Okay, hi everybody. Welcome to the latest edition of the library's Adulting 101 series. This one's called Hit the Road All About Travel. We're going to be talking about Study Abroad. We're going to be talking about independent travel over the next hour, and we've got some really great panelists who will lend their expertise. So, without further ado, I am Ben Alkalai, the library's UCLA library's communications and marketing manager. I'll be emceeing, moderating today and then providing some of my own personal travel tips. So just kind of to go over what we'll cover today. So UCLA study abroad opportunities. Then a Q&A with some past study abroad participants. And as I mentioned, some travel tips and tricks. But first, let's do a quick icebreaker. Don't have a ton of time. So just in the chat, if you would just drop if you could live anywhere in the world, where would it be? Let's see the responses come in for one year.  Good answers. Who else is Denmark? Courtney says Portugal. Sophia says Sweden. I clean that says Turkey. Oh, Holland. Oh, what's going on here. Appreciate it. Okay. Good answers. Thanks for sharing everybody. And without further ado, we're going to go into some study abroad opportunities through UCLA. I'm going to be joined by Magdalena Badagon. She's the executive director for UCLA's International Education Office. And in that role Magdalena oversees all undergraduate study abroad programs at UCLA. Prior to her position, she worked at UCLA Anderson School of Management, where she managed student exchanges and was the primary advisor for international students. She's been working on creating opportunities abroad for all students and her mission is to send out as many UCLA students as possible to experience and learn about our commonalities and differences with the rest of the world. Magdalena studied French literature and languages as an undergrad at UCLA. And without further ado, take it away Magdalena. Yes, as mentioned, I am privileged to serve in the world as executive director overseeing our undergraduate study abroad programs and I am really grateful for this opportunity to reach out to all of you to encourage you to think about all of your opportunities while you're here at UCLA. In this session, we'll go over some of the basics about studying abroad at UCLA, including our top five reasons and why you should study abroad. You should not miss this. This is, I would say, one of the top regrets that we hear from alums is that they didn't study abroad and also for those who did participate. We have a lot of data from seniors over the past few years who say it was the best thing they ever did during their time at UCLA. So I want to encourage all of you to make the effort and to think about where you could see yourself moving forward in the future. To help facilitate the conversation, I'm going to mute my image or just cancel my image and we'll go through the slides together and then I'll come back on screen and help answer any questions. So let me do that, just to get started. So let's go to the next slide. Sorry technical difficulties, let's advance. There we are.  No, let's see. I'm not seeing it. Or maybe I've got the review on here. I'm still seeing the original one. All right, let's go back here. One more. No go. Yeah, I don't see it. What screener is everyone seeing. I'm trying to get to the second slide, which will be, let's see, the start here go anywhere. Are you seeing that? Yeah, that's okay. Okay, so there we are. Okay, sorry about that. So just to get started, I just want to, you know, talk a little bit about our office in terms of who we are and what we do. UCLA International Education Office, aka the study abroad office is the one stop shop for all our undergraduate students considering studying abroad. We also have a few US based programs like in New York City. So there are a variety of options for students to consider we think that the exposure to another part of the world is important and really essential for students to, you know, position themselves in the best way of possible for their future careers. We have a team of advisors and coordinators to guide students through their program planning and exploration. We have pre departure readiness and support while abroad and upon reentry. So we're really trying to, you know, create a full life cycle of a student's experience once they decide to study abroad from nurturing you and guiding you in the decision on where you're going to go throughout the time you're abroad and then on your return working diligently to create more resources that you can apply for your job interviews and partnerships with the career center so that you gain as much as possible from that experience abroad. We do also hold a number of events throughout the year some are major specific sessions. We hold financial aid and scholarship workshops as well as our annual study abroad fair in the fall, where students can learn a little bit more about the variety of options that are available. We also have students, you know, a group of students were developing an alumni study abroad participants to connect you with peers who've already gone abroad on this journey to hear what it's like from them directly on what their experience was like and to gain some insights or tips from them as well. Let me go to the next slide. So these are the top five reasons to study abroad. And it's also helping you understand there are a lot of preconceived notions that students have when they think about studying abroad we hear a lot from students who say, Well, studying abroad is for people who study languages or particular field. I want to make sure you know that any major can study abroad. It doesn't matter what your major is. There's not just one but a handful of study abroad programs for you and UCLA students are not obligated to pick a particular one, or even one that's tied to your particular major.  It could be at a Japanese University conducting fieldwork in the rainforest of Costa Rica, or among other options we have something for you to make progress towards your UCLA degree. You also have general elective requirements that you can satisfy abroad you don't have to take a major specific course. You have a lot of opportunities within your time at UCLA to decide where you want to go and what you want to study while you're abroad. So we really encourage students to think early as early as possible about what can they do in another part of the world and that can be again a general election, general elective, a diversity requirement or a major specific requirement as well we also have some really interesting non traditional opportunities. You know the meaning of study abroad UCLA has expanded beyond just traditional language and culture programs that study abroad used to be known for our programs are no longer just for humanities and social science majors. At UCLA stem students can equally participate in study abroad. In fact, we are extremely proud that our university is now one of the only UC campuses that officially sends more undergraduate stem students on study abroad than non stem. And that's really unique. So we see this as a huge achievement here at UCLA to provide opportunities for our stem majors. While we do still have programs that focus on language and culture we also have opportunities for students that conduct research and labs in Singapore field work off the coast of Australia and secure an internship in global health in Colombia. You can study at the United Nations in New York over a summer, or take courses that are not in Mexico City in Spanish, among other local students. Again, possibilities are really plentiful and almost endless. We have programs in English. And this is another myth that we often hear as students are concerned that maybe they don't know a different language and that they will have difficulties while they're abroad. The vast majority of our programs are in English, even though they are placed in another part of the world. So first of all there are about half a dozen destinations available to you, where UCLA is already the primary language and those are countries like New Zealand, Australia, UK, South Africa or Ireland so the country there already has English as a primary. And those are always easier options for students who are concerned about immersing themselves in a country where a different language is spoken. But we do encourage students to push themselves and especially if you have any language training or education in your under your belt, you can certainly polish that while you're in another part of the world. You will also be able to get around and learn a lot of the local language in countries where another language is spoken. It is just a natural skill that you will pick up while you're there. And again, there are lots of programs with lots of different majors and opportunities for you to explore. The affordability of study abroad often there is also a lot of concern about what does it cost it's too expensive. I know I thought that when I was an undergraduate. And, you know, admittedly some of the study abroad programs can be more expensive than staying at UCLA. But I do want to make sure that everyone understands that there is quite a bit of, there are very many variables that can help you manage the expense. Things like the duration of the program, when you go abroad, and also to assure you all that our study abroad programs because they are for credit, they are helping you make time towards your degree are funded by financial aid. So you can get loans for these programs there are also scholarship opportunities for study abroad programs. Every year we have a large communication to the campus about study abroad scholarships. There are, you know, a variety of ranges some of them pay for half of your program fees some of them might pay for all of your program fees. You know you're encouraged to not not give up just because you're looking at a cost for a program. Come to our office to talk to us because we can help you find a way to get there. I mentioned making time to degree the sort of fifth reason is because you will graduate on time. We've done a lot of students studies and run a lot of data over the last few years. And to affirm this, most of the students, vast majority of the students who participate in study abroad, graduate on time, if not a little early. Again, and that's because you are earning credits that you will use for your degree you're satisfying major, minor, or general education requirements, and so it is not going to set you behind for participating in study abroad. You can sometimes see students more and more choosing to do more than one study abroad opportunity, a summer and fall abroad or maybe two summers. So you are earning those credits and continuing your progress towards graduation. Next slide please. So just kind of the essential steps for you to think through as you are preparing or considering studying abroad. The first is really kind of exploring where do you want to go. It's a big world, lots of different options and narrowing, you know, what kinds of things are you looking for to expose yourself. I think on a sort of larger scale is a good idea. Planning wisely, you know, being realistic about your plans do you want to be abroad for a year for a term for summer.  