Good afternoon. I would like to welcome University President Dr. Donald Farish, honored members of the University Board of Trustees, honored members of the University School of Law Board of Directors, honored guests, faculty and staff of the law school, university officials, alumni, parents, family members, friends, and last but not least, the graduates to the 2014 Commencement Exercises. Please rise as Renee Delegarza sings the National Anthem. Oh, say can you see, by the dawn's early light, what so proud at the twilight's last gleaming, whose broad stripes and bright stars through the perilous fight, o'er the ramparts we watched were so gallantly streaming, and the rockets' redding in air gave a flag, was still, oh, say, does that star-spangled banner The years ago, this year, a handful of courageous men and women, including our namesake, Roger Williams, secured a royal charter for the colony of Rhode Island and Providence plantations. Central, to the founder's vision, was their commitment to a lively experiment, which meant, among other things, path-breaking recognition for the freedom of conscience. 20 years ago this year, another lively experiment began when Roger Williams College became Roger Williams University by opening the state's only law school in Bristol. As a result, Commencement 2014 represents the completion of our year celebrating the 20th anniversary of what some wags have called the miracle on MediCon. To recognize this institutional milestone, we have asked representatives from all of our previous graduating classes to process, with the class of 2014, a tangible link of our past to our present. I would now like to introduce representatives from the classes from 1996 to 2013 who have taken their place in the legal community. Please hold your applause until the entire group has been recognized. Kimberly Ahearn, class of 2009. Jamie Bichant, class of 2012. James Bagley, class of 2012. Carly Bovay-Iefrati, class of 2000. Nicole DeLude Benjamin, class of 2006. Stephen Bernardo, class of 1998. Dina Castricone, class of 2002. Ann Corvo, class of 2001. Diana DeGroove, class of 2003. Don Oyer, class of 2010. Michael Field, class of 1997. Mark Gemma, class of 1997. Jenna Wimms-Hashway, class of 2011. Christy Heatherington, class of 2002. Genevieve Alera-Johnson, class of 1997. Mike Just, class of 2008. Anthony Leone, class of 1997. Steve McGuire, class of 1996. Eric Miller, class of 2001. Elisha Morris, class of 2004. John Pagliarini, Jr., class of 1999. Matt Plain, class of 2005. Adam Ramos, class of 2006. Margarita Vellucci, class of 2007. And Will Ray, class of 2013. Please salute these wonderful alums for the round of applause. Then three years ago, another group of intrepid men and women, the class of 2014, gathered in Bristol to begin their journey to the law. A trek through legal research and writing, memos and rewrites of memos, exams, moot court, seminars, clinics, and externships, guided by the world-class educators who make up the Roger Williams University School of Law faculty. They have graduated from the top schools, clerked for leading judges, and have had distinguished careers in a broad range of practice settings. Their excellence is manifest in the books and article they place in top law reviews. They also play an important role in the advancement of the rule of law in Rhode Island, on the national and even on the international stage. But every single one of them understands that their most important job is the training of the next generation of leaders for the bench and bar. They do this by providing a rigorous but personal education. My faculty colleagues know the law school classroom is the foundation for successful legal career. And so at this moment, I would like the entire faculty to now stand and accept the appreciation of this commencement crowd for all they have done to launch the RW law class of 2014. Faculty, please stand up. At this point, I would like to single out several people for their special contributions to student success. First, each year the graduating class selects two professors of the year, one from the full-time faculty and one from the amazing group of lawyers and judges who make up our adjunct faculty. The winner of the award for the best full-time professor in 2014 is Emily Sack. Emily, will you please rise to accept the congratulations of the commencement crowd? The award for adjunct professor of the year goes to Andrew Spocone. Andrew is of counsel to the leading firm of Adler, Pollock, and Sheehan. He can't be with us today. He's probably billing some hours. But we offer Andrew our congratulations and our appreciation from the class of 2014. Additionally, the Student Bar Association recognizes a member of the staff who goes above and beyond helping students on their voyage through law school. The recipient of the SBA staff member of the year this year, 2014, is Barbara Sapsky. Barbara, will you please stand? I want to announce award that's given out very rarely, the Dean's Distinguished Service Award. The colleague who will receive it this year has been an invaluable administrator, twice serving as interim dean, as well as a stint as director of clinics. He is creating and led our mediation clinic and has served on myriad committees and projects, not to mention regularly teaching and overload. In short, he's the perfect colleague. I am pleased to award the third Dean's Distinguished Service Award to a founding member of the RW Law Faculty, Professor Bruce Kogan. Over to Logan. The journey from law school orientation to law school commencement takes years. It's a journey marked by challenges met and results today in a dream achieved. We know that few law students make this arduous journey to the special day without the help of people outside the halls of the law school. So now is the perfect time to recognize some of the other people who provide valuable support to the graduates we honor today. First, it starts with parents. I would ask all the mothers and stepmothers and fathers and stepfathers