute of a stone by manifesting it through man-made means. To conclude, my approach to Byzantine enamel introduces a new understanding, not only of the medium as a material, but to the medium as an alchemical process. In objects and in texts, enamel embodied and offered proof of Byzantine knowledge of how natural materials and their physical qualities worked. Knowledge, understanding, and reproduction of nature was a primary goal of Byzantine alchemy. This allows us to understand enameling as a technological process of artificial reproduction, one that was part of a larger endeavor that could take nature itself and rearrange it into new configurations that highlighted Byzantine control and mastery over the natural world. Thank you.


Welcome to The Journey. Today we're going to talk about how to get customer feedback for your business. Think about collecting customer feedback for your business, which I hope you're doing all the time. It's easy to get overwhelmed by the volume of possibilities, trust me. And with so many customers and so many ways to connect with their feedback, it can be hard to know where to start. One thing is clear, taking a proactive approach to collecting customer feedback ensures that you never stray too far from the needs of your community, even as those needs evolve. Feedback is a powerful guide that can give you the insights that chart a path forward for every aspect of your business. A customer survey helps you harness those insights and identify ways that sets your business apart. It's important to develop customer surveys that collect and show useful insights and also get shoppers and clients to complete your survey. And we're going to show you how. Let's start with tips for creating a customer survey. Well, before you start writing your customer survey questions, here are some best practices to keep in mind. I would say, first off, know your goals before you begin. It's good advice for most beginnings and don't blindly create a customer survey just because you know you should. Instead, consider what you want to gain from the survey. And I would suggest outlining, you know, your business goals before you begin and ask yourself a few questions. First off, hey, do you want to gather feedback to improve a product or service? Ask yourself, do you want to learn about your customers? Another great question. Do you want to improve the customer experience? Knowing what you want to learn will help you to craft better questions for your survey. Exactly. You also want to consider all question formats. As you develop the questions for your survey, think about what format might help you collect the best feedback. Use a mix of multiple choice, yes, nos, open-ended and even sliding scales. To gain additional insight, also be sure to ask follow-up questions. Definitely consider that. Also make sure that they're relevant. So, for example, follow up with a yes or no question with an open-ended why question. Another way to gain better insight is to ask how questions. Don't ask, hey, do you like the product? Instead, ask how well would you rank the product? Or maybe you could ask, how would you describe the product? So, Nealey, I got a how question for you. How well am I doing explaining how to get customer feedback for your business? Is there a non-answer or other? Wow, Nealey. Well, it's a good segue. Speaking of non-answers, this is something you should also include in your survey. If a survey taker must select an answer, allow them to choose a non-answer if they just don't want to or maybe they can't respond. So, essentially, add options for like, I don't know or other. Kind of like what Nealey wanted to do right now with my question. So, my answer remains, I don't know, Emma, but don't lead on survey takers is our next tip here for you. To really get the most useful and truthful answers, you're not going to want to put words in the mouth of your survey takers. Positive feedback is great. We love to hear it, but only when it's true. So, don't use leading questions that guide your users to say what you want to hear because if there's information that you need to change and you'll never hear it, you'll never change and your business will never improve. Also, another great tip, be specific. You know, you can gather better insights if you are more specific in your question. And I have an example of this. For example, don't ask, how would you rate the overall customer experience? That's pretty broad. I would encourage you to be specific and ask something more like, how would you rate the onboarding process? Now that we know what goes into the survey, let's talk about how to get people to actually fill them out. You can spend a lot of time perfecting this questionnaire, but there's still a good chance that not everyone will want to fill it out. So, let's make sure you're getting a fair ROI. Starting off with just keep it short and sweet. You don't want your survey to feel like work for your customers. People have a lot of other things that they need to do, so keeping it short so it doesn't take too much time and energy to complete. And when you conduct a survey, try to keep it to about five to 10 questions max. Great point, Neely. Also, be upfront about the length and time it'll take to complete. I know for me when I see a survey, oftentimes I'm a little hesitant, but if I see, oh, this is only going to take five to 10 minutes, done. Happy to proceed. And right away, you know, you're letting customers know what is expected from them before that survey starts. So, in other words, you're setting expectations, let them know even how many questions are included and how long it will probably take them. Love that. Let me know upfront because I don't want to go on and you're like, all right, next page. Next page. Next page. How long is this thing? But with your survey, make sure you show the benefit. Show how this survey is going to benefit your business and ultimately, them too. Let them know what you're going to do with the results and how you will use their insights to improve. So, consider using language like help us improve. Fill out this survey to tell us how we can get better at, insert your service here. How's we wondering, Nealey, do we have a survey for the journey? I want to know, how are we doing? Which takes us to our last point. Ask at the right time. Be mindful of the customer's journey with your brand as you decide when to present shoppers or clients with the survey. Time means everything, right? So, for example, you might not want to push a survey on a customer only a few minutes after they do business with your brand for the first time. By following these tips and best practices, you can create a customer survey that will help you gain valuable insights about your business. You can use what you learned to improve your brand and create better products, services, and experiences. You just learned how to get customer feedback for your business and your journey is not over there. So, be sure to check out over 200 additional videos that will help your business thrive. This is a journey.


In SOFA statistics, you can either enter data by hand or you can import it from a spreadsheet program. You might wonder why you would ever want to enter data by hand. If you've given a survey on paper or you need to enter the results of an observation that you did in the field, entering data by hand could be your best choice. To enter data by hand, click Enter Edit Data. You can then add a new data table by clicking New. Enter the name for your data table. In this case, Survey will be a reasonable name. Now you can start designing what goes into your data table. The first field is going to be the gender, so instead of R001, type Gender. That's going to be a letter, not a number, so go to the Data Type column and select Text. The next field will be for the age, so click the empty field name and type Age, then tab over and select Numeric. Click outside the menu and then press Tab to create another field. You don't want the height in feet and inches as separate fields, so you'll combine them into the total height and inches. Enter the field name Height, which is numeric, and I got that quickly by pressing the letter N. Click outside, press Tab, and now the next is the number of pairs of shoes, so the field name is Shoes, and the field type is also numeric. The field for the favorite color, which we'll call Color, is going to be Text, so I'll select Text from that field, and the last two questions are coded as Numbers, so the fields Campus and Pets should both be numeric. Make sure you press Tab to create an empty row, or your last field won't get into the table design. Finally, click Update. You'll see a message that your new table has been added to the default SOFA database. When you click OK, you can start entering your data. I'll enter a couple of rows just to show you how that works. I enter the gender for the first person, M, and use Tab to move from field to field. The person's age was 23. His height was 5'7", so I convert that to 67 inches total. The number of pairs of shoes he had was 10. His favorite color was red. He thought the Campus was very beautiful, which scored 4 points, and agreed, but not strongly, that everyone should own a pet. That was scored as a 3. When I press Tab, it immediately goes to the next row. The next person was a female, who has aged 22. She was 5'3", 63 inches total. She owned 14 pairs of shoes. Her favorite color was green. She liked the Campus a little, but disagreed that everyone should own a pet. Again, make sure that you press Tab at the end of every line, so that the data gets entered into the table.  And then when you're done with that, you click Close again, and you now have a data table entered by hand, ready for you to use.