We look at it quite regularly and we try to keep thing information very current, but you're always welcome to come to our office. So virtually, lots of different ways to connect with us we have a lot of info sessions as I mentioned, and a really key point is to apply early students are often surprised how early you need to apply for a program and that's because there's a lot of planning and coordination on the ground logistics that have to be prepared for students arrival in any part of the world.  And that really will give guarantee you the most opportunities. Most of our programs are first come first serve. So the competition are your peers, and we do see a lot of interest in study abroad, especially as we're resuming sort of life. You know, after our shut down or lockdown. I think a lot of students are interested and curious and wanting the opportunity to explore and get back out there. And we of course are encouraging of you as well. We do keep safety in mind with or without coven in the picture we are always concerned about making this experience as safe as possible and we spend a lot of time on the ground here and throughout the time you are abroad, ensuring that you have resources and that all of the risks have been mitigated to the best of our ability. Next slide. I mentioned there are so many programs options in the UC offering UCLA students can experience a spectrum of study abroad programs. We've got 200 plus programs and 45 countries. We have a variety of types of programs we have the program which will go into that's our largest program offering that's a UC system wide offering so you can take classes with other UC students on those programs. We have our own campus based programs like the travel study program or global internship program exchange program and our global cities program. Next slide. So we talked about different kinds of program types. We have what's called the traditional university immersion that's where you study at a local university with local students. And in this particular example we would. Say the UC EAP program is the program that has the majority of these types of program, and that is a system wide program where you can take a class with a Berkeley student or Riverside student, San Diego all UC programs applied in the same experiences overseas. We have friends and other campuses. It's a cool experience that you can maybe meet up with them in a different part of the world. You are immersed you are taught at a local institution by the local faculty there so it's really more of a traditional thing to say. And then we have our campus based programs which are programs that we develop here at UCLA specifically for UCLA students. We have our UCLA faculty led program, which is really what I call a classroom abroad. It's a program or course that is a UCLA approved course for UCLA credit led by a UCLA faculty member. And it's just situated in a different part of the world. And the range is you know everything from language to we do have an engineering class we've got food studies we've got lots of different courses that satisfy again major minor. Or to grade just general election general elected requirements. So please do check those out. Our newest program is the global internship program. And this is an opportunity for students to do internships or research abroad for credit. We have the global internship program is one that we've designed, and the EAP program also has research opportunities abroad, and again dozens others so you know you're encouraged to maybe think through the type of experience you want as sort of an initial step for determining where you're going to study abroad. Next slide. It's all about planning. It can't be said enough. It is, I would say 96% of students who study abroad receive major credit. And that's really a great opportunity for you so that you can again explain this to your parents. It's evidently that you are not wasting your time and understand the value of these opportunities is more than just the experience in a different part of the world. It's actually working for you. It's helping you graduate on time. Again, if not early. Studying abroad takes careful academic and financial planning. It's possible to find a program that helps in your progress towards your major or degree requirements. In most cases, there is no formulaic way to study abroad. Students think about their personal academic and professional goals UCLA degree requirements major GES foreign language technical breath, and there is no one right time to study abroad. Some first year students study as soon as their first coming summer and others go in the spring of their senior year. We are seeing more students choosing to study abroad earlier. And that makes me very happy because I think again there's an opportunity to really influence your field of study and help you in your development. You know, you can really ruin and help you in your career, you know, goals as well. Sometimes we see students who go abroad and decide, maybe they want to change their major, or maybe this experience helps them think about a different career path. So again, you don't have to wait until junior year, but planning is essential so plan as early as you can. Next slide. So when you study abroad. Again the planning the earlier is the better. Even as your first year as I said you can you can certainly start to do that. I encourage you to connect with us it doesn't have to be any particular time of the year. We have virtual advising sessions, and you can come and talk about your major maybe what you're thinking maybe the program type and we'll help guide that conversation for you. And that's really kind of giving you a visual timeline of what that can look like for you in terms of preparation for getting to that study abroad, boarding the plane and getting to your the new destination, where you're going to have an amazing experience. Again petitioning for study abroad courses, if you need to, and then finalizing the exact courses that you're getting getting all of your financial aid and time. And your housing all of those things take time and so just again, give yourself all of the opportunities to succeed and plan the best opportunity for yourself. Next slide. So funding for financial aid is available for all eligible students for all of our programs again because these are credit bearing programs. You can use financial aid here at UCLA to fund them if you're eligible. But there are a number of ways, if that you can, in general finance any of the opportunities, other than financially there are scholarships as I mentioned there may be personal funds. Students sometimes tell us that they will ask their family for money or opportunities like airfare tickets instead of Christmas presents and they know they're working towards that goal of studying abroad. And so again that's this is where that planning really pays off in very literal ways. So again there are many ways for you to get there. If I can impress anything for all of you on this call is, you can do it, you can get there and we can help you. There are many, many opportunities that we're eager to share with students and it isn't always obvious to you and until you come in and talk to us and we can help you try to map that out a little more clearly. Next slide. Just to give you a cost comparison of what it might look like. Don't panic when you look at these numbers. I'll sort of walk you through what you're seeing here. When we look at what program again no program costs the same and that's because it's determined by location, by the duration of the program, and the type of experience that you're having. So just to convey here is just give you a couple of programs to look at and see how expensive or affordable these programs can be. The programs can be either more expensive than the ones you're seeing here or less expensive. There's again a very broad range and again determined by all of those variables that I just mentioned. There is no resident tuition for summer and again a quarter at UCLA ballpark give or take is about almost $10,000 and a semester abroad, which is two quarters is about $20,000 abroad again but that's two quarters. That includes housing that includes a number of other excursions and so you need to kind of really think through and weigh out what these numbers mean for each of these experiences. When we look at the cost comparison on the screen you have a granada summer program that's five weeks. That includes your housing that exclude includes excursions to museums, some transportation on the ground getting to and from venues. And so your program fee the cost that you will pay for that program is $6,000. Then we have suggested budget for you in terms of how much money to bring along. And then you factor in your airfare as well so you know ballpark it would be about 10,000 for that experience but it's not just tuition it's housing some meals excursions. That is is factored into that final number. And then you look at the contemporary Mexico summer program also five weeks but look at the units you get 11 units. And the program fee is about just under $5,000 which is pretty reasonable. Again