of the class of 2014 to please stand and accept a round of applause for all you have done to bring us to this great day. One of the great safety nets is an extended family. With the siblings, aunts, uncles, great uncles, great aunts, cousins, and last but surely not least, the grandparents of the class of 2014, please rise. And finally, many law students are married or otherwise involved in long-term relationships. And these people also provide invaluable support. With all the domestic partners and spouses, common law or otherwise, plus their children, please stand and accept our thanks for all their help. The School of Law is part of a vibrant university. And we are lucky to have at our helm Dr. Donald Farrish, the 10th president of Roger Williams University. Besides being a distinguished teacher and scholar, Dr. Farrish brings deep experience as an academic administrator. The law school community is especially pleased that Dr. Farrish has a JD in addition to his PhD in biology. So he, like us, has survived the rigors of the Socratic Method, Moot Court, and the Bar Exam. I've worked closely with Dr. Farrish during his three years here, and I know that the university and the law school are very lucky to have him at the helm. I give you Dr. Donald Farrish. Thank you, Dean Logan. Distinguished guests, faculty and staff of Roger Williams University School of Law, alumni, friends and families of the graduates and the graduates themselves. Good afternoon and welcome to your day, commencement for the class of 2014. This is an exciting day for you, the graduates, because you are finally finished with your formal coursework and training and are ready to move on to the next stage of your lives. But it's also an exciting day for me because I was present when you took the oath of ethics administered by Chief Justice Satell of the Rhode Island Supreme Court at the beginning of your first year of law school when I was newly arrived at Roger Williams myself. Allow me a brief digression which kind of picks up on what Dean Logan said a moment ago. In the 1970s, I was on the biology faculty of the University of Missouri when I decided to study law. I had a bad day. And after finishing my degree and passing the bar, I accepted a position at the University of Rhode Island as the assistant dean of the College of Arts and Sciences. I was looking forward to spending time in the law library at the law school in Rhode Island. I hadn't bothered to check whether it was that you or I had a brown. But I wasn't particularly concerned about the location. Well, this was 1979. And as you know, at that time, there was no law school in Rhode Island. It hadn't occurred to me that there would be a state with no law school. So the joke was on me. So fast forward all these years, and I find myself at the University of the Law School. And that's only because Roger Williams University decided to create one. During my first year as Roger Williams president, Judge Salia told me something of the history of the school and how he had been approached to do a feasibility study to create a law school. And he declined because he didn't believe at the time that a law school was necessary in the state given the availability of law schools in nearby states. Judge Salia now admits he was incorrect, as were many others in the law profession at the time. We have seen in the 20-year history of this school how profound the impact of a law school has been on the workings and culture of the state of Rhode Island. Something Judge Krause, who will receive an honorary degree today, was speaking about an event at my home just last night. In addition to obvious improvements in the quality of legal practice and the much greater availability of legal services to the public, there is also no question that our law school has played a significant role in how to put this delicately enhancing ethical behavior on the part of the public officials. Corruption has certainly not been eliminated, but it is on the wane in Rhode Island because the Rhode Island bar and bench and the public itself have greater expectations that public officials will behave ethically and lapses in such behavior will incur both public and judicial condemnation. And those thoughts lead me to this next point. It is traditional at commencement everywhere for the president of the campus to charge the graduates. And since we at Roger Williams enjoy honoring academic traditions, I have the responsibility of charging you now. You have entered an honorable profession. As a nation that follows the rule of law, we depend on some of each generation's sharpest minds, becoming lawyers and ensuring that the rights of society are balanced with the rights of the individual. We need talented lawyers to protect society from those who would cause it harm, to defend the individual against overly aggressive regulation and governmental bureaucracy, to ensure that all members of our society have access to legal representation when they need it, in short, to keep society's machinery running smoothly. These activities must be carried out of the highest ethical levels so as to ensure that society never becomes mistrustful of the legal process. These are significant responsibilities, but asking anything less of you is to diminish the importance of the roles you will play. Graduates of the Roger Williams University School of Law Class of 2014, go forth with confidence and resolve. And as you do so, remember your professors and those who helped you get to where you are today and make them proud. Thank you. As a young law school, it is imperative that we bring leading lawyers and judges into our orbit. Mark Mandel is one of America's leading attorneys for injured people. He has served as the president of both the Rhode Island Bar Association and the American Trial Lawyers Association. Some of you may recognize his name from his work as counsel for the victims of the Hurvick Station nightclub fire. Mark is on the podium in many capacities today. He is a trustee of Roger