Yeah, I'm 24. All right. Let's do it. No. I'm not that young. Danny Granger. I saw a dancer. Trill Spreewell. I don't know. I like this one. 24 a different century. All right. All right. Maybe sad. All right. Yeah. Oh, mine is perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect. That's perfect.


Hey everybody, Lisa Kearney here with Fandle, and today we're gonna talk about simple horse bets. Simple bets or straight bets are pretty straightforward. And the simplest of them all is a win bet. You just have to pick the horse that's going to win. Another really easy bet to make is to bet on a horse to place or show. A place bet means that your horse can finish in first or second. While betting to show means your horse can finish anywhere in the top three. And if you like all three of those options, another easy bet to make is an across-the-board bet. This means your betting win, place, and show all at once to cover all your bases. So simply put, that's how you place simple horse bets. If you liked this video, be sure to check out the rest of our Racing 101 series.


OK, zelo povedaj. To je pravda, da imamo Korina Učigaraj, kaj imamo povedaj, kaj je zelo povedaj, in Lagrangi Spektra. Prečo, prečo, povedaj, da se povedaj, da je na webače google doc, document, kaj vsi vse povedaj, da se povedaj, nekaj zelo, nekaj se povedaj, je to, da se da vsegače. Na sada je, da je bilo v tomom, da se prišli početno, in bomo razgledati, da se pače. Če vidim, da se tače, da se prišli početno, da se početno, da se prišli početno, da se prišli početno, da se prišli početno, So, as some of you know, I'm actually originally from Trieste, so it's a great, even though I left when I was 18, so before I had scientific contacts with the city and the institutions of the city. So it's a great pleasure for me to be back in Trieste and give a talk in my home city. So I've been to some of the dynamical system conferences that have been given here, so what I do is more dynamics than geometric theory, but I try to find some topic which has more geometry and some groups in the talk. Ok, so first I want to remark that the work that I'm going to talk today about two works, which are joint works. The first is a joint work with Diana Davies and Irena Pasquinedi, and this is a work on cutting sequences on which surfaces. I'll tell you what this means in a second. And the second is a joint work with Mauro Artigiani and Luca Marchese. So I'm going to put the names here, I should have done it earlier. So these two works, both are works on which surfaces, on dynamical properties of which surfaces, both of which involve the study of the which group action on the space of affine deformations of a which surface. So that's why I'm going to talk about them together. And the outline, I will spend some time at the beginning because this is not a common background for this type of audience. I'll spend some time about basics on what is a which surface and which group. Then I'll introduce the first result on cutting sequences. I'll introduce and motivate the second result on Lagrange spectra, which are also in the title I had penetration spectra. It's actually the same thing, Lagrange penetration spectra. And then I'll try to give you some ideas of the proofs of the two results. So you will be back and forth between the two topics and you have several locations to wake up and catch up if you got lost in some part. OK, so time to do some background. So I know that the students who were here last week had a course with Pascal Ubert. So you're probably all well-prepared on what our translation surface is. So actually the best description for me is the easiest. So you take a polygon which has pairs of parallel sides, isometric parallel sides, glued together like A with A, B with B. So in this case we have a surface of genus 2 glued out to a regular octagon. So this is the regular octagon surface. And whenever you get a surface out of gluing polygons, your surface naturally carries a flat metric. So apart from possibly finitely many points which come from the corners of your polygons. And this is the set of singularities, sigma. So outside is finitely many points. In my case all the vertices of the octagon are a unique point on the surface. Outside this point you have, other point has a locally Euclidean neighborhood and changes so coordinates in your manifold definition are given by translations. In complex coordinates z goes to z plus c. So this is why we call those translation surfaces. And the singularities, the special points, we have an excess of curvature because we are on the surface of higher genus so there should be some negative curvature. And these points are conical singularities called, what the model if you want is z goes to z to the k. So it's a k cover of the plane. So on translation surfaces there is a well-defined notion of lines. So the geodesics are flat lines and you can talk about directions. So I can say there is a line in direction theta. Direction is well-defined. Apart from the singular point. At the singularity, so what's happening? I'm right too far, yes. At the singularity there could be many lines in the same direction which hit this point. So this point you can think of it as a saddle on your surface. So these are how linear trajectory looks on the surface. And again we are thinking of surfaces not polygons. So two polygonal presentations of the same surface are two polygonal presentations are equivalent in the same surface if you can cut and paste your polygons by translations. For example, in my picture you see you can take this surface and cut and paste the polygon and you will get again the same flat metric on the same surface of genomes too. OK? OK. So if we have two translation surfaces we can look at affine difiumorphisms among them. OK. Maybe I should have said that affine... I missed something. It's a continuous map. It's a homeomorphism which actually maps singularities to singularities. And it's affine in each chart. So it's a linear. It has a linear part plus constants. And the linear part is independent on the point. And I can look at self-affined maps from a surface to itself. Affine automorphisms of S. And the group... This is a group of affine automorphisms. And the which group... I'm too far. The which group is the group of linear parts. So I look at these affine maps and they look just at the linear parts. So a linear part will be given by a matrix. And this matrix is... Affine difium... automorphisms have to be area preserving. I'm not assuming necessarily they are orientation preserving. It's not standard. So there are actually matrices in SL2R plus or minus. So 2x2 matrices will determine plus or minus 1. Possibly I can also use reflections. OK? So let me give you an example. So if you take the square torus like here I claim that the matrix 1101 is in the which group. So if I apply 1101 that's actually a shear. Or if you want it's a full then twist in the square. But after I shear my square I can actually cut and paste it back to square square. Parallelogram can be cut and pasted back to a square. So this shows that this gives me a self-automorphism of the torus into the torus with derivative 1101. Here's a more sophisticated example. So if you take the regular octagon I claim that this matrix is in the which group. So this is also a shear where here the upper entry is here. So if I apply to my octagon this matrix what I get is a shear octagon. It's this black figure here.  Because I want to show you that like in a jigsaw puzzle you can rearrange these pieces of the shear octagon back into the standard octagon. I'm wondering whether Pascal Ubers show this example in the course. So this matrix gives you a fine automorphism.  