includes housing excursions your credit. Your costs, your expected costs there Mexico is less expensive than Spain so it's you see that reflected in the number. And your airfare obviously also it's a closer destination it's much cheaper. And so the cost for that summer experience you see a significant difference there. And the last one we're looking at is the summer physics program in Ireland. Again that's a longer session that's eight weeks in Europe for 12 units so those are going to be variables that you see bring up the cost. When you consider 12 units that's a whole quarter. In one summer. And again that includes your housing excursions. Some meals I believe as well. And so it's important to see what you understand what you're comparing. What would a session at UCLA for summer cost you or any other term of the year at UCLA when you factor in housing excursions and all of those things and I think you'll start to see the picture. It's really kind of normalized and in some cases, you might find it even less expensive than attending UCLA. Next slide, please. And sorry about the way that I just wanted to check and say, we're actually running a little behind schedule. I don't want to rush you but all incredible information but I just want to plant that seed I'm sorry. Thank you no worries I'll go, I'll move a little quicker. Let's go to the next slide. See. Okay. We again just recommend coming to our office or or sorry the virtual advising is available to everyone. We have study abroad advisors, you can also talk to your major and school advisors. We have peers from prior participants that you can check in with and certainly family and loved ones see if they have any advice maybe they did study abroad as well. Next slide. Again key is applying early, as early as possible but no later than four months ahead of time it is competitive and that it's first come first serve so it's difficult for us to tell you what it would be for a specific program. You can question yourself in the best way possible. Just to give you a point of reference applications for summer 2023 open in the fall of 2022. And again that's a first come first serve. And let's go to the next slide. Safety is important to us as I mentioned earlier so we've got a lot of resources on the ground. We spend a lot of time planning we will have staff on call. We will ensure students for most of our programs, and we are dialed into a lot of resources on a national and state level to make sure that students are cared for in the event of any crisis including corporate. And next slide. We also feel very strongly in our program that all all students should have the opportunity to study abroad and that it very much touches on diversity and how we can continue to diversify the pool of students who have these opportunities. We've developed a number of resources on our website, and you'll see the bitly link there. I encourage students with, you know, intersection intersecting identities to check them out we've spent a lot of time on that, and we want to assure everyone that study abroad is for you. Next slide. This is our contact information and again we are available for virtual advising. Please check out our website, shoot us an email. We are eager to talk with you and to help you plan your next adventure. Thank you very much. Thank you, Magdalena. I pangs of regret here. As I told you and just earlier I didn't take advantage of the opportunity to study abroad. So I hope this information is really valuable for our students in the room who may be on the fence. And now maybe considering studying abroad. Are there any questions for Magdalena before we move on to our Q&A panel. I'm just hand raising and I will happily unmute you. I don't think I see any. Just scanning through here. So let's let's move on and hear from a pair of Bruins who are previous study abroad. Participants with us. We have we have Giselle Rios. She is a UCLA library employee she works in the Charles Young Research Library where she sits with collection development of Western European materials, as well as overseas efforts for her department outreach efforts for her department. She's simultaneously earning her master's degree in library and information sciences, and she expects to graduate the end of this quarter congrats Giselle. After this role, Giselle worked at UCLA summer sessions, where she scheduled programs and classes for academic departments on Giselle study Latin American and US history as an undergraduate at Georgetown University. And while she was a Hoya she studied abroad in Denmark. We're also fortunate enough to be joined by Jacqueline Tafoya, a 2021 graduate of UCLA with a degree in Chicanx and Central American Studies. In her junior year. Jacqueline studied abroad in Mexico City through UC study abroad. And she recently began working as an intake advisor. She was in the Atlanta in UCLA international education office after previously serving UCLA destination college advising core. So without further ado, let's hop into our Q amp a and if anyone here has any questions for Jacqueline or Giselle, please just raise your hand but we have a few preset questions we're going to ask both of them. One. So what compelled each of you to study abroad. So I was very fortunate enough to have a professor who really encouraged me to study abroad at first being first generation college student I had no idea I had this opportunity. I'm also not from LA. I grew up a very small town. So those options were never open for me I never knew anybody from my community who had done it. So when I went to office hours I had a professor who really encouraged me after I was sharing about my passions and my interest my family is from Mexico. So I did highlight that and that's where she suggested I study abroad. Then I came for an appointment here at the study abroad office and the advisor here was very kind very nice and detailed everything for the program. And that further encouraged me to go ahead and apply and that's what compelled me so I had both my professor and then the advisor who compelled me to study abroad. Actually before we go into just so we had a question in the chat from Courtney said do you need to be fluent in Spanish to live in Mexico city and they seem to mean also study abroad in Mexico City. Though I am fluent my specific program was at a university in Mexico City so my classes were taught in Spanish, however, there's different programs where you do not have to be fluent, where the classes will be in English. So, definitely encourage you to come to the office or have an appointment set up with one of our counselors and they can further detail the programs that we have, but the short answer is no. What compelled you to study abroad. So I studied abroad because when I transferred to Georgetown I as a first generation student I was on scholarship and that's when I realized that I had about two years to enjoy Georgetown and as much as I really wanted to spend as much time as possible on my new, my new school. I took advantage of that scholarship and decided I was going to study abroad for an entire year and so that's what I did I studied abroad for two semesters in Denmark and I think it was definitely the best decision I could have made. Because, although I kind of disrupted my timeline I could have graduated a little sooner, but by studying abroad I think I gained so much more. That's great. That's a great question. So there, as Magdalena kind of went over there, different types of study abroad experiences. Tell us about your particular study abroad experience and why did you pick your program. I originally wanted to go to Spain, however, it all came down to me specifically with finance and Mexico, the best was just cheaper and I just really wanted to make the most out of it. So that's why I ended up choosing Mexico City. As I mentioned I am Mexican American so I always wanted to explore my culture more and I always had a dream of being there for Independence Day and celebrating my family's country of origin, which I was able to do things to studying abroad. Personally, I had several reasons for picking my particular program which was the Danish Institute for Study Abroad in Copenhagen. One of the main reasons was that back when I was looking for programs to study abroad in. They didn't have a lot of options in Scandinavia and I had a general idea that I wanted to, you know, explore and live in Scandinavia and the only program that they had was in Copenhagen. Now, upon closer inspection what stood out to me about my program was that it was really unique and that you could concentrate in a specific subject area each semester and based on that specialization you would study abroad with. They would actually take you on a trip with your entire class. It was included in the program and it just seems like a really well set up program so it wasn't just the fact that I wanted to be in Scandinavia was also the program itself, and the fact that it offered so many different housing options and opportunities that stood out to me and made it more appealing than other programs that I was looking at. Third question. What was one academic and one non academic highlight of your time abroad. An academic was, I always share this with people that I ended up learning Mandarin in Mexico City. The school there just had a lot of different opportunities, one being being able to learn another language. Again, the best was very cheap so my books ended up being like a dollar or so. So I just took the opportunity and I began to learn Mandarin, and that eventually opened the door for me to explore more languages and so now I'm learning Korean little by little. And so that was a really awesome academic highlight abroad. I also was able to develop my Spanish, particularly my academic Spanish academic Spanish, but definitely Mandarin because I was also able to meet other people. The school that I was in specifically is very well known within Latin America, it's called Nam. And so there is a lot of different international students who arrived there as well to study abroad and so I met students from China and so while I was also practicing my Spanish I was also trying to practice my Mandarin, which I thought was very awesome. And then