Williams University. He is the chair of our board of directors here at the law school. And he is a popular member of our adjunct faculty. To bring greetings from the board of directors, I am pleased to introduce to you Mark Mandel Esquire. I want to share, and Dean Logan and President Farish is welcome to you all. I want to just thank both of them for the great honor they've given me to be able to work with them as chair for the last three to three and a half years, and even before then with Dean Logan. I especially just want to talk to the graduating students just for a few minutes today. Please know we're honored. We're really honored that you came to law school here. And we're honored to have had you as our students. And I'm talking to my own students as well as everyone. We're proud of what you've accomplished and of your being here today. And we look forward to sharing in your success and your accomplishments in the future. So just a few thoughts from somebody who's been practicing almost 40 years now. A few years ago, my wife Yvette, who's one of my law partners, and I went to see a movie called Saving Private Ryan. And it's a movie that takes place at the beginning of the end of World War II. And it's a simple but pretty compelling story.  Command decisions made in Washington, DC to save the fourth brother, James Ryan. And he was an airborne ranger. He had been parachuted behind enemy lines the night before the D-Day invasion. So they picked a squad of eight people, eight men, to make this rescue mission. Each of these eight men questioned why so many people had to risk their lives to save just one soldier. And each squad member was willing to give their life, to sacrifice their life, to stop national socialism in the world, but not to save just one ordinary man. And as it turns out, only two of those eight rescuers survived themselves. But they were successful. And at the end of the movie, Sergeant, excuse me, Private Ryan is with his wife, children, and grandchildren, his having survived the war, in the St. Laurent military graveyard, paying homage to his rescuers. So perhaps the most critical scene in that movie is in a battle over one of the last intact bridges. The Nazis, the Germans, vastly outnumbered the few Americans who were protecting the bridge and Private Ryan. And at the end of the battle, the captain who led the rescue patrol, played by the actor Tom Hanks, is dying. And he has Private Ryan come close to him, and he whispers to him, earn this, James, earn this. So Tom Hanks, the captain's message, is pretty powerful and pretty clear. Private Ryan needed to earn the gift of life he had been given by living the balance of his life charitably, honorably, and ethically. Only by doing that could he have made the sacrifice of lives worth that rescue. And as he stands later in that cemetery with his family, he asks his wife of many, many years, am I a good man that I lead a good life? So to me, one life is worth saving. Also, all injustice is worth preventing. As Martin Luther King said, injustice anywhere is a threat to justice everywhere. We, all of us, must earn the right, and now you have to earn the right, too, that we've been able and will be able to practice law. To me, it's a sacred profession. I've been doing it 39 years. It is an absolutely sacred profession because the principal goal is to attain justice. And it's the only profession I know of where that is the principal goal. I've been in the presence of justice, thank God, a few times at least, and there's almost nothing like it. So we earn our place by living honestly and ethically, by acting as role models for others about everything that can be decent in this world, by dedicating ourselves to public service and public interest work, and by ensuring that our families are safe and your children are wiser, more understanding, and better than we are. We, and you all, we all need to look beyond ourselves. The only good, the real good we do in this world is the only measure of our ultimate wealth. The legacy, you all, that all of us will lead behind is gonna be determined directly by our charity, by our compassion, and by our understanding. And each of our epitaphs will be written based on how we live our lives, not by any calculated effort we may engage in to construct that epitaph. We'll be remembered not by what we have, but by what we have given. And as Anne Frank said, nobody, not anybody, ever became poor by giving. So as we teach you, our students, we need to know that you can be our teachers as well as our students. We need to listen to you as intently as we tried to teach you. And we need to know, all of us need to know, that in fact, in reality, you are our legacy to the world. So we all wish you absolutely the very best in everything that happens to you and all good fortune in your lives. Thank you. Thank you very much, Mark. Just a little ad lib here. Today is the last day that Mark Mandel will be chair of the board of directors of the law school because at the board of trustees meeting earlier today, the board accepted the nomination from the law school board of directors that the new chair of the board would be federal judge Will Smith. And I just ask Judge Smith to stand, if you would please. Or the school of law is indeed fortunate to be able to award honorary degrees today to Robert M. Barge, the Honorable Robert D. Krause and the Honorable Judith S. Kay. Would school of law board of director Robin Steinberg please escort Robert M. Barge to the podium? Robert Mansfield Barge, it has been 35 years since you first arrived in Rhode Island, a recent graduate of UCLA law, to work on the front lines as a staff attorney for the non-profit law firm, Rhode Island Legal Services Incorporated. And it has been nearly 25 years now since you became that organization's executive director. Under your dedicated leadership, Rhode Island Legal Services has dramatically enhanced its representation of low income Rhode Islanders in a wide variety of civil matters. Nor are these profound