So we want to focus today on gravitational surfaces which have plenty of self-affined automorphisms. Those are vich surfaces. So a surface we say that it's a vich surface or also lattice surface if the vich group is a lattice in SL2R. So again same two examples. If you take the torus you can add to this you can either add the other then twist, vertical then twist so this is actually a rotation and you get the full SL2Z which is a lattice in SL2R. And for the octagon you can take for example, you can take here I took two, I like these reflections which are determinant one. This is a horizontal reflection this is a slanted reflection so all isometries of your polygon will in particular give you a fine self-automorphism. Izometries and then I can put this non-trivial element that we saw together and this will generate a lattice in SL2Z. So the regular octagon surface is a vich surface and this one. So this is a little bit of introduction because it motivates people. People have, like molar dynamics are very interested in vich surfaces because they are interesting from the dynamical point of view that it is via dichotomy that all trajectories are either by infinite trajectories are either periodic or dense and uniform and distributed. I don't want to spend time this is just their motivation. And also if you look at the SL2R action they generate closed orbits known as tec molar curves. I'll tell you more later on in the proof part. And people have tried to list, classify vich surfaces so we saw that the square and the octagon and more in general any regular polygon will give you vich surface and this is a seminar paper by vich where he defined vich groups and discovered vich surfaces. There are some examples which were found by Ward and then a little bit more than 10 years ago there were two infinite families of vich surfaces which were discovered one by Boh and Moller by Boh and Moller this is the Boh-Moller family and another family was discovered independently by Calta and McMullen it's a family of L-shaped vich surfaces in genus 2 and the Boh-Moller surfaces I will focus on them today so I will tell you more about them and then Mokamel gave a talk on Tuesday here I think he didn't say he talked about the gothic locus and in the gothic locus there are infinitely many new vich surfaces which have a cathedral shape and there is lots of research and people working on proving that there are infinitely many with fixed complexity genus and singularities as I said apart from these classical examples the Boh-Moller surfaces have vich groups which are triangle groups while these L-shaped family surfaces have more complicated vich groups as I guess Ronin has studied so I want to focus on the Boh-Moller after regular polygons they are the nicest family and they are obtained by gluing a collection of polygons which are semi-regular so this is all these polygons together glued according to the numbering that you see one with one six with six this is one surface and this is another surface and let me first say pat Hooper polygonal presentation so Hooper discover this simple polygonal presentation of these vich surfaces so first of all vich surfaces Boh-Moller surfaces are parameterized by two indices M and N so the SMN surface the MN Boh-Moller surface is made by M semi-regular polygons each of which has a symmetry of order 2N so this is S4-3 so there are four polygons which have order 6 symmetry so these are semi-regular hexagons so the angles are equal but the sides come in two lengths and these are triangles but they have still order 6 symmetry and I order 3 I think I said it wrong and this is S3-4 where there are three polygons with order 4 symmetry and you have to pick the lengths specifically so they are not any two semi-regular you have to pick them carefully because you want the surfaces to have a global shear like the one in the octagon that I showed you before and maybe I could have said about the octagon that this shear you can think of it as a simultaneous then twist so the octagon has two cylinders and you are doing a then twist in each but the cylinder modular are commensurate so you can do this to simultaneously and here you have cylinders and you want to be able to do a simultaneous then twist in each cylinder to get a globally defined shear on the surface ok so this is as much as I wanted as an introduction I got too far and then I cannot click ok and I want to close maybe ok so now it's the first result I want to talk about it's about cutting sequences so we have a Vich surface and you want to first of all give names to the pairs of sides which are glued for example in the octagon I am using the alphabet A, B, C, D to label the sides and if I have a bi-infinite linear trajectory what am I doing wrong with this clicker I think ok this is an example of a bi-infinite linear trajectory you go out and come back and I want to associate to a linear trajectory a sequence of symbols to kind of give a symbolic coding of the trajectory and this is a sequence in the alphabet A, B, C, D called cutting sequence and the idea is just that you look at the sequence of sides that you hit as you travel along your line so in my example I hit A so I record an A then I hit a B then I hit another B then I hit a C so that's the cutting sequence of a trajectory and I only want to look at bi-infinite trajectories that give me bi-infinite cutting sequences so the problem I want to talk about is how do you characterize these cutting sequences among all sequences in the alphabet and what do I mean maybe I should say that cutting sequences are rare I mean if you flip a coin actually you cannot flip a coin with four letters so if you draw a tetrahedron dies and at random you will get random A, B, C, D but here the sequence that you get is very far from a random sequence because it has a lot of structure and it has actually very low complexity compared to a random sequence so if I give you a sequence how can I tell if it is a cutting sequence and how can I build cutting sequences by drawing the lines by hand and really have issues with this ok, so just to say that this is a problem which in the case of the torus is very classical because in the case of cutting sequences for the torus are actually well first of all you can think of them as just how a line hits sides of a square grid where you record by A and B horizontal and vertical sides and these cutting sequences are called Sturmian sequences they have been studied by headlong and Morse more than a century ago and they have the lowest possible complexity among non-periodic sequences I don't want to tell you what that means so apart from Sturmian sequences there are nice references for their characterization there is one way to see geometrically by Caroline Siris I strongly recommend this paper on mass intelligence and I give it to many of my undergraduate students because it is beautifully written or there is another characterization more combinatorial by Arnou that I will tell you about today and then a few years ago in joint work with John Smiley we managed to give a similar characterization for octagon cutting sequences and two endgones, polygones with even number of sides inspired by this series work and then also Ferenci characterized the language of this cutting sequences for regular polygones and after our work there was kind of a surge in interest so Diana Davis and Diana Davis with daughters they looked first at the pentagon and then at octgones so you have to take two polygones with odd number of sides to glue them together so double octgones and found that our kind of characterization goes through and there were some other works Diana Davis had some partial results on this bone-molder family so what I'm going to talk about the result that I'm going to talk about is that it's about the bone-molder family so which is in some sense the next step up from regular polygones because for regular polygones which group is a triangle group and for bone-molder you have a triangle group with two elliptic points so I will see that later and I think it gives a fairly good intuition of what would be the general picture so the substitutions these characterizations will be in terms of substitutions and what I call esadic systems that's the language that people in work combinatorics like and especially people in Marseille like Karnu also Valerij Bertha ok so let me tell you what the substitution maybe you've seen in your life I don't know so it's a substitution on an alphabet A it's a map which to each letter of the alphabet associates a word so for example these are the two Sturbian substitutions sigma 0 and sigma 1 so sigma 0 maps A to A and B to AB so to each letter you associate a word and this is sigma 1 and substitution then the action extends in words or sequences by just a position so if I want to find sigma 0 of sorry this should be AB I change 0 1 to AB so sigma 0 of AB is sigma 0 of A and sigma 0 of B so it's A AB and so on so you just apply to each letter to each letter you substitute a word that's another example applying sigma 0 squared and then applying sigma 0 squared to sigma 1 cube and by applying sigma 0 actually this is a piece of a Sturbian sequence ok so people often look at sequences which are fixed by a substitution so sigma of omega is equal to omega but if you have a word which is a fixed point of a substitution you can actually produce it as a limit by taking some letters you have to choose the initial letters suitably and applying the substitution many, many times so as you apply the substitution more and more times you get longer and longer blocks sigma to the N of L and you can imagine if this longer and longer blocks stabilize and converge the limit sequence you can converge towards the limit and you call this kind of the what do you mean here I am just talking about substitutions so there is a linear flow I am going to study maybe I am going to