making topics over a simple lunch of soup and bread. And this event will be recorded today. You can find the recording on Marquette University's YouTube page. Our sponsors for today's event are Marquette University Center for Peacemaking, Office of Mission and Ministry, Peace Studies major, and Campus Ministry. And Pat Kennelly, the director of the Center for Peacemaking, will be introducing our speaker today. So I'll pass things over to Pat now. Well, thanks, Amanda. And I just want to thank everyone for being here. One of the silver linings of the pandemic is we've been able to really expand programming. And this year at Marquette, one of the sort of climaxes of the year is Marquette Mission Week. And so I think doing things virtually, we're really grateful that so many people from around the country can join us. And I want to thank Amanda, the student who facilitates super substance. It's a weekly series. And so if you like what we hear from Sister Kathy, and are interested in these topics, I'd encourage you to come back each week. As Amanda said, my name's Pat Kennelly. I'm the director of the Center for Peacemaking at Marquette. And we're really excited to have Sister Kathy Sherman with us today. I've gotten to know Kathy a little bit over the past couple months and realize that not only is she a gifted singer, composer, and activist, she also really challenges people to be present, to grief, while also at the same time holding on to hope within sort of that beloved community that Dr. King and others have spoken about, and certainly the prophets have written about, and the sacred texts. Sister Kathy's passion is writing music that fosters hope and healing. And I think where that conviction lies is not only her faith, but her belief that all life is permeated with the sacred. And she's deeply committed to working for nonviolence, racial justice, and advocating for the protection and healing of our planet. As some folks heard as we were talking before the event, Sister Kathy is a member of the Congregation of St. Joseph. And the CSJs have been just dynamite partners with Marquette and the Center for Peacemaking through their generosity and frequent visits by different CSJ, both nuns, but also affiliates. We've been able to reach literally thousands of youth throughout Milwaukee and carry those charisms of care for the earth, and justice, and peace, and nonviolence to the next generation. So we want to say thank you to all of the CSJs. And I'd be remiss too if I didn't recognize Parisa Shirazi from the Center for Peacemaking, who worked closely with Sister Kathy to bring this event together. And so without further ado, please, if you have a second, if you want to unmute, and we could give a warm round of applause to Kathy to welcome her sister Kathy. Thank you so much. Thank you, Pat. And thank you, Parisa. And thank you, Sherry was also a person I spoke with when we were talking about my possibility of my coming to be with you and Amanda for all the work she's done. I'm really thrilled to be with you today. And certainly your mission and the work that you do is so important. And so very understandable why you are connected with the Sisters of St. Joseph and our mission of unity. So I'd like to just begin this morning with a little mantra that I wrote for today. It's really the starting premise of what I'll be sharing with you today on that challenge of being present to grief while at the same time holding on to hope within the beloved community, as Pat mentioned. And that starting premise is being rooted in a greater love than we could ever know. A love that is God. A love that God has placed in every human heart. So we'll just center ourselves for a moment in that reality as we listen. Grounded in a greater love than we could ever know. Grounded in this love you've placed in every human heart that gives us purpose and courage to begin anew. I name this hope. I name this me. I name this you. Grounded in a greater love than we have ever known. Grounded in this love you've placed in every human heart that gives us purpose and courage to begin anew. I name this hope. I name this me. I name this you. So this past year of pandemic and loss has been a daunting challenge for me, and I'm sure for you as well. So right up front, I need to just tell you that I don't claim to be an expert on grief or on this topic. I can simply share with you some of the learnings that have come to me through the reflecting and the heart and soul work that I've done personally but also in community these past months and that I continue to do as I try to process how I've been changed by this pandemic and how I'll continue to evolve moving forward. So I share with you an experience I had on the night before the presidential election. As you might recall, the night before that election, there was a national call to go outside and light a candle at would be six o'clock hour time to remember and to pray for all those who have lost their lives from the pandemic. What grief? More than 400,000 people at that point had died. And I cried as I stood there holding my candle. For days, I had felt as though there was this dam inside me holding back a tsunami of feelings and emotions. And as I stood there alone with my candle, I was present to the grief that I felt. And though I stood all by myself, I didn't feel alone.  And then I went inside to get on a Zoom where I was going to lead an inauguration vigil prayer. And about 300 people showed up for that prayer. I was completely overwhelmed. This Zoom community was a powerful affirmation of holding on to hope, even while experiencing grief. And I could do both. I could grieve. And I could hope. Because I was a part of a beloved community, a community of faith, and a community of hope that lets me know that as long as we hold on to each other, things will be all right. It's all right. I am here. And I will never leave you not today, not tomorrow. I will not ever. And in hand will be strong. Though the journey may belong through darkness of night into the brightness of day. You and I together we will walk. Walk as one. It's all right. I am here. And I will never leave you not today, not Oh, none of her hand in hand will be strong, though the journey may be long, through darkness of night into the brightness of day. You and I together, we will walk, walk as one. It's all right, I am here, and I will never leave you not today, not tomorrow, not ever. Hand in hand will be strong, though the journey may be long, through darkness of night into the brightness of day. You and I together, we will walk, walk as one. You and I together, we will walk, walk as one. I've had this image going on inside of me these past couple weeks of this image or metaphor of grief and hope holding hands, both offering gifts. I think it's because I strongly believe that as we move through grief and attempt to hold on to hope, we need community, a beloved community. And I just love this term that of course was coined by John Lewis, the great civil rights activist and legislator who died just this past year. This beloved community that holds on to each other and sustains each other, or perhaps another way of describing a community of faith rooted in the love of God and in the gospel. Grief is intense sorrow, and grief is a deep emotional response to great loss. These losses may have to do with the death of family or friends, or lost sense of purpose, or loss of relationships or dreams, or the devastation to the environment that causes grief, or grief such as what we witnessed at the capital of our country on January 6th. We've all experienced loss, especially this past year. And we've also witnessed tremendous loss in the lives of others in our families, communities, country and world. How often during these months have our hearts been broken at what we've seen, at stories we've heard, times when we've been overcome with fear and uncertainty, or been worried for our jobs, or our families' well-being, or even the future? I think it's important to remember that it's both appropriate and necessary to grieve and to weep about these losses and painful realities. I once read, quote, we need to pay attention to our tears, because tears tell us what's important to us. End of quote. And I have found that to be true in my life. But making space for grief in our culture can be difficult and not easy to embrace. As Westerners, we are generally more concerned on finding a quick fix or a more expedient path to feeling better and as quickly as possible, rather than companioning our grief. We find it hard to trust in the possibility that grief can lead us to healing and wholeness. And it will. But only if we take the time to honor our grief and to move through it. There's a wonderful Persian poet and Sufi mystic whose writings I love by the name of Rumi. And I came upon a poem of his called The Gift of Tears that for me images this paradox. The poem goes like this. The cloud weeps and then the garden sprouts. The baby cries and the mother's milk flows. The nurse of creation has said, let them cry a lot. This rain weeping and sun burning twine together to make us grow.   