improvements just a matter of opinion. The Legal Services Corporation, the nation's single largest funder of civil legal aid for low income Americans has specifically recognized your leadership and commitment to a high level legal services program and commended your programmatic vision and innovative initiatives. Prior to your appointment as executive director, you developed a deep expertise in the area of housing law, eventually rising to manage and direct Rhode Island Legal Services inner city advocacy project, an outreach project serving the minority communities of Rhode Island. Attorney Barge, in addition to your outstanding direct legal contributions, you have advocated change throughout any number of community organizations and civic panels, including the Thurgood Marshall Law Society, the Tsetzi Gallery, and the African American Project Directors Association. Attorney Barge, for your tireless efforts in the cause of justice and equality for all, we are honored today to confer upon you the degree of Doctor of Laws, honoris causa, given this 16th day of May, 2014. Would trustee Gerald Divine please escort the honorable Robert D. Krause to the podium. Judge Krause, since your appointment to the bench in 1986, you have presided over hundreds of trials and administered Supreme Court calendars in every county of the state of Rhode Island. The experience and expertise you acquired along the way, steadily preparing you for your present role as senior associate justice of the Rhode Island Superior Court. You have also served as chairman of the advisory committee on the Rhode Island Superior Court Rules of Criminal Procedure, and you were a member of the state's first judicial nominating committee. In your earlier career, you served as assistant U.S. attorney for the district for Rhode Island, and prior to that, you were assistant U.S. attorney for the Southern District of California, where you were also chief of the appellate section, managing all of that district's appeals before the United States Court of Appeals for the Ninth Circuit. Judge Nacrause, above and beyond your extensive legal and judicial pursuits, you have earned a reputation as a skilled photographer and digital artist whose works have been featured in the Providence Journal, displayed in a local gallery and museum, and not to mention in a number of Rhode Island courthouses. Several of your photographs were selected for a publication in a book on Americana. Judge Nacrause, for your lifelong service to law and justice in Rhode Island and beyond, we are honored today to confer upon you the degree of Doctor of Law's honoris causa, given this 16th day of May, 2014. Would Professor of Law, Emily Sack, please escort the honorable Judith S. K. to the podium. Judge K., we honor you here today, not only as the first woman ever to serve on New York's highest court, but as the first woman to ascend 10 years later to the position of chief judge of the state of New York, a role you fulfilled with consummate skill, compassion, and innovation for more than 15 years. From that powerful perch, you developed a national reputation for your groundbreaking decisions and innovative reforms to the New York court system. You wrote important decisions on a wide variety of contentious legal issues, from rights for gay couples to the death penalty. You also left your mark as a creative reformer, streamlining the state's jury system and establishing specialized courts addressing such issues as drug addiction, domestic violence, and mental health. And you created the Adoption Now program to better serve children and foster care in their families. Your reforms have since been implemented by many other state courts nationwide. Part of a judge's duty, you once said, is to, quote, serve the public wisely and diligently, soundly and innovatively, preserving and carrying forward the best of the past to meet the demands of a rapidly changing society, end quote. By these standards, your career has been an extraordinary success. When you retired as chief judge in 2008, you said, there can be no doubt that I've had the role of a lifetime, a privilege beyond description, to labor in the cause of justice alongside the greatest people on earth. Judge Kay, for your lifelong service to law, justice, and the betterment of society, it is our privilege today to confer upon you the degree of Dr. Law's honoris causa, given this 16th day of May, 2014. And I now invite the Honorable Judith Kaye to make the commencement address. Thank you so much, President Farrish, Dean Logan, honored faculty, administration staff, honored graduates, and guests. Thank you so much for the privilege of addressing you on this extraordinary occasion. By that greeting, of course, I have in mind first and foremost, the fabulous graduates, class of 2014. Congratulations to each and every one of you, and congratulations as well to your families and your friends. So many of them gathered here under the tent with us in every sense, no limit on invitees. The people who have worked so diligently and lovingly to help you reach this great moment.  And of course, by extraordinary occasion, I have in mind as well that this is the climax of an entire year celebrating the 20th anniversary of Rhode Island's first and only law school premier in every single way. Here stands Roger Williams University School of Law at the brink of 21, now into its third decade. Now, that's full maturity in every sense. And imagine, more than half of this law school's wonderful history has flourished under the baton of your remarkable dean, David Logan. Congratulations, Dean Logan. Now, you have surely earned a return to your first love, which is full-time teaching with our boundless gratitude. And yes, indeed, we have a lot to celebrate today, and I'm so pleased to be here celebrating with you. Believe me, it is entirely my honor to be with you, an honor I'm pleased to share with my distinguished co-honorary degree recipients. That's two Roberts we have with us, Robert Bard and Robert