far ahead let's talk about it ok but I don't want to talk about fixed points of substitution but I want to talk about sequences generated by finitely many substitutions so these are acidic so as I take s to be a finite set of substitutions and then look at products of this finitely many substitutions so you can look at limits where I take some pick from this set substitutions and apply them and then I get longer and longer blocks I can apply however I want my substitutions and if I get something which stabilizes and actually I do if you grow your sequences towards the middle so you you will get sequences which actually stabilize and converge towards the limit so if a word has a limit can be obtained by longer and longer products of these set s substitutions I say that the sequence has a acidic expansion so acidic expansion so sturmian sequences can be characterized by acidic expansions so a word is a sturmian word or it's a cutting sequence of the square actually I should say it's in the closure so we always characterize the closure of the set of cutting sequences if there exists a sequence of integers such that the word infinite sequence has an acidic expansion and here I'm alternating sigma 0 and sigma 1 sigma 0 and sigma 1 are the sturmian substitutions that I showed you before and I alternate first I apply sigma 0 a 0 times then again sigma 0 a 2 times and so on and this is if I can write every sequence as a with an acidic expansion I say that this is an acidic characterization and I should say this word acidic was due to ferensi and it reminds of Varshik and this ak actually have a meaning dynamically they turn out to be the continuous fraction entries of the slope of your line in the plane and they can recognize them combinatorially ok so this is the type of characterizations I'm gonna look for and as I said the first main result is about bone moller cutting sequences so if you take a bone moller surface and label the sides in this joint work with Diana and Irene we show that you can actually write n-1 times n-1 substitutions and we have a recipe to actually produce them so that a word infinite sequences in the closure of the cutting sequences on the bone moller surface if and only if there are in the actually this I parameterize these substitutions on two indices for reasons that maybe we'll be here later and if I have an acidic expansion with the substitutions so this is an acidic characterization and you can use it to actually have a recognition algorithm to know if a finite block is actually a part of a cutting sequence or not and you can recover this indices can be recovered combinatorially and you can interpret them in a continued fraction generalized continued fraction ok so this was the first result I hope you are still with me so you can wake up if you are lost because there is a new topic so which is the second result on which surfaces and then we'll go through the tools in the proofs so this is the Lagrange spectrum part so are a generalization of a classical object in number theory but I'll first give you the general definition that I want to use and then tell you what it is in the classical case so it's again a geometric object for me so I take a translation surface and saddle connections and the translation surfaces are lines which go from a saddle to a saddle from a singularity to singularity without hitting singularities in the middle and if I have a saddle connection I can represent it in the plane like a segment and the displacement vector or holonomy vector is just the difference between the end point so it's really a this vector in the plane so this is the holonomy of a saddle connection or displacement so if I fix a direction on my translation surface and I have a saddle connection I want to define one more thing the area of saddle connection in that direction so I'm gonna take this area to be an area of a rectangle which has the saddle connection as a diagonal and the sides are parallel to theta and theta orthogonal so this area is the area of the saddle connection in the direction theta the light blue area and if you want you can rotate the direction to make it vertical and then it's real time imaginary part ok so what is the Lagrange spectrum so if I fix a direction theta I want to look at saddle connections which are more and more in the direction theta and look at the areas of the saddle connections and take the lean soup of one over the area so maybe I have one better picture so as I take saddle connections which are better and better approximations and look at the rectangles which I can inscribe and take these areas and I want to take the smallest areas as I grow so to make the lean soup big I need the area of one over I need the area to be small so I'm searching for saddle connections in that direction with small area and this is the Lagrange value in the direction theta the Lagrange spectrum is the collection of Lagrange values as theta changes so it's going to be a set of numbers yes of course I should have said that so these rectangles when I draw a rectangle I mean so there are no actual flat rectangles so there are no other singularities but actually actually maybe you don't need to require it here but in reality the saddle connections which minimize the area are the ones which come with an embedded rectangle ok ok, so there is a more general definition for a saddle to are invariant log chi of translation surfaces that we gave in joint work with Kuber and MacCazer but no this is a special case and this is a generalization of the classical Lagrange spectrum so if you take your surface to be the square the unit square surface then it turns out that this is a well studied object which has also the definition in terms of dieofantine approximation and that's maybe the you can look at denominations of continuous fractions there is a dieofantine approximation and there is also a geometric meaning in terms of penetration spectrum so if you look at the modular surface with hyperbolic geodesic flow actually you can take a ray geodesic ray in a certain direction theta and you can look at excartions into the casp so you want to know how far you are going in the casp and you can measure excartions through a proper function which goes to infinity in the casp and I want to use the saddle connection flat system so this is the shortest length of a saddle connection on the torus and these values can also be computed as the lim sup of 2 over sisto squared along my tec molar ray so I look at 2 over sisto squared so remark actually now that the typical geodesic will go arbitrarily far in the casp so the system will go to zero and this number will go to infinity so this value is actually infinity for a typical geodesic the only geodesics for which it's finite are bounded geodesics or closed geodesics so closed geodesics stay in the casp and then what you are measuring is the maximal asymptotic excartion so you are kind of measuring the diameter of the limit set of your geodesic so this is why this spectrum is called the penetration spectrum ok this is, if you didn't get it, it's ok it's just a motivation about the classical case so what is known about the classical spectrum this classical Lagrange spectrum can be studied for more than a century so we said it's a subset of real line union infinity and for most values it's actually infinity but people, I mean it's closed and it's the closure of quadratic irrationals which correspond to closed geodesics on the modular surface and the minimum it's square root of 5 it's called Hurwitz constant so there is a minimum it's my blue dot to the left in my line real line minimum is there and the beginning up to 3 there are discrete values so those are called Markov numbers then that's what I want to care about today from some point on this Lagrange spectrum contains a full interval so from a value actually the best value is the Friedmann constant from here on the full semi line is contained in the Lagrange spectrum and what about in between the discrete part and the interval interval is called Hall Ray and in between the structure is quite complicated it's like a counter like structure and there are recent results by Moreira on the household dimension so this is the picture of the classical Markov spectrum discrete beginning, Hall Ray and complicated middle so there were many generalizations of Lagrange spectrum other fuchsian groups higher dimensional hyperbolic many folds, bianchi groups by many people and for translation surfaces some properties we proved in this first paper with Kuber and Marquesa and for example the Hurwitz constant was recently announced by Bošanica and Delacroix so what I want to talk about today is the Hall Ray so what we proved with Mauro, Artigiani and Luca Marquesa is that this Hall Ray phenomenon persists for any Vich