And I would add that as you remember this truth that Rumi speaks of, that you also remember that the God of hope is also the God of grief and remains with you and each of us at each and every moment in juncture of life. And will be there in each and every future moment. And so as you listen to this song called All You Need to Know, I invite you to be with the God of your heart, your journey and any grief that you may be holding. I was there right with you. Never did I leave you. I was your strength, your peace of mind. All you hoped that you could be. And though perhaps it's hard for you to grasp or understand, all you need to know for now was I was always there. I was there right with you. Never did I leave you. I trusted me to be for you. The song of courage you were needing. And though the mysteries are for you to grasp or understand, all you need to know for now was I was always there. I will be with you for all of nothing to fear. Love you, protect you, and keep. I was there right with you. Never did I leave you. You had me a mystery that you would never be alone. All you need to know for now. And what about hope? In his 2007 encyclical, Saved in Hope, Pope Benedict the 16th wrote, the one who has hope lives differently. The one who hopes has been granted the gift of a new life. I would probably add the one who tries to hope lives differently because I know I'm not always good at it. And especially this past year, hope sometimes has not been very easily. Has been not very easy. It's also easy to confuse hope with optimism, but we know that hope is different than optimism. Hope is not a fleeting emotion or an attitude that fades when life is hard, but rather a resilient stance towards life, marked by trust, confidence, and perseverance. Hope empowers us to live differently because faith's understanding of hope is rooted in the unshakable conviction that God loves us and wants our good. That God's great love is the very true and energy that has strengthened, buoyed up and sustained doctors, nurses, chaplains, educators, essential workers, first responders, and so many others all these many months. It affirms Paul's message in Romans, if God is for us, who can be against us? Do we dare to believe this? Do we want to believe this? If we desire to live with hope, in hope, and for hope, then taking these words to heart can change our lives in creative and surprising ways. A first century philosopher who is unnamed wrote something that it's very poignant to me, and especially as I consider it in light of this past year. He said, or she said, I'm not sure, but was a man or a woman, when I light a candle at midnight, I say to the darkness, I beg to differ. When I light a candle at midnight, I say to the darkness, I beg to differ. Jesus also said that while he was in the world, he would be the light for the world. As long as he is in the world, he is the light of the world. Look through the Gospels, how many times and how many situations do we see the light that is love and justice and compassion and healing, that Jesus dispels the darkness because of what, who he is and what he does. Jesus also calls us to be light in the darkness. As you think about, as you listen to this song, as long as I'm in the world and Prisa is going to put it up for you, I invite you to think about the ways that you also are light.  As long as I am in the world, I will be a light, a light for the world. I will be a word of truth, a song of hope, a handout stretched for healing. As long as I am in the world, I will be a beacon of love for this weary world. So shine with me, oh shine with me and I will bear the beams of your holy light. As long as I am in the world, I will be a light, a light for the world. I will be a word of truth, a song of hope stretched for healing. As long as I will be a beacon of love for this weary world. So shine, oh shine, shine, shine even. More recently, I was inspired as I heard President Biden talking about the losses in his own life. And he was doing this as he comforted those who were grieving the loss of loved ones from COVID. And as he was talking, he said that what kept him going through all the loss and grief in his life, that of losing his wife and daughters at a young age in a car accident and then losing his son to cancer. He said what he needed to do was to dig deep, to tap into his passion and rediscover his soul's purpose or call. Never, never, never, never denying his grief, but strengthen in his resolve because of his grief. It was the sense of his life's purpose, his life's more expansive purpose that compelled him to keep going. Not only for himself and his family, but for the beloved community, a life poured out in service to his family and to his beloved community. That purpose, even in difficult times, before I do that, I wanted to invite you just to take a minute to yourselves and to get in touch with your own purpose. What is it that gets you out of bed each morning? Even on days when you'd rather stay in bed. Who is the beloved community that is blessed by your life? Who is the beloved community that is blessed by your life? And if you're not sure, then ask for the grace to discover it. It is there for you, waiting. That purpose, even in difficult times, that purpose that gives us hope, it's like the mustard seed that Jesus uses in his parable. It's like the mustard seed in that even though it may be tiny, it is full of potential and possibility. Everything it needs to grow, to flourish, and to thrive is already there inside. Everything is present in the seed, ready to become what it already is, ready to become what it already is. My sister is a social worker, retired social worker. And she told me the story that when she was first getting a job in Florida, they wanted someone with a master's. And when she had her interview, she told the people, she said, you know what, I'm as good as a person. I'm as good of a social worker as someone with a master's. And she got the job. And she was a social worker in a high school for 40-something years.  He was moving away and wanted to thank her. And he was going out the door. He turned around and he said, Miss Gillies, can I ask you a question? And she said, well, of course. And he said, well, I know you get paid to do this work. He said, but it seems like you really care. When I listened to her tell that story and connected it back to that seed of long ago, I realized that she knew from the beginning that she had become what she knew she was from the very beginning. During these long months, it's often been the many small yet significant gestures, words, actions of others that have sustained me and probably you and kept hope alive. Martin Luther suggested that just as love transforms the lover into the beloved, so hope changes the one who hopes into what is hoped for. And this next little song I'd like to share with you is a new song of mine.  And it was actually inspired by the Jesuit, Alfred Delp. As you probably know, he was a German Jesuit who was part of the Catholic resistance group during Nazi Germany, during the Nazi regime, and he was executed when he refused to renounce the Jesuits. But in his writings from prison, he encouraged people to be heralds of good news. To do whatever they could do, however they could be, in order to bring good news to others. And so as you just listen to this little song, I invite you to think about how have you been a herald of good news and let the memories flow because I'm sure you have many. I'm doing the other song first because I forgot to do that one first. So this song is called, If I Had Faith. If I had faith the size of a mustard seed And could plant it in the earth of my heart Maybe I could grow a love so deep and wide That the world could find shelter in me That's what I pray to be If I had faith the size of a mustard seed And could plant it in the earth of my heart Maybe I could grow a love so deep and wide That the world could find shelter in me That's what I pray to be If I had faith the size of a mustard seed And could plant it in the earth of my heart Maybe I could grow a love so deep and wide That the world could find shelter in me That's what I pray to be And now think about how you've been heralds of good news. Is there a song that you can sing? A comfort you can bring? A dream or an imagining? That heralds love and peace? Is there a smile you can give? A hand you could extend? A quarrel you could end? A heralds forgiveness? Be a herald of good news An angel of light A voice of tender love That holds the darkness of love Is there a promise you can make? A risk that you can take? A dream, a vision, a love to heal And make the world one? Is there a place that you can go? A truth you long to know? And seeds of hope you can sow To grow a tomorrow? The heralds of good news Voices of tender love That holds the darkness of love So as I was preparing for today I started to think about what are some of the things that have helped me to be with grief and to sustain hope and so I kind of came up with this list of things that's certainly not comprehensive and I'm sure you would all have your own to add but I thought maybe it might be helpful to share it with you so if Prisa would do me the honor of putting that up that would be great and I'll just say a little bit about each one of them so practices that sustain hope and help with grief so one of the big ones for me is being present to the present moment to know that in the here and in the now I have what I need that now is all I have and now is all I need again not always easy and probably one of the most difficult spiritual practices there is getting back to this one about paying attention to our tears honor your tears remember the tears tell you what's important to you they're messengers messengers for you to keep you attuned to your heart and alert to your purpose connect with God yourself and others so critical to remember that we're all connected and tethered together as one sacred community you just can't forget that take some time to be with nature spend time with nature John O'Donohue the wonderful Irish poet talks about especially in times of grief how important it is to be with nature because it's the only thing that's large enough to hold our grief and it can also be such a source of healing and comfort reclaim your purpose each day you are meant to be here you have a presence a purpose and gifts that will contribute to the healing and unity of our world and planet and revisit this question often who do I want to be in the world and how do I want to be in the world this one's kind of also you have to have a sense of humor surrender to the reality of good days and bad days and I'm going to share a little song that goes along with that one it's called tell it like it is