a non academic highlight definitely the friendships as I mentioned not only was I able to communicate with my native language in Spanish with other students from China for example, but I was also able to have friendships from develop with students who went to like UC Davis UC Santa Cruz and so I was also able to kind of network in a way, which I definitely enjoyed. So for me an academic highlight was learning Danish, it was a so as students in the Danish and stupor study abroad we were required to take Danish every semester that we were there. So I ended up taking Danish to semesters and it was a language completely different from any that I've studied before but I now thankfully can actually read it write it and speak it a little. And then a non academic highlight was just them living with my host family so I was fortunate to live with a family who lived 30 minutes north of Copenhagen. And that was the most rewarding experience they became like my second family and you know we still keep in touch I've seen them a couple of times since and to me that was definitely the highlight of this experience. So final question for our panelists. Do you have any study abroad of regrets. I'd say my only regret is not having more fun. Again because I'm first generation I was very scared that I was going to mess up somehow that I maybe wasn't going to pass my classes even though the classes are I was able to choose them so I, I knew what classes I was going to be I think it was just more the nerves where like the first months that I was there. I was just a very applied to all of my like academics where I had a lot of friends who were able to explore more of Mexico. So that's definitely one regret. Definitely recommend having fun if they invite you to go places definitely say yes, because this is a once in a lifetime opportunity especially because I personally do receive or had received financial aid. So, having most of the program being paid for is obviously a blessing. So definitely have fun. Don't think about it too much. Yes. Did you go to a Puma's match while you were on campus at, you know, can you repeat that. Did you go to a Puma's football match. There you go I did not go to a Puma's game which I should have but I did toward the stadium but unfortunately I did not. Okay. And I have and I didn't and I spent all of three days in Mexico. So it's all about priorities right. I actually I kind of echo what Jacqueline said, definitely wish I had just taken advantage of being there more and gone out more made more friends. I, my biggest forgot personally from that and do you feel like I was just in a bubble at times just going back and forth for my routine from class back to my host family's home and traveling on the weekends but it really enjoy the city as much as I would have liked. But thanks for both sharing your insights and believe incredible experiences you had in two very different parts of the world but I'm definitely a huge enriching part of your college experience. Do we have any questions for Jacqueline or Giselle before moving on. Keep an eye on chat and hand raises. Think we are good. I know that everyone has made it through the really valuable enriching portion of our adulting 101 session now I get to take over a little bit and share some of my tips and tricks from 20 plus years of independent travel and as I mentioned earlier I did not study abroad. And I alluded to it a lot of college students they'll come at the end of their UCLA experience and say really shy to study abroad I didn't figure out so several years later, several years later that I really should have studied abroad so I'm making up for it as I can with independent travel. Let's put together a quick little slideshow of me through the years. If you want to all dream that's many, many years ago, Sydney, Australia. Anchor what temples in Columbia, Columbia, Cambodia. It's doorway in Morocco. You go Sioux Falls in Argentina.  I was in Hong Kong a couple years ago scuba diving in Hawaii. This is South Korea where I stopped over for one night to visit a friend from high school with their soccer as I alluded to as one of my passions here I met a match in Guatemala city, rooting on the United States, Guatemala one that day which is okay. They were very happy. Petra in Jordan, and my friend who's a really good photographer took that shot I'll never probably look that artistic again. Just a couple more Rio de Janeiro several years ago, writing the Children's Railway and Budapest it's actually a train line that every employee is 14 years of age or under, not the actual train conductor but all the employees who have experience thing. And that's in New Zealand on a sled type of thing they have going on. And then finally, on the beaches in Tobago in the Caribbean which is a place I always really wanted to go finally found a cheap ticket and my partner and I went several years ago. I even dorked out and I put together this map of every flight I've taken in my life to the best of my knowledge. Added up to 530 flight segments, almost 800,000 miles and 113 days of my life spent on airplanes, which was kind of crazy when I added it all up it's three and a half times to the moon and back. So what I'm going to do and it's not about me it's about everyone else and inspiring a love of travel and doing so cost effectively efficiently trying to travel longer so I'd love to share some tips and tricks with everybody. First I just want to talk about like traveling with intentionality and again if you have your own tips, feel free to raise your hand feel free to put them in chat. This is not a be all and all list. Wherever you go in the world, learn the language basics. And I've found that just these five words or phrases go a really, really long way to being appreciated by locals. Obviously hello, obviously goodbye. Thank you. Excuse me, I'm always bumping into people are trying to navigate past people so I say excuse me a lot and whatever language I use. And this is a big one where is the toilet. I do haven't traveled abroad much. Most of the world they, they don't go with for a restroom or bathroom or washroom they just cut straight to the chase, and they use the word toilet so figure out the phrase where's the toilet wherever you're going in the world. To is to research your trip in advance, but don't overplan it. So guidebooks I love travel guidebooks guidebooks travel journals and narratives movies set in the destination you're visiting music from there. Those are all great resources to really get acquainted with your destination. And as a shameless plug here in the library and Powell room one 145 we actually have a collection of travel literature. And so if you know you're going stop by Powell you can either check out a book or more and or just sit and read and get acquainted with your, your potential study abroad destination. However, some people really have a tendency to just overplan everything down to the minute I have my itinerary set from sun up to sundown. I really recommend against that obviously you want to do some sort of logistical planning you want to book your accommodations especially for your first night in a city so you're not bleary eyed and jet lag. You want to reserve tickets for popular traction you don't want to just walk up to the Eiffel Tower and expect to be able to climb to the top, and you want to reserve that in advance, but definitely allow for some serendipity in your travels, somebody may invite you to their home for dinner. Somebody may have no the cool event going on you don't want to have to turn that down just because you've planned every minute of your trip. And I think some of those spontaneous moments will be among your most treasure. There we go. Number three, most major cities in the world have a free city walking tour that they offer. Something's going on with my advancing but usually they're led by young adults they usually people who are training to be full time tour guides and they need to have a certain number of hours under their belt of giving tours. My advice is sign up for a free walking tour take it on your first or second day in the city. That will help orient yourself and kind of learn some of the historical context of where you are, some of the modern day realities and you know they're usually about 1520 group tours so you'll meet some fellow travelers along the way. And then yes they are free. It's customary just to provide a tip at the end $10 usually suffices it depends where you are in the world but tip is always appreciated by the guy. And then number four, I'd say prioritize experiences over sightseeing. So the people you meet along the way and actively engaging in a culture will resonate a lot more than ticking off a list of must see destination, you know, must see attractions on your trip. Plan around things like festivals concerts sporting events, or like a long height multi day hiking track. And then travel is an incredible opportunity to learn something new you have a block of time or you're not working you're not in school sometimes you are in school, but you have this unencumbered time. So try to learn something new take a cooking class perhaps get scuba diving certified. That's something I did in Honduras. It's a great week long trip. Next I just want to go into flights because I think it's my galena lead to that's probably gonna be your single biggest outlay of cash. So forgive this internet clip art GIF that I found. But my first flight tip is avoid backtracking. So, learn the phrase open jaw. That means when you fly into one city and back from another. When you want to go to South America, you want to start your trip in Lima, Peru, and ended in Rio de Janeiro, you don't need to book a flight to Lima and then back home you can do what's called an open jaw ticket, where you start in Lima and fly home from Rio. And the advantages to that are it often rarely increases the cost of your international ticket sometimes it decreases it. And then you also save the time and money about not having to backtrack to your original destination. Sometimes think of an open jaw ticket is allowed or you can fly LAX to Lima and then for some reason you need to be in New York City so you can fly Rio to New York. Flight tip number