Crass, Judge Crass. We now also become alums, we become alums. And you know, I think I'm going to be substituting RWU Law School, class of 2014, for NYU Law School, class of 1962. Makes me feel an awful lot younger, more energized, more invigorated, more inspired to go out and change the world right along with you. Right. Now, having spent more than 50 years as a lawyer and judge, oh my God, it should not surprise you to know that I like to follow good precedents. So I'm going to take a cue, one cue, from famed civil rights attorney Morris Dees by echoing his words at last year's commencement ceremony. Like Morris Dees, I too can say unequivocally today that I am so very proud to be a lawyer. And I will take my second cue from remarks recently published by your great dean. A lot has changed in the legal profession, particularly in the last few years. Some of it good and some of it not so good. But I'm going to begin with the unequivocally good news. Welcome to a great profession, one that has helped assure over the past two plus centuries that this indeed remains a land of justice and equal opportunity dedicated to the values of our founders. Indeed, if you think back on our nation's beginnings, this has been a law-oriented society from day one. Do you know that more than half of the signers of the Declaration of Independence were actually lawyers, as were many of the delegates to the Philadelphia Convention, many of the leaders in the revolutionary movement and drafters and signers of the Constitution? Throughout history, our most pressing social and economic and political issues have been framed in law terms with profound problems of social order resting in the hands of lawyers and judges. I know no better evidence than the fact that we gather for your graduation on the eve of one of the nation's landmark events, the 60th anniversary of Brown against Board of Education, a powerful unanimous decision of the Supreme Court of the United States issued on May 17, 1954. I remember it well. That was the year of my graduation from high school, changed my life and lots of other lives, to declaring the right to equality of education. In just a very few eloquent pages, and I urge all of you to have a look at Brown against Board of Education, the Supreme Court overruled a longstanding president and boldly reaffirmed an enduring foundational principle of this great nation. We are not separate but equal. We are equal in every sense, equal in every sense. And to this day, the lawyer stands as the essential intermediary in our nation's social progress, the resolution of every key issue from global internet privacy to neighborhood individual property at the core determined by statutes and rules and regulations and judicial and administrative decisions. In all my decades of experience, I have yet to discover any other calling that offers so many opportunities for serving society generally and helping people individually, whether by writing a will or closing on a home or a massive corporate merger or defending against criminal charges or seeking compensation for a slip and full or for catastrophic injury. Without any particular artistic or architectural skill, lawyers can make a symphony go forward or an exhibit open, a low income housing project go up or a blight on the environment come down. Now within this profession, I have been able to enjoy both a stimulating career and a very fulfilling family life. I even had the exquisite pleasure some years ago, like so many of you in the audience today under this tent of attending my own child's graduation from law school. Now that happened also to be the place and year of the graduation of a very close friend of my daughter, your wonderful professor, Emily Sack, she's my daughter too. Teacher of the year, Emily Sack. Now her life, Emily's life at Roger Williams as I have kept in touch throughout the years has dedicated to social justice, offers yet another example of our opportunity to make this a better world. Just think, she's a dedicated teacher. We've talked a lot about her love of her students. She's a dedicated leader in our fight against domestic violence. She's the co-author of the terrific book in the field. She's the president of the board of a local domestic violence shelter and an advocacy group as we have together, all of us together, lawyers at the forefront, led in the transformation of a vital social and legal response to the scourge of domestic violence. Yes, most definitely I agree with those bright, positive, optimistic thoughts that brought you to the doorstep of this law school. You indeed can help to make this a better world. And that brings me to the doorstep of Dean Logan's mindset or opinion article on the changing, challenging face of the legal profession today in Rhode Island and might I say far, far, far beyond Rhode Island. And this is the mixed news part of my remarks today. Perhaps the most obvious change and we have only to look around us is the very long overdue growth of diversity in our profession starting right here. At Roger Williams and I've studied all of your statistics, I know more than half the class is female, more than a quarter from groups traditionally underrepresented in the profession, including older graduates, first generation college and law school graduates, people embarking on a second career. Now I have to tell you, forever unforgettable for me are the days when I was one of barely 10 women at NYU Law School. That's why I'm moving over to RWU Law School. I was barely one of 10 women in a class of 300 at NYU Law School. That was a time when firms openly turned us away with the line, our quota of women is filled and the quota probably was zero. When court appearances, rare court appearances were often greeted by the words lady, get a lawyer. They were days of sharply unequal pay and corporate policies mandating, mandating, firing in the seventh month of a pregnancy. So cause for celebration as we look around us? Absolutely, absolutely. But as the dean points out in his