surface so take any Vich surface look at the Lagrange spectrum then the Lagrange spectrum of any Vich surface will contain a half ray so there will be an R from which all values are achieved actually the proof uses Vich groups and arguments by Hall and symbolic coding but I hope to get there a little bit later ok, so now I finish the statements part so we have a result about cutting sequences on Vich surfaces especially Bo-Moller surfaces and result about Lagrange spectrum ok so now I want to give you some tools from the proofs any questions now? yes? so the so actually I am kind of lying because there is something called Markov spectrum which is similar to the Lagrange spectrum but instead of soup you take a soup and people have studied the Markov spectrum for other Fuchsian groups and there is actually maybe I should have said this, I should really say so there is a result by Schergon and Schwitt where they prove the Hall Ray for Markov spectra but there is quite a big difference ok, there are different definitions so say that you want something which has casps so if you want to talk about penetration spectra but ok, if you have a Vich surface you always have casps but yes so for this Markov spectra there are results about the Vich but in some sense if you want to achieve a soup it's easier than if you want to achieve something which gets to that value infinitely often so this actually creates some confusion so they say they tried to understand Hall Ray for Lagrange and they couldn't solve ok yes so this is true also here so we are also talking about penetration spectra ok, you will see which on what on the Teckmuller curve absolutely, so both motivation the diophantine motivation also generalizes it's a diophantine problem for interval exchange transformations analogous of rotations ok, that's another thing that I like to do but it's also the geometric motivation is penetration spectra generalize that's why we studied this Lagrange spectra ok, so the first key picture if you have a Vich surface in the Vich group you want to look at the space of a fine deformation of your Vich surface so you might have heard from Ronan from last week, you can act on a translation surface by SL2R by applying the matrix to each polygon and gluing after applying linear matrix parallel sites are still parallel you can still glue them so there is an SL2R action and I want to look at the space of a fine deformation and I'm gonna define it like this I'm gonna think of marked a fine deformations as triples of from a fixed S0 to S and I'm gonna say the two triples, two marked triples are equivalent if I can find a translation equivalence which makes this diagram commute something is wrong, ok, it doesn't matter ok, so the space of marked a fine deformation is actually isomorphic to SL2 and SL2 is like the unit tangent bundle of the unit disk picture yourself this disk the unit tangent bundle of this disk as all possible a fine deformation marked a fine deformations of surface, you can actually place the standard triple your surface you can think of it as the center of the disk and as you apply a matrix in SL2R you move on the disk and the which group acts on a marked a fine deformation by changing the marking and it acts actually on the disk by maybe those transformations and you can take the quotient of this disk by the action of the which group and this is an example of a tassillation of the disk where the fundamental domain is the which group and the quotient is the space of a fine deformations and this quotient will be the unit tangent bundle of a hyperbolic surface with casps and this hyperbolic surface with casps is the type molar curve associated to your which surface so I want to think of type molar curves as space of a fine deformations of a which surface and remark this is not the case if I take a random translation surface then the space of a fine deformation will be much larger it will be the whole the closure will be the closure will be typically the whole modular space but this is true for which ok so this picture will be useful and I want to do some kind of use the action of the which group on this disk to understand my dynamical properties so I want to explain some ideas from how you characterize cutting sequences first so the first three slides will not have the disk but then I will show you the disk how it enters into play and I'm gonna do a simplification as I said the result I want to talk about is bone molar classification but I want to tell you first some ideas from the octagon case from our work with Miley which are easier and then I will tell you the difference and novelty in the bone molar case ok again you can start listening from here if you are lost for now so if I give you this octagon so I'll talk about the octagon and then explain the difference with bone molar so let's go back to the regular octagon and to cutting sequences on the regular octagon so first using isometries of the regular octagon I can assume that the linear trajectory I want to study has slope in the sector 0 pi over 8 so this is this blue sector I can always apply an isometry and bring it back there so first I want to do an elementary observation that there are clear restrictions of what a cutting sequence can do so I want to look at possible pairs of letters or transitions so which pairs of letters can follow each other given that my trajectory has a slope close to horizontal for example if my trajectory hits the side A and has slope in this sector it's bound to hit D after A so I cannot go anywhere else than D so from A I can only see D but if I hit this from the side D and it depends on how I come and what my slope is I could either hit A or I could hit B so from D I can go to A or to B those are both allowed transitions and similarly you can record which other restrictions there are and you get a little diagram which I call transition diagram and by elementary geometric restriction in this sequence it has to give me an infinite path on this diagram because it's forced to of course there's far from being a characterization but it's a first necessary condition and the idea is that renormalization allows you to multiply this condition on different scales so first you have to do your exercise if you have other sectors you can get just permutations of the same diagram so these are all diagrams one to the eight possible sectors the other half is symmetric so I don't need to care about and if I give myself a cutting sequence just by looking on which so it will give me an infinite path on at least one at most two of these diagrams so you can kind of recognize the sector already by just this elementary observation then there is a non-trivial operation so I want to say a combinatorial operation which is called derivation and derivation will be an anti substitution so I remember substitution grows the lengths of words derivation will shrink the lengths of words and we will find substitutions as antiderivations so if I have a cutting sequence on the octagon I can say the derived sequence consists of the sequence of sandwich letters what are sandwich letters so sandwich letters are letters preceded and followed by the same letter so here A is sandwiched between D's B is sandwiched between C's and I can run through my sequence and search for sandwich letters and keep only those so the derived sequence consists of only the sandwich letters in my sequence which reduces lengths of finite blocks and so in Smiley we show that the derived sequence is again a cutting sequence if I start with a cutting sequence the derived sequence is a cutting sequence so it has to be admissible on one of the diagrams and conversely substitutions are like anti derivation so if I know the arrival sector so if I know in which diagram the derived sequence is leaves then I can reconstruct my original sequence by a substitution so there are 8 substitutions so that if the derivative leaves in the I's transition diagram the original sequence is substitution I of the derived sequence so derivation is not uniquely invertible but if I know where I land I can invert it through a substitution ok, that's the key step that I want to bring for word for on goes and they're not really substitution they're actually substitution on transitions so you can think of them as you have to change alphabet if you want to really call them substitution and in our first paper we don't talk about substitution you can think of them as morphism of the transition diagram so an arrow is mapped to a sequence of arrows so this combinatorial operation of derivation is nothing else than the symbolic effect of an affine automorphism actually of the affine automorphism that we saw before the stretch is the octagon so this is a geometric meaning of derivation remember this matrix that we saw that sends the