because sometimes I think on the bad days I have to remember in my prayer to be bold about that too and so it's called tell it like it is like it is tell it like it is when you're talking to God just pray and tell it like it is don't keep me out of the distance I heard God say to me I'm in the story you're living each day and that's where I'll be so tell it like it is tell it like it is when you're talking to God just pray and tell it like it is you may not think I'm listening I might seem far away but I promise you I hear every word that's what I heard God say so tell it like it is tell it like it is when you're talking to God just pray and tell it like it is I'm everywhere and in everyone yes in everything you see there's just no way God shouted out that you're ever far from me so tell it like it is tell it like it is when you're talking to God just pray and tell it like it is when you're talking to God just pray and tell it like it is so a sense of humor make a commitment to hope being hopeful takes intentionality and we need to choose hope even when it's hard because hope sustains and inspires and shines light engage in creativity cooking, painting, writing rearranging your closet there's been lots of time for that this year rock painting whatever it is planning your garden it doesn't matter what form the creativity takes it's just that research tells us that creativity nurtures our sense of self-worth belong to a community we need the beloved community where we have roots that can't be shaken especially during times like these a community also strengthens and sustains us in our living and work and ministry nurture your inner life this is where we listen to what's deepest inside our souls and where we process our lives and our call nurture take time to be still to pray or reflect however that looks for you because this helps us see the big picture and puts things in perspective practice kindness simple and yet so significant a practice that helps to build up the beloved community give generously and receive graciously your time your gifts your pains and your joys and probably the most important love deeply and wholeheartedly because as scripture tells us let your heart on the greater gifts faith hope and love and the greatest of these is love and love changes everything so before we break for the questions I just would like to share this last song called somewhere in the future because I think when we talk about hope our lives the way we live our lives gives us hope for those who come after us so this song is called somewhere in the future somewhere in the future somewhere down the line children will be asking what we did with our time they'll say were you building bridges of peace and harmony more rays and walls of fear and hate for all the world to see tell me how will we answer the children will we weep in shame at what we failed to do or will we stand before our sons and daughters and say we did our best and we tried to change the world somewhere in the future somewhere down the line children will be asking what we did with our time they'll say was your silence deafening when the planet screamed in pain or did you shout and march and act for the oceans for his planes tell me how will we answer the children will we weep in shame at what we failed to do will we stand before our sons and daughters and say we did our best and we tried to change the world for you somewhere in the future somewhere down the line children will be asking what we did with our time they'll say was your language that of love for all races colors creeds or did you cling to your not me forgetting unity tell me how will we answer the children will we weep in shame at what we failed to do and say we did our best and we tried to change somewhere down children will be asking what we did with our time so now we're going to move on to the Q&A section of today's event so if anyone has any questions they'd like to ask sister Kathy we have time for a few questions you could submit those questions into the chat and then I'll read them out loud but to get started one question I have for you sister Kathy is what advice do you have for Marquette students to keep hope during the pandemic stay connected I think one of the things that I keep saying is hold on to hope in order to hold on to hope we have to hold on to one another we can't go it alone as it were that's why I think the beloved community is so important and to stay connected with the things that give you joy the things that even though I know some of them are it's difficult now because you can't be with your friends necessarily in the same way but to nurture or to foster some of the things that are life-giving for you great Pat has a question he had in the chat he said that the sister or CST has so many important messages what are ways to share CST in a secular context or Catholic social teaching oh I think it's really important to put it in the context of what well this is where I think that faith and politics do go together that we need to be a moral voice with the things the issues that are really important to us to look at them from a gospel perspective and you know there used to be a saying on people used to say what would Jesus do people used to wear back bracelets WWJ or D or whatever it was and and I think that's that is a good question sometimes so that when we're looking at at what's going on the injustices and what's happening in our culture that we need to to not be afraid to speak to take a take a stand from the gospel from Catholic social teaching people are sharing to just thanking you for sharing your gift of music and time today and one question I had to was just how the process looks for you in writing these songs like if you have a certain if it's always the same each time or if it looks different it is it is different Amanda generally lots of times I will hear someone well as an example was that song that I talked about with Alfred Delp I was really moved by what his story and so that got me to thinking about how how can we be heralds of good news so it oftentimes I'll hear someone say something and there are days I wake up and I know that I'm going to write a song it's kind of a weird thing and I don't always necessarily know what it's what the song is going to be but other times it takes me a very long time just it there's a lot of kind of incubation period that goes into an idea but I write about really ordinary things and I write about a lot of so I write a my music isn't necessarily religious in one sense but it's spiritual and because to me the spiritual and the ordinary of life can't be separated that the the sacred is in all of life so that's what I write about Eileen is asking how do you juggle being creative and working on your music with your activism and other responsibilities that's a good question I will actually the activism feeds my creativity and the creativity feeds my so so I you know I try to write music I believe that that music and the arts in general can really affect systemic change and so but music sometimes music can be kind of subversive because it gets into us at a different place so I write a lot of music about issues that are really important to me and knowing about the issue feeds my creativity to writing and and then I try to be able then to share a song that will then deepen that help other people may become to a deeper understanding or embracing of that particular issue another question is what musicians do you recommend that this group checks out in other words what artists does the artist recommend I'm an old folks I really like all kinds of music so I can't say that I'm probably very up on music of your generation I can't probably say that but I really enjoy listening to a lot of different music I love folk music I love musicals I love some soft rock music but what the great thing for me about music is it it it it can speak for any emotion that you have so you can find a song that you want to cry or you want to laugh you want to dance so music's kind of wonderful that way so and I guess one last question we have today is from Mary Jo and Steve artwork behind you to also speak your message of grief and hope to us today. What is the art speak to you. I miss that Amanda tell me. Did you choose the artwork behind you to also speak your message of grief and hope to us today. Because Mary Jo is used to seeing a different picture behind me. Mary observant Mary Jo. Well, I actually just because I wanted a larger picture to took up more space, but I do love this paint it's in our living room and I do love it it's one of Mary Southern and to me it just is all about one you know the orb in the middle and it's the swirling and there's movement and I think that's really what this journey is all about right remembering the oneness and know that everything everything swirls in and around and through that oneness. Did I lose somebody. Amanda are you there. I think I think Amanda got frozen. Is there is Pat are you there or Prisa. Okay, yes. Connection so we can go ahead and end with your last song. Okay, great. Thank you. I'm just going to put one thing up here. Hold on one second here. Well you can put that go ahead and put that. So I just I just want to again say thank you so very much for inviting me to be with you today. I really appreciate it and thank you for all the efforts that so many of you put into making this possible. I'm just going to close with the song called hold on to hope and and you can feel free to join along with me if you like to sing.  They will walk and not be faint. The power of hope fill you with all joy and peace in believing so that by the power of the Holy Spirit you may abound in hope and Jesus's beloved community may abound in hope. So here we go. Lames the world you know hope will see us through and hope.  Just dig deep inside wake up your soul and hope is a good place to start. Hope will see us through. That peace will come in our day.