excellent article, these are only signs of promise, a stake in the ground for equal opportunity throughout our profession, as we set out to spread the message throughout the highest levels of society. This task is far from over, believe me. If you look at the still daunting statistics throughout law firm partnerships, the judiciary and the government for just starters, women for example, are definitely doing a lot better in law school, but years later in actual practice, we're nowhere close to duplicating those sunny statistics. And herein lies one of your challenges, one of the dean's challenges, one of my challenges, a really serious challenge, not simply to study and gather and assemble and mourn the statistics, but to go out in the world and rewrite the statistics. Having had the pleasure of chatting with your classmate, Garrett Marshall, I'll use his word, thank you Garrett for helping me, I'll use his word and describe this as a gateway moment, a time you can, you have to, you must push through the gateway to make your opportunity equal opportunity, genuine diversity a reality. And might I say your law school, our alma mater, mine too, has prepared you very, very well for the task. Supporting, working with the community is a hallmark of the school. Public service is part of your DNA. Throughout the profession and the world, assuring equal opportunity, public service, those are heavy tasks that lie with you, but surely you have been well-primed and well-prepared to move them forward. And here again, here again, I offer another lesson from my own decades as a lawyer and a judge, and I've boiled them down to just two words, hang in, hang in. Having been born on a farm in upstate New York to immigrant parents, I have to tell you that never in my own wildest dreams did I imagine that I could become a judge, first woman judge on the high court of the state of New York, let alone the chief judge of the state of New York, the most glorious role imaginable in the entire universe.  You have to believe in yourself. You have to push back. You have to stay focused to pursue the dream that has brought you to this glorious day. Now that's easier said than done, I know. And you are bound sometimes to make missteps to disappoint yourself even from time to time, personally, to fail. But I also know the group I am addressing. You wouldn't be here today if you weren't believers in push forward, hang in, push forward, be positive, seize the day. Those are the qualities that have brought you to this day and those are the qualities that will take you through the gateway.  That should be your goal because let me tell you, this is a real treat. But perhaps of even greater concern today are two of the dean's reflections and I'm still focused on his article on today's new challenges. And again, he's unquestionably right on target. First of course, in a day of law school debt is the dramatically changed marketplace of legal employers demanding that new hires be equipped to add value from day one. The days I remember so well of tagging along, having the comfort and expectation of constructive mentoring. Today those days have changed dramatically. But being the good news person I am, the positivist, the optimist, I have put this genuine concern together with the dean's reflections on the many today in desperate need of legal assistance and the burgeoning innovative programs, especially right here at your campus and your law school, linking lawyers to work with service organizations to help the homeless, to help at-risk youth, to help low-income families. With genuine concern and with genuine sympathy, I have watched the coming together of these two populations on the one hand seeking out and on the other hand finding new ways to define and redefine our professional values in a most constructive way. I know no better example than your many, many collaboratives here on campus and believe me, we are hard at work all through the profession and through the nation to put the two populations together, to multiply the innovative programs for those that serve people desperately in need of legal services on both sides, need for legal services and opportunity to fulfill those needs. There can only be brighter days ahead, I promise. So that brings me to the dean's final reflection. On a decade of profound change and that is the most intriguing of all, the challenge of improvement in the rule of law. The very purpose underlying the founding of this law school and to use his words, he calls it a lively experiment. The very purpose of this lively experiment was to raise the level of debate and discourse on important issues while also enhancing public respect of law. In other words, this law school was to be and remain and again I'm quoting the dean, an unbossed and unbought source of opinion on the issues of the day. Now that's a tall order too, but looking out on you, having read and learned a great deal about you, I would say you are well up to the challenge. Now just take a look at a few of the hot issues of the day, affirmative action and campaign finance and voting rights if you just look at some of the recent Supreme Court decisions, issues that go to the very heart and soul of our great nation. Has the last word been spoken? Absolutely not, I certainly hope not. I've watched state by state avid taste for death penalty dwindle, certainly in New York, thankfully, and I see changing attitudes about our nation's mindless mass incarceration and mindless juvenile detention. On the subject of change in the law, perhaps most sweeping of all, we've witnessed the recognition of the right to marriage, to same sex marriage. All issues with the rule of law at their very core, at their very center, fodder for graduates who are graduates of an unbossed and unbought law school. On this day, we are infused with pride in you in this law school and in the unbossed and unbought model you now have for earning and enjoying appreciation for public service, for good work. I agree wholeheartedly with Dean