two, I call it just get to the continent you're interested in visiting. This could be really advantageous if you're studying abroad let's say you need to get to Budapest for a study abroad program, but you're researching tickets and it's $1,000 to get to Budapest. Well, here's a solution. You perhaps you find a ticket LAX to Dublin for $600 roundtrip, and then you find a budget airline kind of like Southwest Airlines there's a ton of budget carriers in Europe, and that can get you from Dublin to Budapest for about $100 So it's two separate tickets but you're saving $300 in the long run, and there's a bonus to that, you can then stop over and see Ireland for a few days or however long you want along the way. If you want to travel caveats you'll definitely want to allow plenty of time if you do one of these stopover type bookings, and then watch out for additional fees a lot of the budget airlines just like we have here, they'll charge for seats they'll charge for bags etc. Flight tip number three, and I got a screen grab there from Google flights. You know where you want to go set up a Google fair alert. That will set your route set the dates or approximate dates you want to travel, and then monitor a little bit fluctuations and price. So what I've got on the right here I'm just monitoring a flight to London over Thanksgiving. You can see on the graph, it had dropped as low as 550 bucks round trip. It's currently at $1,100. So it's been that way for what three weeks now so say I get an alert and it drops down to $600 $650. I'll know that's a really good airfare and that's the time to lock it in. And then another trick is to keep your Google fair alert active after you've purchased a ticket. The reason is airlines have become a lot more flexible since coven with refunds and exchanges. So there's a chance you can actually get a partial refund on your ticket or credit with that airline. If it drops even further, you just have to contact them or go online and manage your account. Final flight tip. Book first, decide second. So some of you know that airlines in the US, if you book an airfare, airlines are legally required to refund the ticket within 24 hours of booking. And that's advantageous because if you see a really great airfare, stop what you're doing in a place you want to go, stop what you're doing and book it. And then move by the minute I've done some research and I've hindered how do I want to commit to a ticket do I want to go there and it's jumped up by $100 in the next 10 minutes. So airfares can literally change like that. So then book your ticket and use the next 24 hours to request time off from work. Find a friend who might want to join you and then research the destination find out if it's a cheap place to go an expensive place, see if it really interests you. And if you ultimately decide if you get cold feet and I really don't want to do this trip, you just cancel it online and the money's refunded to your credit card. Last set of tips I want to offer is around accommodations. So first, I think in my 20s, I definitely stay in hostels to save cash. Sometimes that word really scares, scares people who have never done it before. They think it's like this, you're sleeping with 40 people and it looks like army barracks. And occasionally it still does but generally hostels nowadays are nice modern have tons of amenities and the amenities you really want to look for a youth hostel. If you're in a place with a central location you don't want to be 2530 minute bus ride outside of the city or be somewhere pretty central, especially if you're only somewhere for a few days. You want to look for someplace with a big clean kitchen one of the best money saving tools to travel is to go to a grocery store stock up on items and make a make a meal or to make your lunch make your dinner a few times. And I'll digress for in supermarkets are pretty much my favorite attractions or places to visit when I, when I get somewhere. I feel it's the best kind of glimpse into the culture.  I'll digress, but a big clean kitchen scene cook for yourself. A communal space, ideally a hostel that has an onsite bar or restaurant, because that will just be like a magnet for your fellow travelers to hang out, swap stories meet people new I see Jacqueline, you know, nodding a little bit. And then finally, a lot of hostels schedule organized activities I mentioned the free walking towards sometimes hostels schedule walking tours. They have classes they'll go on bike rides around town etc etc. And they usually had a pretty reasonable cost. And then if the thought of sleeping in a 15 person dorm with people you don't know is really scary. The ultimate sweet spot is a private room in a hostel. You get the privacy. You get the benefit of meeting fellow travelers, it costs a little more. And if you're traveling with a friend you split the cost so it's actually not that much more so I would highly recommend private rooms in the hostels. Second accommodation tip is booking a vacation home like an Airbnb or VRBO really that's best for groups and or longer stays and the reason is they all charge a cleaning fee and a service fee. So that tends to make it impractical for a short stay. The cost of that cleaning fee $75 $100 is going to be the same whether you stay two nights, or whether you stay two weeks. And then this is not everybody knows this but you can actually negotiate with Airbnb hosts, same with VRBO. And then you can send them through the platform say hey I really like your apartment. I'll be in Tokyo for a week. Is there any chance you would be willing to discount it x% or just count it down to this amount. And unless it's a really busy time of the year where they know they can fill their, fill their, their space, they might be willing to work with you and drop the price of it and then they'll reset the pricing. Finally, I don't know why that just came up but yeah usually during the low season is when there's been to negotiate. And finally overnight trains and buses. So what's better than getting onward to your next destination. I find it's getting there while you sleep and without having to pay for another night of accommodations. If you do do an overnight bus or train do definitely do your research because they're all privately owned companies and the quality of their service the comfort level can really vary widely. Some routes also the time it takes some routes stop at every town, and what's usually a four hour journey can turn into a seven hour journey. So research how long it takes. And then also paying a little more can be a much more comfortable experience trains have private cabins. Some buses even if you if you travel abroad particularly in Latin America they have these incredible double decker buses with seats that recline almost fully. So it's another great way to save on accommodation is also really efficient thing to get to get to your next destination in the morning and you've avoided the night of having to pay for a hostel or hotel. On that note, we're a couple minutes over. I'm not going to go through these one by one but I just put a lot of flight resources that I use to find cheap airfare. These all be linked in the deck that everyone will receive afterwards and will be linked to on the library website. So just mishmash of other travel resources hostile world for booking hostels trip it.com I love that's an app that's basically stores all your itinerary details, and you can even create a link, send it to your parents say hey if you want to follow where I am on my trip. Just go to this link and you'll find out where I am day by day. Rome to Rio is great for comparing transportation options. Isaac is great for college students because it's an international student ID card. And there's lots of discounts associated with it a lot of museums will give you a student discount you just show your Isaac card, and then Google offline maps. So most people now travel abroad they have their phone enabled. They have a data plan either they buy a local SIM card or you just go through your company here. But looking at maps and navigating can really eat up a lot of data. So Google offer something called offline maps where you set an area you go to say I'm in Mexico City, you set Mexico City and like a 30 mile radius. You download that map and then you're using it offline and it saves a lot of your data. On that note, I know I spoke a mile a minute I was trying to keep our, our session to end as close to three as possible. Thank you all for joining us. Definitely have a couple. A couple of minutes for I can stay after I know about some of our other panelists if you have any questions, just wanted to say thank you on behalf of our group. And then we have a adulting one on one survey, which if I can I will try to drop in the chat. I don't have any questions before we wrap up and move on. I know it's a really busy time of the year so thanks for joining us. Oh, looks like we really covered a lot in an hour. Thank you Magdalena. Thank you Jack. Let's have you pronounce your name I'm so sorry. No worries Jacqueline Jacqueline I'm sorry I had you do it phonetically via text but I still got it wrong I'm sorry Jacqueline. Thank you as well for sharing all your insights about study abroad. Again, we'll we'll make this deck and the recording available. I'm so look for that in your inbox. And again, thank you everybody. Thank you. This is a great session. Bye bye. Thank you.