Thank you very much for the invitation and for being there. So I'm going to tell you about joint work with Peter Josen about e-functions. But before I, so all new results will be joined with him but before I get there, I would like to start with maybe a bit of a long introduction about e-functions, their history, some examples and then the kind of questions we are going to answer. Okay, so everything starts with one of the most beautiful results in transcendence theory of the 19th century, which is about the values of the exponential function at algebraic numbers. And the final statement is, I'm calling it Hermit Lindemann Baistras. And it says that if you take n algebraic numbers and you assume that they don't satisfy any non-trivial Q-linear relation, and then you look at their exponentials, exponential of alpha one, exponential of alpha two, exponential of alpha n, then these complex numbers are algebraically independent. So there's no non-zero polynomial with rational coefficients in n variables that annihilates these n numbers. And well, the particular case where n is equal to one, then being linearly dependent means being non-zero. And then the theorem says that the exponential of alpha, so alpha and non-zero algebraic number is transcendental. And this was first proved by Lindemann, when alpha is, sorry, by Hermit, when alpha is rational in 1873. So this gives, for example, the transternance of the number e. And then nine years later, it was proved by Lindemann for all non-zero algebraic numbers. And at the end of the paper, he said that, well, this statement was true, and then sketched the argument, and it was, well, completed and maybe made rigorous by stress in 1885. And well, also the remark is that, well, this is about values of the exponential function, but you also get the transternance of pi from this, because exponential of two pi i is equal to one. And if pi were algebraic, then two pi i would also be algebraic, and one is not a transcendental number. Okay, so this theorem gives, at the same time, the transternance of e and pi, and we can say that it completely solves the question of understanding the properties of the exponentials of algebraic numbers. So, well, soon there was the question about how to generalize this theorem to other special functions. And it's not at all clear because the proof relies on the property of the exponential that the exponential of alpha plus beta is exponential of alpha times exponential of beta. And you can imagine that this is very useful because the theorem is that there are no non-trivial algebraic relations. So, using this property, you can reduce the question of starting algebraic relation to linear relations and those are easier. And in fact, you can rephrase the theorem as saying that if you have different algebraic numbers, then the exponentials are linearly independent, even over Q bar. Yeah, so that was the situation for a while. And then comes a fantastic paper by Siegel. So, the paper is published in 1929. Here is the beginning. It is dedicated to Max Dehn. I actually was, it was a time where and the mathematicians of getting in were already feeling that there was too much pressure to publish. Publish or Paris was already quite a bad thing for them. So, they had this thing of offering papers to each other and not publishing them. So, this paper, I think was written around 26 and it was sitting in Max Dehn's office for a while. For example, Andre Veil tells in his memories that he could read it when he visited Max Dehn. And it was only published later. So, at the beginning of the paper, he says, okay, so this is when he talks about this theorem by Hermit and Lindemann. He says that it uses this property of the exponential and you can already see from the first page. So, there's a formula for the Bessel function that is going to come back again soon. So, he's in particular interested in having, well, in understanding the transcendence properties of the values of the Bessel function at algebraic numbers. And one reason is that these numbers are very much related to some physics problems. So, one of them is when you study the vibration of a circular drum. So, you're hitting somewhere in the drum and then you look at the function that gives the height of the point x, y at the time t. And it is, this function is governed by the wave equation. So, that the second partial derivative with respect to time is equal to the sum of the derivatives with respect to x and y, up to some physical constant. This is inside the circle and well, the boundary doesn't move. So, there's the boundary condition that u is equal to zero on the boundary. And you can try to look for solutions, well, you can, since it's a circle, you can use polar coordinates and then try to look for solutions in separated variables. So, this part depends on the radius, on the angle and on the time.  And then you see that the part depending on the angle is also something similar as sum of cosine and sine. So, here you have another parameter m. This doesn't appear if you hit the drum in the middle. So, the first thing is m equals to zero. But then you have other modes where it really depends on the angle.  Okay, so it's an equation of degree two, differential equation of degree two. And there's this term where you see lambda and m. So, this equation has a unique holomorphic solution around zero, which is the Bessel function of order m. And it is, but you can look for the power series and the differential equation gives you a recurrence relation for the coefficients and then you find that the coefficients are like this. So, this half minus one to the n and divided by m factorial, m plus m factorial. So, m is the order again. And then set over two to this power. So, this is the power series representation and something that is going to be important later on is that it also has an integral representation. So, actually this is the integral of exponential of set over two t minus one over t against this differential form where you see the order again. And if you, well, if you use Cauchy's theorem, then you need to compute the residue and then there's a two pi i appearing. So, you divide by two pi i. Yep. So, this is the Bessel function, which is the holomorphic solution to that equation. And then the conclusion of the story is that the radial part is this Bessel function evaluated at lambda times r. Lambda was the frequency that we got at the beginning. But then you plug in the boundary condition. So, this is r equals to one. And then this tells you that the gm of lambda has to be zero. So, you find that the frequency is a zero of the Bessel function. And this was actually known much, really much earlier than Siegel's paper. And it appears, for example, in a paper by Bourget in 1866, where he makes the conjecture. So, you can plot the first Bessel function, g0, g1, g2, and so on. And, well, you see that they seem to have no zero other than zero. So, when the other is strictly positive, here you get a positive power. So, it vanishes at zero. But this seems to be the only one. So, and I went to see Bourget's paper and I was surprised to find that immediately after the paper in the volume where it is published, you get the referee report. So, this is the ones who presented the paper and then they explain what he does and why it is good and should be published and so on. Okay, so, that was one of the motivations for Siegel, this question about zeros of the Bessel function and something that it's not too hard to see, for example, by integration by parts or there will be a comological interpretation later on is that there's a recurrence relation between the Bessel functions of different order. So, you look at J M plus K, then you can write it as J of M and J of M minus one. And what appears is a polynomial with rational coefficients in one of our set, which is a explicit polynomial. It's called Lommel polynomial in the literature. And, okay, so now let's assume that there's a common zero to J M and J M plus K. Okay, so this means that these two terms here vanish. So, the other one has to vanish as well. But J M is a solution of a differential equation for the two and the derivative of J M is essentially up to sign J M plus one. So, if these two vanish at the same time, it means that the function of the derivative vanishes and then you can use the equation of all the two to deduce that the function will be identically zero. Okay, so what it's easy to see is that J M and J M minus one does not vanish at the same place. So, the conclusion is that this has to vanish at this potential common zero. And then this says that this zero, if it exists, is an algebraic number. Is there a question in the chat? Yeah, there's a question by Christopher Lloyd. No, sorry, they have absolutely nothing to do it's just a clash of notation that I didn't realize. Okay, so now let's park this for a while. We will go back to it later. So, that's Seagal's motivation and to try to generalize as he can, the mid-Lindemann biostatium he introduces the notion of an E function. Okay, so what is an E function? It is by definition a power series with algebraic coefficients. This is the field of algebraic numbers. It is convenient to write it as A n divided by m factorial set to the n. No mystery about the name. If you put all A n equals to one, then you find the exponential and E is for exponential. And besides having algebraic coefficients, you ask for two conditions. The first one is that it is a solution of linear differential equation with polynomial coefficients. So there exists some differential operator for the n, pn derivative to the nth power and so on such that annihilates the function. You want this to be