Logan. The rule of law will continue to improve in Rhode Island and far beyond Rhode Island as all of you, each of you, your faculty and alumni, and most especially my class, the class of 2014, continue to tackle important issues that make this state, this nation, this world a better place for every single one of us. Congratulations to all of you, especially to my fellow alums, the great class of 2014. I hope that the good sense and the good fortune that have brought you to this moment will be with you all through your lives. Thank you so very much. Thank you, Judge Kay, for your thoughtful remarks. At this point in the commencement ceremony, the 19th of the School of Law, I invite the class of El Victorian, Samantha McCoy-Clark to offer remarks on behalf of the terrific class of 2014. Samantha? Good afternoon, Dean Logan, President Farish, members of the Board of Trustees and the Board of Directors, members of the faculty and staff, parents, family, friends, and of course, welcome to the class of 2014. It's an honor to speak to you today. At first, I wanted to give the speech from the end of Legally Blonde because I already had it memorized, but my family told me not to. Then I thought, why not use famous quotations, time-tested, well-loved by millions, mostly from Star Wars, but my family told me not to. There's a theme here, and it's not just that I turned to pop culture when things get awkward. No, when I need advice, support, or just someone to tell me that singing, let it go at a commencement speech isn't appropriate, my family is there. Now, when I say family, I don't just mean my mothers and fathers, my boyfriend, my uncle, my cousins. I mean all of you, all of us. We're a family, but you're hardened lawyers now. You don't believe me yet. That's all right. I've been preparing my case all week. Here is my evidence. We share memories, we share values, and we share a name. First, our memories, the good ones, like parting a barrister's ball, making outrageous bids at the auction, taking over Aiden's after finals and scaring away the lunch crowd. I promise you, some of our good memories even happened during school hours. We learned an astonishing amount. All of us worked harder than we ever thought we could, forging friendships in 48-hour study binges, writing endless memoranda and briefs, reading every night, attending symposia, debating passionately in class. We made late-night coffee runs and early-morning gym visits, some of us. Somewhere in there, we met five United States Supreme Court justices. We dined with professors at Potluck's, and, of course, we watched Josh Xavier rap to juicy, flawlessly, frequently, and sometimes with backup dancers. It wasn't all Biggie Smalls and Sonia Sotomayor. We had some tough days. All of us felt that same sense of impending doom whenever Professor Ritchie looked down at his seating chart. Many of us remember the first day we survived the sack attack. And, after what seemed like nearly three years of apocalyptic blizzards, hurricanes, and blackouts, we all became masters at studying by candlelight. But those are the stories that we remember, Class of 2014. That's not what's gonna be remembered about us. Instead, others are gonna remember our values. They'll remember our commitment to service, those late nights at the ACI, the spring breaks and summers that spent providing legal services to those who needed it most, the compassion shown by all the student attorneys and the clinics, the internships, externships, tax assistants, panels, outreach, and more, much more, too much to list. And, often, the work didn't have a name or a class credit associated with it. We just did it. We did it to serve. That's what they'll remember. They'll remember our drive and our passion, all the late nights, practicing for moot court and trial team competitions, the three editions of Law Review this year out, first time in a decade, all three. Our work at private firms and corporate offices and the fact that day in and day out, we prove that we belong there. That's what they'll remember about us, Class of 2014. So what's left? A family with memories and values and a name. We share one now. When we introduce ourselves and people ask us where we went to school, we'll give the same answer, Roger Williams. And because the Class of 2014 is made up of such an incredible group of people, wherever we go, our classmates' reputations and successes will precede us. Knowing this, I feel privileged that I will share a name with you. But even so, today is bittersweet for me.  To the parents, brothers and sisters, aunts and uncles, cousins and friends, all the loved ones who are so proud of you and came to see you today. And I'm gonna run over to my moms and dads and I'm not gonna be able to figure out who to hug first, but that's okay. I think I've made a solid case to you that no matter what, no matter what happens after we cross that stage, Class of 2014, we're a family now. And so, family, explicitly against your advice, I'm going to end by quoting Legally Blonde anyway. Congratulations, Class of 2014, we did it! Thank you, Sam. Tough act to follow. As you leave the familiar grounds of law school to undertake your career in the law, I challenge you to retain that sense of hope and optimism that you brought with you on your first day of law school. Remember the lessons taught about personal and professional responsibility so that you will always be proud of who and what you are. And keep ever in mind that as a lawyer, you are empowered, indeed, duty-bound to make a difference in the lives of your clients, who you will serve and the society in which you will live. We will now confer the law degrees and present the degree candidates for the Class of 2014. Will the candidates for the degree of a Juris Doctor please rise? President Donald Farish, I certify on behalf of the faculty of the School of Law that these candidates are eligible for the degree of Juris Doctor. By virtue of the authority granted by the State of Rhode Island and Providence