All right, so welcome back next section module finite extensions and plus closure. Um, so in this section are will be integral main, so we don't have to worry about equidimensionality like what was talked about in the questions. Um, so, um, So let I be an ideal and let X be an element of our we say X is in the plus closure by if there exists a module finite domain extension are into s and what I mean by that is that s is an integral domain, and you've got this injective ring homomorphism from our into s such that s is finite as a normal finite as a module with respect to that the structure homomorphism of that as a module, I mean this homomorphism. Um, such that, so it's called a speed, such that the of X is in the of is. And so we write, usually write X is in is intersect R. Right, because if you say X is in the pre image of that about. That extension. Okay, so proposition plus closure is a plus is is a closure operation.  And so what we want to do is we want to find a common module finite domain extension of s and T. And so what we want to do is we want to find a common module finite domain extension of s and T. And you can always do that. And I noted, but not how to do it.  So hence, X and Y are both in you intersect R. And so X plus R Y is to um, so I plus is an ideal. And I claim that extension and order proper order preservation are clear. And so, if X is in I plus plus. It's quite a potence. Um, so you have X is in I plus s for some module plane extension s. But why one through Y K generate I plus but T one through T K, the module finite extensions with Y, I, Y J in I T J find a common module finite sensor of s find a common model to find extension you find each Y is in I U. So X is in I as well. Okay. So, now let's say you have module finite extension of domains. Take any non zero element D of s and then there's some R linear map module map call it G from S R such that G of D is non zero. This is what Mel Hoxter later called solidity. So instead of saying that s is solid. So what do you do? So let W be the non zero elements. And so then Q is equal to the fraction field of R is equal to R W. And for all W and W, we have W D not equal to zero. So D over one isn't zero. SW I guess I should say right so you have R W into SW. And so. So what do you have so then SW is a Q vector space. It's a finite dimensional finite dimensional new vector space with the over one zero. So, then it's part. So then the over one is part of a minimal generating set that is part of the basis. So we'll let the SW to Q the projection onto like that. So, and that's a Q linear transformation, right. So the set of Q linear transformations from SW into Q is what home R W SW R W. And then since S is finally presented as an arm module. That's the same as home R S R localized W. So there's a R linear map from G from S to R. And C and W such that the is equal to G over C. And then that's the same thing as saying that G of D over C is V of the other one is one, which is C over C. And since we're in domains that's the same thing the same. So it was not so. So, why do we care. Well, if you have a modifying domain extension and I as an ideal of R, then I claim that I star is equal to is star. And actually equal is true but all I need is containment. Sorry, I need this. That's my plus. So, proof, let X be an IS intersect R. Let D be a non zero relevant of S for S to R. Zero. Actually, I'm going to do it this way. It's not zero. And q not a power of P, such that DX the queue is in IS to Q, which is I QS for all Q, not, then what you have G, S to R. That's the same thing here. With G of D is not zero. Then by our linearity, what you have C X to the queue is equal to G of D X Q. So that's our layer that's just what we have, but then that's the same as G of DX to the queue, because X to the queue is an R. And then G of IQS, which, again, by our linearity, this is contained in IQG of S rational local ring. Then I claim that R is a normal domain. Okay, so first one shows that R is a domain. So assuming that you knew there was a paper domains. So I'm going to show you the actual bill of R, let alpha be integral over R. Let S be our bracket alpha. It's going to be a lot of finite. Right, that's, that's one of the equivalent definitions of an integral elements, and we'll close your rings. So let's say alpha is equal to a over B. So a and B are in R and it's not zero. So a is in BS intersect R, which is contained in star, which is equal to.  So where you have a equals BC for some CNR. And so you have the alpha is equal to BC, and therefore by cancellation is S. There's probably a slicker proof of that somewhere, but I'm colon capturing test elements and persistence of the power of tight closure theory comes from this colon capturing stuff, which is allows you to treat a system parameters as if it were a regular sequence up to tight closure. So theorem. R be equidimensional quotient, Y bar, the system of parameters for R, then two things, actually a whole lot of things, but at least these two things for all J less than the Y one through Y J colon Y J plus one is contained in Y one to Y J star. And to T for all natural numbers T to take Y one to Y D to the T. You call in with Y one to ID. T minus one is single element. You get inside of the tight closure of Y one to Y. And so this certainly would be true for like variables over over a, you know, how long we're over a field. It's true for regular sequences without the tight closure and so but then the nice thing is that it's true for services and parameters in this very general context. So proof. Well there's proof one. So, first, reduce to the main case. Let P one to P K be the minimal primes of R by equidimensionality. I'm assuming that are as I could mention all the dimension of arm odd J is deep for all J. That's what I could mention is. So the image of one bar is a system parameters in each or larger. So, Y one to Y J colon one J plus one over R. And for each. Yeah, I just die. Y one to Y J bar colon over arm odd. And then with Y J plus one bar is by the domain case. Below containing the Y one bar to Y J bar tight closure and P. So then by reduce to domain case proposition above phase in the tight closure of these matters in. Okay, so there are we may assume now that ours of the main. Okay, so let S in the common quality local with R equal S mod J. And J is prime because ours of the main. But TV dimension of S, then like home the colonists, the height of J. So let's call that H is equal to T minus T. So let X one through X H the elements of J, such that in S localized at J. They become a system of balance for S localized J. So, then X one through X H S J is J S J primary. So, now, now just one question just just for clarification. Just for clarification, you say that your your sequence leaves in J so and you are working in S mod J. And when you are modding always mod J this is become a system of parameters but this becomes zero priority. I'm not working in S mod J. I'm working in S localized. Okay, okay. Okay, okay. Okay. Thank you. So there is. Yes, yes, sir. So there exists an M such that J S J. The end is contained in X one to X stage. That's J. The maximum primary primary and maximum ideal means that some power of the maximum ideal is contained in your ideal. So there exists a C in S minus J with CJ to the end is contained in X bar let's call it X bar. So we're kind of working in S here. So let's choose, and I'm calling it C suggestively, because this is going to end up because that the residue class of this is going to end up being sort of testing, or parameter ideal stuff. So choose q naught bigger than or equal to n, that's power p. Then, you know, CJ to the q naught is also. Okay. Um, so what do we have before we have this system parameters in R. So, let Z one through Z D. Yes, the listings of why one true. Then I claim that for any power q of P X one through X H, along with Z one to the q comma Z D Q is a commutable regular sequence in S. Right, because it's a system of parameters and a comma call a local ring and so it's a permutable regular sequence. So, so now to prove one.  Then what you have is that in in S of R, then S Z J plus one times a is in Z one through Z J plus capital J. So that means that for any q bigger than or equal to q naught. So, Z, Z J plus one to the q A to the q is in Z one to the q comma Z J to the q plus CJ to the q. But CJ to the q is contained in the X is right so this is contained in the one Q, CJ to Q X is. That's what you have C A to the q is in that sequence. comma the exit this colon Z J plus one to Q but this is a regular sequence of colon doesn't make any difference. And that's contained in the one to CJ to the Q colon or plus capital J. So you have C bar a bar to the Q is in Z one to the Q. Sorry. And that's for all and that and C bar is a non zero element of the integral the main R. And this is for all q bigger than or equal to this q naught, and therefore a bar is in Y one, one J star. So that's the one that's the sort of classic colon capturing and the proof of two is very is very similar modulo and exercise. And a remark, any excellent rings the homework the image of a comma collie ring, how a sock to prove that in 2002. And so this is true for any equidimensional excellent local. To similar. And as a corollary that the menomial conjecture in character, but but I didn't prove to so I won't state that. So the corollary to what we did above is an effrational rain in this circumstance has to become a colleague. So that are be effrational and the homework image effrational equidimensional. So proof of a comma collie local ring. Actually I don't even have a colleague. So proof by an earlier result is a normal domain. So the corollary to the national, but why one through YG parameters, and what you have Y12 y j colon y j plus one is contained in Y one through y j star, this is colon capturing, but then this is equal to Y12 y j rationality. Okay, and that's what the common collar. And the system parameters is back to record. So section seven, test elements of tight closure. Okay, so like I said before, this is something that really that makes tight closure very powerful because choosing the C is generally a persnickety difficult thing in a general ring. It's what makes tight closure in some ways more difficult to deal with than my previous closure where you just take the power and it has to be the correct power of an ideal. So right, you haven't really made tests and you don't really know how to test it because you don't even know what elements try to multiply by. Well, so definition. Let's see the element of R naught. And we say C is a test element, respectively a big test element. If brawl ideals i and brawl x and i star, C x to the q is an i bracket q brawl q, respectively brawl submodule inclusions L and M and brawl C and L star M, C z q M is an L q M brawl. So right, so big test element that's, there's, it's like tests for big modules as opposed to just ideals. And then there's this weak test element thing as well. So, and that might seem like a lot to ask for. And in some sense it is, but there's this big theorem by, you know, Hartzmann-Huecky and then there's part of it is in this thing of shark.  