octagon in a shear octagon so the claim is that to see that the derived sequences of cutting sequences are cutting sequences you can prove that if I take a cutting sequence in this basic sector the derived sequence is the cutting sequence of the same trajectory with respect to the sides of the shear octagon so the shear octagon will hit my sequence less often and the sandwich letters are exactly the ones which hit the sides of the shear octagon and then if I want to re-normalize if I apply my affine diffiel which sends the shear octagon back to the octagon and then I open my trajectory but I still have the derived sequence is the cutting sequence of a new trajectory on the regular octagon so I can then repeat my procedure now it's time to see the disk picture so this is the disk that molar disk of the octagon so the center is the octagon which is the symmetry of order 8 and this is the fundamental domain the tussillation given by the which group of the octagon and all the red vertices which have order 8 symmetries are shear octagons so they are cut and paste equivalent to the center to the regular octagons so you can think of them as shear octagons and if you choose as generators of the which group I choose generators of isometries times this gamma this shear and this is the telegraph of moves that I want to use so here I can use reflections or rotations in the group and to move along a segment I use my gamma or conjugate of gamma so from the center these are the possible moves I can do so if I want to study a trajectory of a direction on the octagon sorry trajectory on the octagon in direction theta I'm gonna look at the geodesic ray which points to theta in the tekmolar disk and I'm gonna shadow this ray with a sequence of moves from my tree so I'm gonna look at group elements which approximate my geodesic and it will go to a sequence of octagons it will go to a sequence of vertices and I claim that if I derive my sequence what I'm getting are cutting sequences with respect to this stretched octagons so these octagons which approximate my geodesics will be stretched in the direction of my direction and the right sequences are just describing how I hit the stretched octagons and how do I find my substitutions if I can actually label this tree with number is from 0 to 7 and the path that I'm selecting it's telling me which substitutions I have to use to invert derivation so if my labels will be I0, I1, blah blah then my original sequence is the ends derivative with this substitutions backwards so this is how you find your exotic expansion for the octagons it's a triangle group with actually infinity infinity n so it has one elliptic point and two parabolic elements thanks for this question because now what is the difference with Bohm-Mohler Bohm-Mohler has again a triangle group but it has two elliptical elements so SMM has two elliptic points it's a MN infinity triangle group so it has one parabolic point and two points of order M and order N so indeed that's the why it's kind of harder than the square and actually what are the green vertices 3,4 in my picture there are the 3,4 surface which has order 4 symmetry and what are the red vertices in this disk actually turns out that on the same disk there are also affine copies of the 4,3 so on the MN disk there are also the NM so these kind of surfaces are coupled and the red vertices are affine copies of cut and paste copies of 4,3 surface so the key idea is that this time to use the randomization we are not gonna use which group elements because actually Diana tried to describe which group elements and she couldn't get so they are kind of hard to describe so the idea is to break your task into simpler moves that are not made by which group elements but I want to go I want to use this tree of randomization which goes from a green vertex to a red vertex so this is not a which group element will go from green to green so how do I go from green to red well I want to go from MN surface to an MNM surface so by the way sorry this is octagon and two squares and this is hexagon and two triangles and if you shear them a little bit this is octagon and two squares so it's SNM and this is hexagons and two triangles so how do you go from one to another so Hooper proved that there is an affine diffio from one to another so let me show you one cartoon movie you can straighten this picture change diagonals from red to green in this rectangular grid cut and paste some triangles to get the rocket, this is the rocket formation and shear it to get back that so that's how you go from one to another and you see what you have to understand it's a flip of diagonals so this is the last slide that have on not the last, the one before the last so what do we do? we also define derivation combinatorially I'm not gonna tell you what the derivation looks like combinatorially but our derivation goes in two steps I want to go from MN to NM and back so first time I derive I'm gonna send a cutting sequence of a trajectory here to a cutting sequence of a trajectory there so derivation with map cutting sequence is here to cutting sequence is there and it comes from with the recipe for substitutions so if I know where the arrival the sector where the arrival direction is and I can recover this combinatorially from transition diagrams like I did before then I can also invert and write down explicit substitutions and we give recipe to produce substitutions for every MN so I can find substitutions to go back substitutions in quotation marks because technical details so what's the picture on the disk this is the last picture and then I have only two slides on hall so that's the picture of the disk so let me remember green vertices are one surface and red are the other surface this is my tree of wolves I want to study trajectory in direction theta I'm going to look at a ray geodesic ray going to theta and I'm going to select a pass in my tree which approximate my geodesic this will give me the normalization moves I have to do and again like before I can interpret these combinatorial derivatives as cutting sequences of the same trajectory to this shared polygonal presentations and here the polygonal presentations for M and odd for k odd they will be the formation of one surface for k even they will be the formation of the other surface I always jump from one type to another type and again you can label this graph explain how to do that and use the labeling of this graph to read off which substitutions you want to use and if you like you can interpret the action of these three of wolves on the boundary as a continued fraction on directions and these substitutions you need to use are governed by the symbolic by the entries of this generalized continued fraction for bomb holder yes actually this is really I'm taking my translation surface I have a direction and I'm just looking at the geodesic flow which shrinks that direction so I'm rotating myself in that direction and that if I start from the whichever convention you pick I start from the center of the octagon and you're gonna give me a geodesic ray so in reality with this convention you know theta goes from 0 to pi you do a full angle so this will be like if you pick reference point will be like 2 theta or something explicit but it's really I want to pick the geodesic which renormalize my linear trajectory ok so now I have just two slides to finish with some ideas about whole rays so break the last again and there's a last recovery point for the last idea so again here we want to work with this like molar disk of renormalization moves and we want to show I will remind you what the Lagrange spectrum is because it will have forgotten so I take any which surface here so I don't know the which group I can take a fundamental domain for the which group but we saw before that it was very nice to have ideal tussillations so what I'm gonna do if there are elliptic points I'm gonna look unfold the elliptic points and look at the super tussillation which is ideal so I notice I don't want to pass to a finite cover of my which surface which I could I want to study my original surface but I'm gonna use this super tussillation to call to the geodesic flow so my picture has an ideal tussillation which could be a finite cover of my fundamental domain and this ideal fundamental domain there are gluings given by elements of the which group and if I want to study the Lagrange value on a certain direction again I'm gonna pick my geodesic ray which shrinks that direction and I want to call the geodesic ray by the sequence of sites of my tussillation hits or by the sequence of which group elements which glue the sites hit and because this is an ideal tussillation equivalently I can look at my endpoint of my geodesic and there will be a sequence of arcs boundary arcs boundary