non-zero, non-zero differential operator. And this amounts to saying that actually the coefficients A n over n factorial satisfy some recurrence relation. They satisfy a recurrence relation with polynomial coefficients. And then if it is true for A n over n factorial, is it also true for A n? So for example, even if I, here I ask that the coefficients are in Q bar, you immediately get from this recurrence relation that they are in a number field. So coefficients will lie in a number field. And then the second condition is the one that allows one to do a diophant and approximation of transcendence theory. And it is that we want the coefficients, okay, so we want the coefficients to be bounded by C to the n too, so to grow as most geometrically. But the coefficients are algebraic numbers on the one hand and we want to do diophant and approximation on the other hand. So you need to do this in the right way. And the right way that means that you must bound not only A n, but also all the Galois conjugates of A n. So that you want the max of all Galois conjugates of A n to be at most C to the n. And then you also want that the common denominator of A zero up to A n. So this is the smallest integer such that when you multiply by it, you get an algebraic integer. So you want also these to be bounded by C to the n. And if these two conditions hold, then the power is called an E function. It is not going to appear too much, but there's also a, this is quite the previous argument is that's a common zero of the vessel function must be algebraic. Yes, exactly. So in maybe 10 minutes, we are going to use the theory of E functions to rule out this possibility. Yeah. Okay. Yeah, I was going to say, if you delete the n factorial here and you keep the rest of the definition, you get something and this is called a G function. And the reason is again, that if you do A n equal to one, then you get the geometric series. So G is for geometric.  So D n is an algebraic integer. So what does that mean? It is at most C n? No, no, it's just an integer. It's a usual integer. Oh. So, yeah. So it's the, for each algebraic number, I can pick a integer. So it's not a minimal, it's not a minimal way to, okay, thank you. Okay. So, yeah, let me give you some examples. So, well, the exponential function I already gave it to you. And then the motivating example of the Bessel function. So let me just do for simplicity n equals to zero. So then this is how we defined it using the power series. And from here, you maybe don't see it. Well, that we already have the differential equation. So we need to see this growth conditions here. And maybe from this way of writing is not so clear because you need the n factorial to match the exponent, okay? So instead, you can write it as set to the two n. You divide by two n factorial and then what remains is the binomial coefficient. And then, well, it has rational coefficient. So you just need to bound the size. This coefficient, you can see that it is at most one. For example, writing one plus one to the two n and using the binomial formula. And the common denominator will be four to the n. So you can take a c equals to four in the theorem. Okay. And it's not too hard to prove from the definition that this class of functions is stable and there's sums, products, derivatives. And you can also take the primitive that vanishes at zero. So out of one example of a funny function, you get many of them by doing these operations. Okay, so I told you that the motivation was to generalize this kind of transcendence results. And here is how Siegel did it. So Siegel did it for the Bessel function. And with a technique that in principle could apply to all E functions, but there was some technical condition called normality. And at the end of the day, it was maybe harder to prove normality than to prove the theorem. So Schiedlowski in 55 found a way to bypass this normality condition. And then the very precise statement I'm going to give here is due to Andre and Boyker's in, so the final statement is due to Boyker's in 2006 and relies on the structure of the differential equation of E functions that was understood by Andre. So the statement is the following. We are going to look at the N E functions. And we assume that the solution of the vector F1 to Fn is a solution of a linear system. This should be a cube bar, sorry, and actually everywhere in this slide. So they are solution of a linear system with rational coefficients. So this is not really a restriction on a given E function because we know that it is a solution of a differential equation. So you can always take its derivatives and consider the companion metrics of the system. And then the theorem is about the special values. So we are going to evaluate these functions at an algebraic number alpha. And there are two cases that should be excluded. So first one is alpha equals to zero. Well, this is because these are power Cs with algebraic coefficients, so at zero they take algebraic values and there's just nothing to say about it. And more interesting is you need to exclude the situation where alpha is a pole. So this is an ASA metrics of rational functions and you don't want these rational functions to have a pole at alpha. And if this is the case, the theorem says that every algebraic relation between F1 of alpha up to Fn of alpha, polynomial with algebraic coefficients that vanishes at these complex numbers actually comes from an algebraic relation between the functions. So you can lift these to a polynomial Q such that now it's in one more variable and it vanishes identically on the functions. And when you specialize set to alpha, you find the polynomial you started with. And well, something I didn't say that it's implicit here is that because of this growth condition, actually this power Cs has infinite radius of convergence so you can evaluate that any algebraic number. So the question about the pole is to exclude some examples. So you can do the following thing, for example, you take exponential of set which is a transcendental function. So according to this, it should take a transcendental values but then you multiply by set minus one, okay? And then you get a function which is still transcendental but that vanishes at one now. And when you write the differential equation, you will see a set divided by set minus one in the matrix. So this pole explains the fact that you get a zero and the contribution of Andres to understand that all possible exceptions come from this kind of poles. Okay, so that's the theorem. And yeah, I, yes, maybe say it again. So it's a really a very powerful result in the transcendence theory which is completely false if you do the same thing with the G functions. So in general for G functions, you will have many algebraic relations that are not at all explained by the relations between the functions but for E functions is true and it's powerful because it's much easier to prove that a function is transcendental than to prove that the given value is transcendental. So in the case of the exponential, you can, for example, simply observe that the exponential is periodic. And from this, you get, for example, if E of X, Y is a polynomial of minimal degree in Y that annihilates the exponential, then you evaluate that all integer multiple of two pi i. So you, as a polynomial in Y, you will get a polynomial with infinite roots. So it needs to factor and then you find something of smaller degree than annihilates the exponential. And the middle in demand theorem we started with is the particular case where the functions f i are exponential of alpha i times z. So in this case, the matrix A is a diagonal matrix with alpha one, alpha two in the diagonal. And it's easy to see that all relations between the functions come from Q linear relations between the exponents. So it's really a generalization. Okay, so let's go back to Burgess hypothesis. So in the case of the Bessel E function and the Bessel differential equation, the only pole is at zero. Yeah, you can maybe see this from the equation. Where is it? Here, so you just get a pole at, well, the differential equation has a singularity at zero and infinity. So the only thing to exclude is alpha equals to zero. And the theorem says that otherwise and the value of the Bessel function is transcendental. And in particular, we have now approved the theorem that there are no common zeros other than zero to Bessel functions of different order because we saw that if such a zero exists then it must be algebraic. But the Bessel function takes transcendental values of algebraic numbers and zero is not transcendental. So there are no common zeros and it was a big success of Siegel but he doesn't even say it. So he says, well, we have proved this theorem and then there are no common zeros and that's it and he moves to the next topic. Okay, and yeah, this story seems not to be very well known because if you type a Burgers hypothesis on Wolfram Alpha, for example, it tells you that it has been proof for differential for Bessel functions of order one, two, three and four. So it's maybe time to get updated. Okay, so let's move to the new results now. So I gave you a few examples of E functions and just after introducing the notion, Siegel presents the case of hyper geometric functions. So this power series, I'm going to write them like this without the factorial now. So I told you that being solution of a differential equations means that the coefficients satisfy a recurrence relation with polynomial coefficients and hyper geometric functions are the case where this is a recurrence of length one. So it means that a n times a polynomial is equal to another polynomial times a n plus one. And in other