So that's the condition called F-finance. Then, oh, not just that there exists test elements, let's see the any element of R naught such that Rc is regular. So such an element C always exists in an excellent ring of spring, basically by definition of excellence, which has to do with the regular locality being over. Then some power of C is the test element. Is a big test element, C Tucker's lectures, proposition in, and then a proposition. If R mod P has a big or just a test element for every minimal prime P and R is reduced. So you really just have to do the main case, really. So I guess I should mention something about this theorem. You might not know what essentially a finite type over another ring is, so what that means is that there exists an excellent local ring A, and variables x1 through xn, w, a multiplicative set in the polynomial ring such that R is as far to A bracket x1 through xn, localize the w. So that's what's essentially finite. Okay, so I'll just one question. In your previous definition, maybe you can allow that this, in the previous definition you have given about R, maybe you can allow that you can also mod out by an ideal or is like this definition. Oh, sorry, yes, sorry. Sorry, yes, you're right. Or something, you can mod out some ideal there. Yes, you're right for now, I'm sorry, yes. Yes, there's an ideal i such, yeah. So you extend polynomial variables, localize, and mod out by an ideal. And so that's what essentially finds. Okay, good, good. So let's just prove the non-big base of this, that p1 through pn be the middle primes of R for each, let di in R mod pi, get test element, choose ci, ti and c as in the pack element, and c is a test element. And so let i be an ideal and x and i start, and for all i, x bar is in iR mod pi star. So for all q, di times x bar to the q is in iq plus pi, is in just iq plus pi. So ti, ti is in all these other prime ideals, all the other middle primes, is just one example of how this pack element works, and that's contained in i to the q since R is reduced. Right, because it's in i bracket q plus the intersection of all the middle primes of R, but all the middle primes of R hence cx to the q is in iq, since c is for the proposition 7.4, tight closure is a closure operation in such a way. So let R be a ring satisfying, I'll just say a ring with a big test element, then tight closure is a closure operation, not just the finite ones. So proof, we need only established item codes. Let z be in the tight closure, of the tight closure L and M, in M, let c be a test element, and for all, we begin with the q naught, so I'll draw q really, cz to the q is in L star qM, but also c times L star qM is contained in L qM, because it's true for all the elements of L star M. So c squared, cqM is contained at c times L star qM, L star, which is then contained in L qM, but c squared is also in R naught, thus z is in L star. So I said test elements and the persistence of tight closure. So I claim that with this theorem on test elements, you can then show the tight closure persists. So that's like better than functoriality, that means if you're actually changing rings, you can maintain that tight, that if you're in tight closure on ideal, then the image of that element is in tight closure, but the image of that ideal. So let G from R to S, your ring homomorphism. We say tight closure persists across G for ideals, respectively for modules, if for all ideals, if X, let's set it this way, G of I star is contained in, respectively for all sub modules inclusions L and M, G of L star M, let's say G prime, L star M contained in L S star tensor S. So what do I mean by that? So L S is gonna be the image of L tensor S, M tensor S. G prime is the sort of the natural map from M to M tensor S. So you take the closure of L and M and you take the natural thing induced and you take the closure of L and M and you take the natural thing induced from G and you apply it, you get inside of what happened if you first went to sort of tensor everything with S and took the type code out there. So that's persistence. You can see how that would be a useful, and the big theorem here is that is that existence of test elements in the way that we've said implies persistence theorem. Let's see from R to S, your ring map, such that R satisfies the conditions of the theorem on test elements, then type closure persists for ideals of M. So proof is for ideals. So basically you wanna reduce to the case where R and S are domains, G is subjective. It's subjective. So the reason you can go to that case is basically, okay, you reduce to the domain case to show that R and S are domains. If it's an inclusion of domains, then you're okay by an exercise. And so every map between domains factors through a surjection and then an inclusion. So you've got a surjection from one domain onto another and then there's gonna be an induction on the heights of times. So assuming that we're okay with all those reductions and it's all detailed in the notes. So let P be the kernel of G. Prime be the integral closure of excellence. All right, we're assuming satisfies these nice conditions. R prime is module finite. And so by integrality, you can choose a P prime and spec R prime with P prime intersect R is equal to P. And again, by excellence or by integrality, P prime has height one. Excellence of R prime since we know it's a theory and stuff. The regular locus of R prime is open. So there's a C in R prime minus P prime. That's true, but I skipped something. So R prime localized P prime is a DVR. Hence, right, because R prime is normal. It's a very normal domain. So by excellence, the regular locus of R prime is open. So there's some C in R prime minus P prime such that R prime localized at C is regular. Hence by the above theorem on test elements, C has power and C to the n that is a test element for R prime. So now let I be an ideal then X is in the tight closure of IR prime because persistence is okay for inclusions of domains. So for all Q, CX of Q is in IR prime, right, Q. So going mod P prime, you have C bar X bar of the Q is in IR prime mod P prime tight closure. And C bar isn't zero in R prime mod P prime. So that means that X bar is in IR prime mod P prime. But R prime mod P prime is a mod of finite extension of the domain. So when you have X bar is in IR prime mod P prime intersect R mod P star, which is then, but R mod P is S and that's what we wanted to show. So sorry for skipping all the reduction steps but they actually do end up being quite straightforward. So this has some important corollaries and I'll end with those. So let S be a complete weekly of regular local ring and R a complete local subring such that IS intersect R is equal to I for all ideals I of R, for instance, R is a direct sum and of S will always apply that. This is called secret period. Then R is weekly of regular, it's approved. So I is contained in I star, which is contained in IS star intersect R. This is by a persistence is a direct sum by some end of a complete ring to complete. Oh, actually I didn't even say that I said R is complete. So it's an excellency stuff. And it's done anyway. So this is equal to IS intersect R which is equal to I since S is actually a regular. So now proposition S for regular and R and Ethereum subring such that R to S split and R is common color. And if R is locally excellent, it's also normal. Yeah, and I'm out of time. But it comes from persistence because you can, well, it comes from this corollary above. You've got these basically, once you reduce to a nice case, you reduce to the complete case. So S is a complete regular local ring. Therefore it is weekly F regular, in fact it's F regular and therefore R is going to be. And you also get F rational or general. And so F regular is twice weekly F regular. And that ends up giving you a new proof of the Hoxter-Roberts theorem that if S is a polynomial ring, K of field, then G is a linearly productive group. Then the ring of invariance S G is called the collie. That's because it's direct summit because there's this thing called the Reynolds operator which splits the inclusion from S G into S. Okay, again, sorry for rushing. I guess I put too much in these lectures but it was hard to know what to believe out. I did believe out a lot of things. Anyway, thank you. If you have any questions, I assume there's probably a lot. I have one question in the previous corollary. Of course, if you have that R inside S, if you have that R inside S is faithfully flat, this also works, right? Because the condition I S intersectional F is automatic. This is more general decision. Of course, if this is a direct summit, this is another situation but I think faithfully flat is a bit more general sometimes. Well, I mean, they're independent properties. Yes, yes, yes, you're right. It doesn't imply it's a great summit and direct summit doesn't imply a faithfully flat but they both imply what's called secret purity which is this property. That's right, that's a good comment. Yeah, so that's a good thing. Yeah, that we kept regularity among complete local ranks, descends along faithfully, sorry, along faithfully flat maps. Yeah, so you got all this power from being able to pass tight closure. Katie, you have a question? Actually, there's a comment. Cindy Hochscher, I would say, you're assuming characteristic of the field doesn't divide the other two. Yeah, yeah, I'm assuming that the characteristic, I think that that's part of being linearly deductive, isn't it? Yeah, maybe I need to assume. Yeah, and then the other question in the chat was answered in the chat. Well, it's not quite true. So normal, I guess the definition of normal is different from what was said in the chat. So the question is, what does it mean for ring to be normal? What it means is that whenever you localize at a maximum ideal, you get a ring that's integrally closed, or that you get an integral domain that's integrally closed in its fraction field. But it turns out that if you had an inferior normal ring, it's going to be a finite direct product of an inferior normal