arcs here here nested boundary arcs which contain this point so out of the sites of my tussillation should imagine this fundamental domain reflected and I'm gonna pick the arcs which cross nested arcs which crossed my ray and shrink to that point so this boundary arcs is what I'm gonna use and if you know Bowen and series boundary expansions this is actually the easy case of Bowen series because I'm looking at an ideal tussillation if you know Bowen series ok, so if I have a sequence of boundary arcs which come from this ray I'm gonna associate to boundary arcs pairs of saddle connections on my surface like this which contain the direction theta so if I have a boundary arc it has two endpoints, two angles and I can look at actually turns out that this vertices of my ideal tussillations correspond to directions where I have saddle connections on my surface so in those directions which are endpoints I have saddle connections and I'm gonna pick the shortest saddle connection in that direction this will give me a wedge a pair of saddle connections which contains the vertical and as I go to a nested arc as I go to a nested arc a shrink wedge so I will get a sequence of saddle connections which converge to the vertical direction ok, it's a sequence given to me by renormalization by the Vitch group and it's a sub sequence of all saddle connections which approximate theta but I claim that this saddle connection see the spectrum they see the top of the spectrum so remember the spectrum was the sup of one over the area so saddle connections as they approximate theta so so the key lemma is that there exists for high values in the spectrum after a certain L0 I can just use these wedges I can take a direction theta and the sequence of wedges that approximate it and only computing this area of these two saddle connections right and left and the limit as n grows I am missing some saddle connections but those are the important ones that allows me to compute to the spectrum and this is kind of the part I wanted to tell you this is my last slide from here on it goes back to kind of redoing whole argument in a new setup so whole uses a lot that there is a formula using continual fraction for the spectrum so if you know the continual fraction entries so this spectrum is the lim sup of this formula so I have an, here is an integer and then you have an infinite continual fraction of future entries and the finite backward continual fraction of this parentheses are continual fraction so what we do we define as continual fraction we define continual fraction in our setup and this s stays for a modified continual fraction and we show that these areas can also be computed by a similar formula in terms of continual fractions and then whole has a proof which uses arguments on sum of counter sets you can prove that for bounded entries this tail corresponds to counter set of values these other tails correspond to another counter set but if the counter sets are have large halves of dimension the sum of two counter sets contains an interval so this is kind of redoing the counter set proofs and I have a hidden half slide which explains more but I think you are all tired and this is a good point to stop and I showed you the geometric argument using the tekmuler disks so I hope I convince you that tekmuler disks and which groups are useful to study dynamics on which surfaces thank you


And joining me right now, the head coach of the New York Liberty, first year head coach, Walt Hopkins. Joining me, coach, how you doing? I'm doing well, thanks. Thanks for having me. Let's take you, let's go back to drafts night. What kind of a roller coaster, what kind of an exciting day was that for you? You know, it was, it was, there was a lot, there was a lot to it because we had won the side of it that we, outside of a couple of picks between one and nine that we didn't expect that the end results other than us getting Megan Walker, which was just I think people are going to look back on and be blown away that that happened. Other than that, though, the draft really ended up being exactly what we had slotted out in different locks and things like that, even though that the middle of one to nine looked different, it still ended up being the same results that we were kind of open for. The biggest surprise was Megan. So that's one that I think I'm sure you're going to have some questions about, because that was just just a gift. Yes, we will definitely get to Megan. Let's let's start with Sabrina, though. And just an embarrassment of riches there with her being able to do so many different things on the court. What will that process like meshing, you know, everything that she does well with the style that you want to play? Yeah, I think we really focused on putting the style together that fit her. I think coming out the gate, Sabrina's not going to have to do anything to fit in.  I mean, they all are built Marine Asia. They're built to play in this type of a system. And I think it's really going to kind of take away some of the limitations to what they're good at and add to the things that they do great, you know, because they're just going to be in positions to succeed. And they're going to have shooting and people who can put the ball on the floor and defend multiple positions really everywhere you look. So I think for Sabrina, the, you know, for instance, we were talking through our playbook just the other day, because the first thing Sabrina did after she got drafted was text me and I want a username and password to start looking at the playbook and interacting with all that stuff. And she was like, Coach, this there is there some kind of a mistake? Like this is a lot of the stuff we ran in college. And I'm like, no, Sabrina, that's not a mistake. That's intentional, you know, we're we're doing that because we want you to, you know, one, those are good actions. And, you know, and you were running that system because you had a good coach who knew how to maximize his people. But two, we're built similarly to Oregon. You know, we are going to have fours to shoot the ball. We're actually going to have five to shoot the ball, which is really exciting. And then, you know, every other position creating spacing for her to kind of go into a pick and roll and find the person who the help leads. If nobody leaves, like Stanford tried not leave her and she had, you know, 30 something or 28. And so that's the way that we're built really is going to suit her well. What impresses you most about Sabrina's game when you watch her? I said this before, I think her competitiveness, her her edge is really, you know, Diana Kobe like it's, you know, with blast dance airing, you know, it's Michael like it's it's got she's got a real a real focus on the things that she needs to do and that her teammates need to do. And she doesn't really care how it happens. It just needs to happen. And to see that drive behind somebody, you can see it in her eyes. You can see it in her actions. You can see it in the way that she approaches every game. She she approaches the first quarter different. She approaches the third and fourth quarter like she really is thinking and adjusting on the fly as well as anybody I've seen. And it all comes from everything is informed by her desire to win. And she just does whatever is necessary to win. So some games she might have, you know, 15 rebounds, 12 assists and eight points. Another game she might have 32 with four assists because that's what the defense was giving her and so she capitalizes. So to have that that versatility in her skill set offensively, you know, she couldn't do that if teams were able to play off of her and say, go ahead, Sabrina, beat us with your three point shot. And she didn't have a three point shot. She wouldn't be able to do that. But you can't take anything away from Sabrina without giving her something because she's capable of capitalizing on whatever you give her. And naturally, as the number one overall pick and to have a college career as decorated as she has, she's taken the headlines. But I love hearing you, you know, your excitement about what Megan Walker is going to bring to the table. What will she provide to the New York Liberty? Yeah, at nine and 10, you know, I know that people probably know Megan. The fewer people probably know Jocelyn Willoughby. They are statistically very similar. If you go in and look at their numbers, very similar. They both shoot well over 40 percent from three. Megan's at 45 on like 173 attempts or something, which is absurd to go with the fact that she's not shooting a bunch of just set standing there wide open jumper. She comes off the stagger screen. She shoots off the move. She can shoot him off the dribble and she can put it on the floor. She's long. She can rebound, defend multiple positions. And obviously that Yukon pedigree is, is, you know, without equal.  