words, as a function of n, the question of two successive terms is a rational function. And I'm going to take this polynomials with algebraic coefficients. This is for the general definition of hyper geometric function. And then the situation where you get a nif function is when p and q have rational roots. So you might have an algebraic leading term but the roots need to be rational. And then the degree of the polynomial p is strictly smaller than the polynomial q. And instead of considering just this power series, you evaluate that set powers to the power q minus p. This is similar to what we did here for the Bessel function. For example, here. So you see that you have, well, actually the variable appears to the power two n. So this is the same two n in set to the q minus p and it's important to get precisely a nif function. And yeah, so this is maybe the important property that the coefficients are given by a rational function. But then you can explicitly write down what you get. Shana, yeah, I'm sorry. Yeah, so can you scroll back one page? Yeah, this one. Thank you very much. So the relation in n plus one is that related to the Frobenius isocrystal properties that it's really Frobenius isocrystal and not the power of Frobenius. I mean the step is one and not two or three or four. Oh. You have the whole Gauss-Mann in here, don't you? Not a factor of the Gauss-Mann. No, yeah, the differential equation will be of the degree of order, the degree of q. But it usually appears as a factor and not the whole thing. What is the answer? Is that a factor or is that the whole Gauss-Mann? Usually it's a factor. Oh, it's a factor. Yeah, for example, the most classical case is q will be, boys, you get a differential equation for the two, but then it appears on a family of curves of higher genus. And this is still one. So step n, n plus one is still one. Yeah, yeah. So suddenly it's likely not going to be the explanation. Yeah, maybe not. What you do here when you put set to the power q minus p is to get the slope one and then your differential equation is a Fourier transform. Oh. So maybe this, yeah. Well, here I'm doing at the same time, this time I'm talking mostly about the regular case. But that one will be a Fourier transform of the regular one. And it's, yeah, probably it's not the explanation, but I'll have to think about it. Oh, I apologize for the interruption. No, no, thank you for the question, Ellen. Yeah, okay. So, yeah, I was saying that you can explicitly write down a formula and you get the questions of pohamer symbols which is a generalization of the factorial. So you do x times x plus one to x plus n minus one. So if you do the pohamer of one, you get n factorial. And yeah, there are p rational numbers upstairs, q rational numbers downstairs. This should be a not a negative integer, sorry. So because otherwise you get zero in the denominator and then you put the variable to the power q minus p and then you still have the freedom to multiply by some lambda, okay? So, and Siegel proves that this is an e function. So the differential equation follows very easily from this relation, it is of degree of other q. It's not hard to bound the absolute value of the coefficients, so they are rational numbers. And where there's a bit of arithmetic content is in the denominators. So essentially you need to play with the periodic evaluation of factorials. And at the end, it relies on some weak form of the prime number theorem. For example, in the shape that the less common multiple of one to up to n is at most three to the n, let's say. So he proves that these are all e functions. And then he asks the questions, is it true that, yeah. So from the operations, some derivative products and so on you get many, many more. And then he asks the question whether it is true that all e functions can be obtained this way. So as polynomial expressions in hypermetric functions. And this happens to be true if the equation e for so further one or two. So this is the case of the two functions we saw, the exponential and the Bessel e function. It was proved by Gorel of in 2004. And then in 2020, there was a very beautiful paper by Fischler and Rivo Al, where they prove that if one assumes the Grotendig period conjecture or maybe a variant of it that I will explain a bit later, then it should be false starting from e functions which are a solution of a differential equation of order three. So there's such an e function which cannot be written as a polynomial in hypermetric functions. And let me maybe say that what is very easy is to construct an e function which is not literally equal to a hypermetric function or to a multiple of it, because the property that the successive questions are rational function of n, this is very special. So you can easily produce one that doesn't satisfy this but then you need to take polynomials, products, sums, you can vary all these parameters here, pq, a, b, and then you lose control over this rationality, okay? And yeah, so this is my collaborator, Peter. And here we are working in the Island of Tattoo on this question. And we managed to prove, I mean, actually during COVID time that the answer to Seagal's question is negative. So there are e functions that are solution of a differential equation of order three, the minimal possible and that are not in the algebra generated by hyper-parametric. e functions are not even algebraic over this algebra. And actually most of them are of this type but it's not one of these generic situations where it is hard to write down an example. So there's a machine to produce examples that start with a polynomial of degree four and just to give you one. So this power series where it's a power series set to the end and then the coefficient is given by this which is, well, still very close to Po-Hammer symbols and factorials but you mix them in a bit of a complicated way. So this is an e function and it's not of the, it's not a polynomial expression in hyper-parametrics. Okay, so the way we, now I'm going to change a bit the topic for a while. So the, this is the geometry part of the title. So the way we got to this question and we could solve it is because we were studying for other reasons, exponential periods. What's the reason for the theorem? Yeah, I'll try to explain to say a few words about why this is true at the end of the talk. So now I'm going to try to explain how we found this function, okay? So we will have a look at exponential period functions and then at the end I will say a word about why it is not hyper-parametric. Thank you. Yeah, so we were studying for other reasons, exponential periods. So let me say very quickly what periods are first. So periods are the complex numbers that you can write down as integrals. Well, I need to be more precise about the integrals of what. So you integrate an algebraic differential form with omega with algebraic coefficients and you integrate on a subset of C to the N which is defined by, which is semi-ial algebraic which means it's defined by polynomial inequalities and you also want these polynomials to be with algebraic coefficients, okay? So two examples are the number pi when you express it at the area of the circle of radius one or for example, theta of three, the special value of the Riemann's theta function, if you develop the geometric series and integrate three times, you find that this is the integral on the unit cube of one over one minus x, y, z. And from a more sophisticated point of view which is the one which actually allows you to prove anything about these numbers, what we are doing is we are looking at the coefficients of a pairing between two vector spaces associated to algebraic varieties. So you can see an algebraic variety x over q. So this will be given by the shape of your integral. For example, if you are looking at this one, well, it's an integral in three variables. So typically you will start with a three, the affine space of dimension three and then you want a differential form. So you will remove the locus where it has poles. So your x in this case will be a three minus the zero locus of, well, the locus where x, y, z is equal to one. Yes, so this is your algebraic variety. And then you have two vector spaces, the cube vector spaces in this case associated to it. So one is algebraic, the ramp homology, where you do like in differential geometry, except that your differential forms have polynomial coefficients. So it's the homology of the complex of scalar differentials. And you can also, so this is purely algebraic. And you can also consider the complex points of your algebraic variety. That will form a nice topological space and then consider the singular homology of this space. So in this setting is usually called Betty homology. And you will have a pairing where you take a class of differential for omega and the class of a form sigma and you integrate omega alone sigma, okay? So if x is smooth and affine, all I'm saying is totally correct. And then by a theme of growth and dig, this is a perfect painting. And if, well, in the either non affine or non smooth case, you need to be a bit more careful what it means to take the class of a form and integrate and so on. So in such a way, you get a class of complex numbers and it will essentially be the same that we defined before except some, well, minor subtleties. Like for example, here you are integrating on the unity hypercube, which has a boundary. So it's not exactly a cycle in singular homology, but all this can be solved and then you get exactly the same class of numbers. And one advantage of this point of view is that you expect to have some kind of Galois theory of periods that will generalize that of algebraic