ring. So that's in the basic theory of integral closure of rings. So that answers those chat questions. So for instance, if you're local, then you're really close in your fraction field. Then normal is the same thing, it's a neural domain that's integrally closed in its fraction field. And that the invariant stuff and the direct summand stuff, I think Lin-Chuan Ma is gonna talk more about that in his lectures, most likely. So in your proof that an efferational ring is normal, it seemed like really, I mean, is this factoring through, yeah, sorry, that was quite a while ago, is this really factoring through somehow in fact that for principal ideals, tight closure is equal to integral closure? Yeah, yeah, I mean, I guess that's equal. Yeah, that would be another way to say it. I tried to make these lectures without having to mention a closure of ideals. So yeah, I mean, it's essentially because of that. Yeah, there it is. So I guess it's something like a further name and a domain where every domain where every principal ideal is tightly closed than your normal line. So yeah, so here's the sequence of the implications. And some of this is gonna be done in the Leonson-Skoda lectures. I forget who's doing those, maybe Ian Averbach. So for principal ideals, tight closure is the same as integral closure for those ideals. And so, and in general, whether you're characteristic key or not, a ring is integrally closed in its total ring of fractions, if and only if every principal ideal generated by a non-zero divisor is integrally closed. So therefore it follows that an effrational, not an effrational, it's an effrational local ring. But yeah, that's right, that's right. I mean, the thing is that the integral closure, but that also factors for the fact that the integral closure of a principal ideal is in some sense a much simpler sort of object than the integral closure of a general ideal because an integral closure of a principal ideal is just the same thing as go up to the normalization and the trackback, about to the integral closure of the ring in its total ring of fractions and the trackback. And that's not true for non-principal ideals in general, like two-generated ideals. It's like really false a lot. And but so this like principal ideals are simpler than other ideals, no surprise there. Yeah, thanks. Okay, so we want to thank Neil for his nice lectures. Well, thanks for bearing with my pacing issues, but I hope this is useful stuff to understand the basics of tight closure and how to work with it and the other lectures going forward. Thank you, Neil. I'm saying it's factorial. Okay. Yeah, yeah. I think there's a lot of definitions of integral closure of modules and it's like, but I think in all the existing ones, that they are actually factorial. Okay, there's a question in chat. What's the relation between Frobenius closure and integral closure? Oh, yeah, I'm gonna, so I was gonna get into that later in the lectures. Frobenius closure is so pretty, integral closure is bigger, is the short answer. But in fact, you have a whole sequence of closures. So for any ideal I, the Frobenius closure of I is contained in the tight closure of I, which in turn is contained in the integral closure of I, which in turn is contained in the radical of I. And just like tight closure has characteristics zero analog, does Frobenius closure also have characteristics zero analog? I don't think so. I mean, I've never seen it. I mean, I guess you could try to do reduction of characteristic P. I mean, the natural thing would be to say that in all the characteristic P models, I'm sorry, that in infinitely many characteristic P models, that sort of the characteristic P version of the element would be in the Frobenius closure of the characteristic P version of the ideal. But I mean, yeah, I've never seen anybody do that. I mean, there's reasons like, I mean, for a long time people thought that tight closure would have this property that if you know, through the tight closure in infinitely many characteristic P models, that you're going to be able to tight closure in almost all of them. But then Brenner and Katzman have a counter example to that. So that's not true. And it, but it was known from early on that that wasn't true with Frobenius closure. I have an example in my lecture that was at the end. So, but yeah. I guess you could make that definition of Frobenius closure. But in general, Frobenius closure is considered to be more poorly behaved in many ways than tight closure. That's a segue into another question that I have. And this, I guess, is more related to material from Craig's talks. So I feel free to defer. But I'm very interested personally in my own research is studying Frobenius closure. And in particular, I've been trying to develop the right, you know, and maybe it's out there, but the right map theoretic notion to get integral closure of ideal, or sorry, Frobenius closure of ideals. So is there a notion of, you know, maps to the right kinds of rings that when you can track back, you're gonna get the Frobenius closure? Oh, yeah.  Yeah. So there's this thing called the, so I don't know if it's maps to them, but I mean, I will say there's this thing called the perfect closure of a ring, which is the, and you probably know about it, which is basically all the R to the one over Qs at once, like the union of the R to the one over Qs in that ring K that I showed, assuming the rings reduced. And one way to define Frobenius closure is extend to that ring and you track that. And similarly, you know, you look at the, for our module, if you have a module M, you look at the natural map from M to M tensors with what's called the R-perf. It used to be called the R-infinity, but everyone says R-perf, you know, for perfect. So you look at the map from M to M to M tensor R-perf and you look at your submodule L and you look at the elements that in that map land inside L tensor R-perf. I mean, that's the elements of M that land in L tensor R-perf and you call, and then that's the Frobenius closure of L again. But I don't know if that's what you're asking for. A little bit. I mean, so R to R-perf is an example of a purely inseparable extension, right? So I'm wondering if something like, you know, you contract I back from the recurrence of the extension of even at certain flavors that you'll get in Frobenius closure. But anyway, this is a research question that's not made enough for me to decide. I bet that'll work. Oh, okay, yeah, interesting. Yeah, maybe I'll follow up with you some other time. Thanks. Can I ask you something, Neil? Of course, I can. Yeah, this is not regarding what you covered, but like, is this first part of your talk will broadly follow your paper on like closure operation in community algebra? Yeah, yeah, that's sort of where I, that's when I developed this approach is, yeah, that's in there, yeah. Some version of that's in there, but I mean, some version of that's in these lecture notes too that they go see soon enough. Yeah, you know, I wanted to do a survey and closure operations and I was, and I thought, well, how broadly defined are these things and like really broadly, they're even more broadly defined than this. After Moore did his stuff, then like, you know, a post set theorists said, okay, well, you can sort of define this on any post set. It doesn't have to be the post set of subsets of the given set. It can just be any partially ordered set. And then you, you change your containment relations to, you know, the less than or equal to sign that exists in that post set, you still, and then it's still, if it's a post set where all arbitrary meets exist, then you still get this one-to-one correspondence that Moore pointed out in his paper 112 years ago. So yeah, I mean, it's very, it's very general. And that's, you know, I think they used that somehow in the lattice theory in a way that I don't, that I've never looked at. Any other questions? Yeah. I'm sorry. But yeah, Kyle, I'll happily defer your question to anybody that might know more about it than I do. Any more questions? I've looked and it seems like no one has written about, I mean, much less has written about Phrenius Bogey's closure, which is where I'm trying to make money. That's what he's doing. No, it's great.


Appalachian Wireless has some of the best data rates in wireless on our plans and not to mention an amazing deal on one of our hottest smartphones, but that's all the time I have to explain. See store for details, better service, bigger savings. That's today's Appalachian Wireless. On Tuesday the partnership between Berkeley Energy, Edeland Strategic Ventures and EDF Renewable Energy for a solar farm near Pikeville was announced. On Wednesday, Ryan Johns with Berkeley and Adam Edeland came to town to talk about the project. Well let's make no mistake, this isn't a foreign company coming in here to do this project. This is a couple of Kentucky boys who had an idea to try to help an economy that's struggling. Johns and Edeland said they've received positive feedback locally for the planned solar farm, which will be located on a former surface mine at Top Ford Mountain at Johns Creek. It was really good to get to see people's reaction to what we're trying to achieve here. I believe that most people are excited about any opportunity to bring jobs and stuff into the area. We've been overwhelmed by the response. Obviously there are a lot of people who are really curious, but I think one thing that everybody needs to understand is that we've seen the economy here decimated in eastern Kentucky over the last 10 years and Appalach even beyond that. So I think people are hungry for new opportunities that will give them the ability to provide for their family or have enough opportunity to keep their kids from having to move out of the region to find work. Johns and Edeland said they plan to use the local workforce to construct the solar farm. I'm really pleased to say that EDF has totally embraced the vision of Berkeley Energy and Ryan Johns and I own a piece of that as well to make sure that this project is built for the benefit of the people who live in this community. The project is planned for construction sometime in 2018. In Pike County, Chris Anderson, EKB News.