So she moves well without the ball, which is going to be really important for us as teams, you know, are focused on pick and roll action on the weak side. She can either stay spaced and spot up to knock down threes or she can cut if you lose sight of her, if you're worried about the stuff going on with Sabrina and Zowie on the other end.  And Jocelyn is really the exact same. She's, she's six feet tall. She's an inch shorter. She's probably as versatile or a little bit more versatile off the dribble. She really excels getting to the free throw line. She gets all the way to the rim. She's a good finisher, but both of them, I mean, embarrassment of riches is the exact terminology I use. So I, I totally agree with that. All right. We talked about three of the draft picks. So the Liberty never had this many picks in one. WNBA draft. Quickly, your thoughts on the other ladies, Jasmine, Kylie and Leon. Yeah, that, that group, they all provide something different. The thing that's, that's a steady factor with all of them is their, their buy-in. And one of the things that we did, you know, that our assistants, the staff, when they called all these players, they wanted to get a feel for them. And everybody had a different approach, asking different types of questions. And they all brought their genuine personalities to these interviews. And then when I came in from, from the interviews that the assistants told me about, I was able to kind of tailor mine according to whatever questions we had based on that pooled beta. And so getting to talk to all these players, you know, with a player like Megan, for instance, I really went straight, straight with her and just said, listen, you know, this is what you're going to need to do for us. If that's not a fit for you, then we get it. It's not a big deal, no hard feelings. But if you do buy into this, you could be phenomenal in this system and doing the things we need you to do. And she was all for it. And as we went through every single one of these players, they all said the same things. They just, they're bought in. They understand that it's going to be a process for them to learn and grow. And they know that they're coming in as rookies. Nobody has expectations as to what they deserve and things like that. And that's essential for a rookie. I mean, it's absolutely essential because when you have somebody coming in who's entitled and, you know, thinks they deserve X, Y and Z. And then they're upset in the background and things like that. I mean, this group, they all want to play for one another. Getting to jazz and Kylie and Nay, the three of them are all really versatile. So if you look at jazz, she's a phenomenal defender and athlete. And she shot a really good percentage from three. Just didn't shoot a ton of them, but a ball of energy. She's just amazing. Kylie is built for our system. You know, she's six, five defensive player of the year. Led the ACC in blocks, shoots 37, 38% for her career from three. She's perfect for what we want to do. And Nay is a player who literally can play any position as she continues to develop her athleticism off the charts. And she's another just great person. So the way that we see them fitting in is her versatility is going to be huge for her as we need to fill plug different holes throughout the season. You know, as things happen, she'll be there and she's going to be able to do that kind of in that, in that 11th or 12th player role. Last thing for you, coach, I know with you, still things are still fresh. You were announced in early January and you're all about making these connections. I'm sure you're chomping at the bit to get this group together. We had a WNBA virtual draft. Will you be having doing some virtual coaching? How will you, you know, establish player development at a distance right now? Yeah, I mean, part of it's our online playbook, you know, we're doing, we've uploaded everything and we have this, we have some really cool features on the site that allow us to quiz the players and they can interact with it, you know, in a more intimate way than they would have if they were just like looking at a diagram and memorizing, you know, rote memory, all these plays. That's one way. And then we as a staff, we actually have a call tomorrow to really now that the draft is over to lock in on some creative things that we can do with them that are, you know, that aren't corny, you know, that we want to, we want to really bring them together with things that they're, they're adults, they're pro athletes, you know, we're not going to, we're not going to be doing, you know, like rock climbing team building stuff anytime soon. So yeah, we're going to, we're going to really just do our best to do something creative, but, you know, authentic and genuine that really helps them. The excitement continues to brew for this New York Liberty team. Coach, thank you so much for your time. Thank you very much. I really appreciate it.


I don't really see the operation done, but these, do you know about these? Yes, I think we get them in the past, hopefully. Yeah, but do you know where they are? No. They're nothing but plain salt and crackers, regular crackers.  Right. And I just love them. They come out. And I don't know if anyone's ever thought of it, any place else I've never seen or heard of them, except here at the White House. Really? Salt and crackers. They're a different one. I wonder who you were referring to yesterday. It's a hat field of Kennedy that the leaders of the Free is moving, so it's sort of moving. I did not have any Americans in mind. Did you have the AKB in mind? Did I? Thank you. You're important. I'm going to wait for you. You'll never answer, but... That's a non-answer. You'd be surprised what we're gonna do. I'm not battering with the energy about it. You ever said? Yes, sir. About the session, while you're better well-aid. Give me half the chance. I won't do that. I hope Mr. Khabib has an optimistic before. Vice-Place, thank you. Thank you. Thank you. Nice to see you. Nice to see you. Well, I think that they want us for a picture, and the press is going to come up with this. Mr. President, we deem it a great honor to be received today in this blessed house. And I want to thank you for the opportunity to meet you, the President of the United States and the leader of the free world. We feel a deep sense of indebtedness for your peace efforts throughout the world, and especially for that part of the world where peace and goodwill among men was first proclaimed. We know the days and nights you spend thinking, discussing with your worthy assistants and even praying that a solution may be found that would guarantee peace with justice and freedom. In recognition of your outstanding and dedicated services in the cause of peace, allow us as the highest religious leader in the land of the Holy See and as patriarch of Jerusalem to bestow upon you the great cross of the order of the Holy Sepulcher and to assure you of our progress that he who was nailed upon the cross for the reconciliation of man with God and men with men, our Lord and Saviour Jesus Christ, may ever protect you and strengthen you so that your peace efforts may be crowned with ultimate success. To bestow upon you, therefore, Ronald Wilson Reagan, President of the United States of America, the Holy Cross, which is the symbol of victory and peace, justice and love. We proclaim that this be given in hope of peace and justice throughout not only the Americas but the entire world. In conclusion, we ask you, Mr. President, to kindly accept this declaration for it expresses the wishes and expectations of all the world to live and experience the blessings of peace. This is for Mr. Reagan with the blessings of the Holy Sepulcher of Jerusalem. It contains in it a little particle of the real cross when Jesus was crucified. This cross also contains a little particle of the cross. God may watch over you and protect you from any evil. Thank you very much. Thank you and thank you for your attitude. It is very nice to have you here. He tells me that he feels also a deep emotion because of the arm that was missed upon him to be seen by the great King of the Free World. I am honored and I know Mrs. Reagan will be with me. Thank you for her. She, her father just recently died and she is out with her mother. Well, I can't thank you enough. I'm greatly honored and certainly more determined than ever to bring about peace. Your work in current for all this time and your efforts may reach that target. Especially because you are representing the most peace-loving country in the world. Thank you very much. Please extend to Mrs. Reagan for my personal greetings. And she will be most appreciative. Thank you very much.