numbers. So in general, these numbers will be transcendental, but you can imagine to act on such an integral. So G will be a linear automorphism of beta homology and you can imagine to act on the integral by simply acting on the cycle. And the problem of this is that it is a well-defined action on the writing as an integral, but of course, usually you can write a complex number in many different ways as an integral and this will depend on the way of writing. So the grotendic period conjecture that I mentioned in one of its incarnations tells you what are all the possible ways of writing the same period as different integral representations or how to pass from one to another. And this is very totally out of reach, but it's amusing that we can use this conjecture to actually define this Galois group and to get an action which is well-defined at least on symbols. Okay, so that's the very short summary about periods. Many numbers we like are periods, but some other numbers that we also like very much are not expected to be periods. So one of them is E that we already had and another one is Euler's gamma constant, which is usually defined like this as the limit of differences between the partial sums of the harmonic series and log. And we, well, there are good reasons to think that they are not periods in the previous sense, but they are both exponential periods. So exponential periods is a generalization of that notion where you integrate omega along sigma as before, but you are also allowed to put the exponential of an algebraic function. So the new ingredient is that there's an algebraic function F, omega is still an algebraic differential form, and sigma is what is called a rapid decay cycle, which is something that the main property is that it makes the integral convergent. So typically you, if sigma has a boundary, you want the real part of F to be very positive along this boundary, because then exponential of minus F decays rapidly, which means it decays faster than omega, which is a polynomial differential form and then the integral converges. A typical example of this is the square root of pi, which is not expected to be a period either, but that you can write as exponential integral of the Gaussian integral, so integral of exponential minus X square on the real line, which is a rapid decay cycle for this function. And in the same way as before, it's possible to think of exponential periods as the coefficients of a pairing where now you have the RAMc homology, which is associated to the variety, but also to a function and a homology group, which is this rapid decay homology. And the pairing is given, so classes here are non-compact cycles, so you write them as limits of sigma t's. There should be a t here, but they only go to infinity in the directions where the real part of F is very positive. And there's a question. Yes, the notion of rapid decay cycle depends on the function. So actually what we define is this space. So given the variety and the function, this will be unbounded cycles on the complex points of the variety and the directions where they go to infinity depend on the function. And again, this is a perfect pairing, so this is the first proof by Deligne for a polynomial on A1 and then Bloch and Eno for curves in a more general setting. And in general, it's at the end of in Rukeol. So this is exponential periods. And now I try to go back to E functions. And one nice thing about exponential periods that you don't have for useful periods is that it's very easy to deform them. What I mean by deform is that starting from exponential of minus F times omega alone sigma, you can introduce a variable Z and simply consider exponential of minus Z F. This is of course the case where F is equal to zero and then you are back to the wall of classical periods and they are just writing down the constant function given by a period. But in general, F is non zero and then you will get a function. And since the important property that defines the rapid decay cycle is the real part of F, if you take a set to have positive real part, then the integral still converges and then you will get a holomorphic function on this half space. And these functions are not E functions for two reasons. The first one is that even if you assume they are holomorphic, it extends to an entire function. Then in general, they do not have algebraic coefficients. And we already see this in the example where you take as a variety the complex plane minus zero. Complex plane minus zero. The function is X minus one over X. You can integrate on the unit circle. So this is an exponential period function. But when you compute using Cauchy's residue theorem, there's the two pi I missing. So this is almost the Bessel function but not quite is two pi I times the Bessel function. Which makes sense because this is a situation where you can evaluate the function at set equals to zero and then you get the integral of the X over X on the circle and this is exactly two pi I. So this is one problem that arises when the function is entire. But in general, it's not even an entire function and then there will be monotronomy around zero. And what we prove with Petter, so well, the theorem is proof the in progress refers to the writing is that these kind of functions on the half plane real part of it said positive. There are actually linear combinations of E functions with monotronomy. And the monotronomy is as easy as possible. So it's quasi unipotent monotronomy. It means you have rational powers of said and integer powers of the logarithm and then E functions. So in other words, well, this comes from the fact that the differential equation that is going to solve the, well, this function solves a differential equation and that one will have a regular singularity around zero and the monotronomy is quasi unipotent. So you get linear combination of these kinds of things and this settles the question of monotronomy and linear combinations where? So they are linear combinations with coefficients in, so the coefficients can be algebraic numbers but we also saw that you need periods because this is the case where f is equal to zero and then you get a function constant equal to a period. And then the other thing you get is the gamma constant and this one appears when you have logarithms. So it corresponds to the Jordan blocks of the monotronomy and you also get values of the gamma function at the rational numbers and these rational numbers have the denominators the denominators of the rational exponents. They come from the finite part of the monotronomy. Okay, so it is a general property and it really gives a source of new functions that would be hard to come on with to come up with. So for example, you can consider the same integral as before but instead of integrating on the unit circle. So this is a case where the homology has dimension two. So one cycle is the unit circle but there's another cycle which is actually the positive real line except that you need to go a bit around zero to avoid the singularity. And if you do this you apply the theorem to this situation so you find an E function which is the Bessel function we had. It comes with some logarithm and some gamma constant and then you find another power series which is in this case is this one. So HN is the harmonic number one plus one over two up to one over N. And this is a new E function that is, well it's not obvious to figure out and to prove that this is an E function. You will arrive in this particular case but it's solution of a differential equation of order four. You need to bound the denominators and so on. So this is the geometric source of E functions. And the easiest geometric situation that you can consider is your variety is the affine line. An algebraic function of the affine line is a polynomial. So we take a polynomial. I'm going to take it monic so that the real part is just governed by the leading term. And we consider this kind of integrals. And this integral is going to satisfy a differential equation of order the degree of the polynomial minus one. Because for example, you can compute the dimension of comology and if you differentiate under the integral sign and then you get different classes in comology and after doing the degree of the polynomial you have a linear relation. So if we want something of order three, I'm going to take a polynomial of degree four. Then this will give a differential equation of order three and in this case there's only a finite monodromy. So you get a linear combination of powers of set with the denominator of four E functions. And in this case, the coefficients are just values of the gamma function at one over four and two over four and so on. And so this produces E functions. And what we had observed with better before is that the only situations where you get polynomial expressions in hyper-dromatic functions are very symmetric. So the polynomial F has a symmetry property which is in general not true and is the following. So you consider the critical values of F. So this is F computed at the zeros of the derivative. So there are three zeros. You compute F, you get three complex numbers. And if they are not on the same line or if the triangle they form is not equilateral then these functions that appear here are not going to be a polynomial expressions in hyper-dromatic functions. So then if you want to write down an example you just need to take the easiest polynomial that satisfies this. So we took X to the four minus X two plus X and you compute the integral and then this gives exactly the function I was showing to you when you expand the exponential inside the integral. Okay, and that's probably a good moment to stop. Thank you.


A question here is, how do I create the segments? Do I edit my course videos? No, I don't edit my course videos. If you want to know about video editing, I'm not a pro at it, but I'm a very amateur. I think I know just enough to be dangerous. I use capwing.com, K-A-P-W-I-N-G. You can go to my YouTube channel. I'll just show you real quick.  You will find my capwing videos, they're public. This is how I use capwing. I have it on my channel. And so I know enough just to be dangerous and just to teach you to be dangerous with it. But the reality is for my course videos, I don't edit them. I don't have time. I don't want to give the time. I'm sorry to edit my course videos. So how do I create these segments by using the Zoom recording? So when I'm in Zoom, when I click on record, when you're in Zoom, when you click record, you notice that at the top, it says recording. And then there's the pause button and then there's a stop button. Have you noticed that when you record in Zoom? There's the pause button and stop button at the top next time you record in Zoom, you will see it. If I click on the pause button, which I will do now, if I click on the pause button, it stopped. As you see how suddenly my arm went from here to here, it's because it stopped recording, but it's still keeping it in the same segment. So I'm gonna, I could do stop, I could do clay nation, right? Like I'm being silly here, but you can see what I mean. It's like as you click, whenever you click pause, it'll stop the segment and it'll continue with the same video file. But then once you click on the stop button and your Zoom recording, it stops it and puts it into its own video file. And if you click record again on Zoom, it starts a new video file. So this is how I record in segments on Zoom and this makes it very convenient. And if you go, gosh, George, what I don't like about Zoom is every time I click start and stop recording, there's the verbal announcement that comes in. This is recording pause or recording stop, recording continue or recording started or whatever. And you're like, George, I wanna turn that video, that verbal announcement on or off rather than go to my YouTube channel and search Zoom verbal and press enter. And there's my video, how to turn off Zoom's verbal announcement of recording in progress, recording stop and that will stop driving your students mad and driving yourself mad as well. So I hope this helps.


Hello and welcome to the session. I am Asha and I am going to help you for solve the following question which says find the sum of n terms of the AP whose k term is 5k plus 1. Let us start with the solution. So let k term ak, so this is given to us as ak plus 1. The first let k is equal to 1 then the first term will be 5 into 1 plus 1 which is equal to 6. Now let k is equal to 2 then a2 will be equal to 5 into 2 plus 1 which is equal to 11. Further let k is equal to 3 so the supplies a3 is equal to 5 into 3 plus 1 which is equal to 16. Therefore ap series is 6, 11, 16 and so on. We find that the first term and the common difference is equal to 11 minus 6 which is equal to 5. And according to the question we have to find the sum of n terms of the given AP. So the sum of n terms will be n upon 2 into 2a plus n minus 1 into d is equal to n upon 2 into 2a is 6 plus n minus 1 into d is 5. So this is further equal to n upon 2 12 plus 5n minus 5 which is equal to n upon 2 5n plus 7. The sum of n terms is equal to n upon 2 into 5n plus 7. So this completes the solution. Take care and bye for now.


don't often film worthy park run seeing as it's my local so I do it so often I don't really bring the camera along with me but I thought I would do today just to show you because it's a lovely day today. So he's one of the fastest in the country it's pretty windy today though so there's gonna be quite a breeze from behind so that's gonna be good coming back so let's get to the start and get going. So I know we've got, here we go. First kilometer 401, two minutes bogging, that's the second kilometer done in two minutes not two minutes but I wish three minutes 58. Okay that's the turnaround wind on our backs now. Hopefully. Three kilometers done that was 357 bit too fast for my plan but nevertheless 358. Four kilometers to number four. Into the last one. Last 200 meters. Thank you. So under 20 minutes and 20th place today. It's hard sometimes isn't it? I got 1942 what are you 1930? Okay very good. Considering this guy has had a major medical episode recently. That is amazing come back. What was that? Yeah that's what I was aiming for. I was aiming for under 20 and I got 1942. It wasn't quite 5k 4.99 something so not far off. There we are, Worthing Park Run done. I want it to get under 20 minutes that's what we got so very pleased with that. The fitness continues to improve after all the long distance running we've been doing this summer. So we just need to get a bit of speed back although one more 100 to do before that and then we can really get back into the speed work. So thank you very much for watching if you're not subscribed to the channel and you enjoyed this video please do consider subscribing and if you would like to watch the video I did last time I did park run then that's that link there click that and we'll see you on the start line next time bye bye.


What is karma and why did you write this book? Karma means action. Whose action? My action. Whose responsibility? My responsibility. Karma is the most dynamic way to exist in the sense. Karma essentially means, when you say, my life is my karma, it means my life is my making. The moment you see it's my karma, your successes and your failures are yours. The moment you see it is all yours, you will do the best you can. When you leave your destiny from your hands and think it just happens to you, we can call that fate. So crafting your destiny is possible, only possible for one who sees, my life is my karma, whether he or she uses that word or not. In so many ways human beings around the world have realized that it is their thought, their emotion, their actions, their focus and orientation within themselves which gets them where they want to go. Others think it is happening from somewhere else. So those who think it will happen from somewhere else will live accidental lives and they always have somebody else to blame for things that did not happen in their lives. For things that they are not, they think it's because of somebody. You could be in the best place and be absolutely miserable. You can live in a palace and be horrendously miserable. You can live on the street and be exuberantly joyful. Both are possible. Human experiences caused from within, pain and pleasure, joy and misery, agony and ecstasy happens from within us. So when it is happening from within us, at least what is happening from within us must happen my way. What is happening from within me must happen my way. What is around me is not all mine. Everybody has their piece in it. Maybe I also have a piece. I can push for a bigger piece, but that's it. There is never going to be any time for any human being where the external situations will be 100% the way you want it. It never ever will happen. But what is within me must be 100% the way I want. Otherwise I'm a lost case. This is what karma means, that you take, you take charge of all that you can take charge. There are a few things that won't happen in our lives. If we do not do what we cannot do, that's not the issue. But if we do not do what we can do, we are a disaster. Once you see my life is my karma, you will not be such a disaster. That is the important aspect.