Plantations and delegated by the Board of Trustees of Roger Williams University, I confer upon you the degree of Juris Doctor with all the rights, privileges, and responsibilities pertaining thereto. Will the candidates please come forward? The Associate Dean for Academic Affairs, Diana Hassel, will present each candidate. Samantha McCoy-Clark, Summa Cum Laude. Letitia Cunha Tavares, Cum Laude. Josette Rocha, Jocelyn Rocha, Cum Laude. Susanna Santana, Alexandra Davis, Summa Cum Laude. Elizabeth Ann Blank, LMMA, Magna Cum Laude. Jay Sokolov, Horse Bloom, Cum Laude. Thomas Leonard Moran, Cum Laude. Drake Benjamin, Lee Roberts. Paul Carmody Jr., Elizabeth Christum. Lynn Sarah Smith, Caitlyn Virginia McGuire. Anne Hamilton, Caitlyn Mary Wollstonecroft, Cum Laude. Catherine Robbins, Cum Laude. Jennifer Ann Sherman, Kerlyn Reed, Magna Cum Laude. Rose Parker, Shanna Curland, Magna Cum Laude. Magna Cum Laude, Verdi, Magna Cum Laude. Wilson Christina Murphy, Cum Laude. Ashley Ann Brinkman, Magna Cum Laude. Johann G. Denal, Anthony Jordan Coistra. Xander Liu, Cum Laude. Papaginae, Ray M. Sheehan. Elizabeth Smalley, Cum Laude. Daniel C. Tevedo, George Chacarino, Cum Laude. Grace Stockley, Uncribbin III. Paragamo, Jennifer H. Whitmore. Tasha Karine Ludwig, Alicia J. Perry. Asia Sierra T. Millette, Cum Laude. Peterson, Cum Laude. Richard Fyalkoff, MSC, Magna Cum Laude. Jessica Jody Hale. Stegerson Zildmi, LMMA. Elizabeth McNamara, XLMMA. Marcus Jacob Swift, Magna Cum Laude, Odi Yingbo, Obatista, Tentas Jr., Cum Laude. Elsie Christine Ann Leone, Inc. Summa, Cum Laude. Tenaccio, Anton Chase Schrander, Christina, Christine Diana Hutley, Morgan Rodriguez, Cum Laude. Leslie Jackson, Stephanie C. M. Bartholet, Stina Lynn Connor, Elizabeth McCarthy, Ann Marie Francis, Elizabeth Essex, Jennifer Ashley Roth, Jennifer M. Vosberg, Rose De Dio, Tracy Andrea Deceivo, Alexander Denise Esquire, Magna Cum Laude. Ladies and gentlemen, the class of 2014. Before you depart our lovely campus, I urge you to savor this moment. The class of 2014 has achieved greatness both individually and as a group. And on behalf of the faculty and staff, we look forward to being there for you as you embark on the next phase of your journey. At the conclusion of the ceremony, I encourage the graduates and their families to come back to the law school building for light snacks on a chance to shake hands, hug, do some selfies, some photo bombs, with your faculty and friends. I wanna close this commencement as I've closed many before with a message from Garrison Keeler. Be well, do good work, and keep in touch. Thank you and will the audience be seated and remain seated while the platform party and graduates have filed out.


Next, he's going to be in Sheffield. So this project is, at some point, the paper will come out on this, but it's not finished yet. Oh, and he's watching the line as well. OK, so what's the sort of aim of this whole story? The aim is to develop a lattice theory for, let's say, type 2 degenerations. And I'll just tell you all these things are of K3s. That's, shall we say, natural for doing mirror symmetry. And I'm not going to say an awful lot about mirror symmetry, partly for reasons of time and partly because that's part of the in progress. But I'm going to say a lot about the lattice theory stuff. So the first part is some stuff on pseudo-lacknities. And this is mostly background, but they're not sort of common objects. If you're after some references, this was really sort of introduced by Kuznetsov based on work of a bunch of earlier people in a paper of 2017. That paper is called Exceptional Collections for Something, Something, Something. The 2017 Exceptional Collections is enough. And a paper that I wrote with Andrew Harder in 2020, which is called, I don't remember, probably has pseudo-lattice in the title. The unique paper written by Andrew and I in 2020. OK, so what is a pseudo-lattice? A pseudo-lattice is, and I'm going to sort of abbreviate it to PL, that pseudo-lattice is not piecewise linear. G, it's a finitely generated Abelian group, front and generally free Abelian group. And I'm going to equip it with a pairing, a non-degenerate bilinear form. It's an integral bilinear form mapping from G cross G into Z. And the important thing with this, what's the difference between this and a lattice? The important thing with this is we do not assume symmetric. OK, so this thing could be a more complicated bilinear form, not necessarily a symmetric bilinear form. OK, so I'm not going to spend an enormous amount of time going over sort of background theory. I'm mostly just going to give you a sort of example and say, point out some of the basic definitions in this. So here is the running example for the first part of the talk. Let's say that X is a non-singular rational surface. I only work over C.  So X is a non-singular rational surface. I'm going to assume that the anti-canonical linear system of X contains a smooth member. So this contains all of the delpeto surfaces, but it's also much bigger. I don't assume any positivity. I just assume that there's some smoothness. So what we can do, we can build a pseudo-lattice G, and this is the numerical growth and deep group of the bounded derived category. Well, I've got to get my notation right. Bounded derived category of coherein sheaves on X. And we can equip this with the Euler pairing. The Euler pairing is given by if I've got two classes of sheaves, F1 and F2. This is the sum of minus one to the i, HOM from F1 to F2 twisted by i, or shifted by i. So the shift by i means that this thing is not going to be a symmetric bilinear form. So this is going to be a pseudo-lattice. Now, these things are quite general in general. So we're going to have, I'm going to let P be the class of the structure sheaf of a point. This is what's called. So if I'm looking at pseudo-lattices in general, this is what's called a point-like vector. And the reason it's called a point-like vector is because it looks like a structure sheaf of a point. There's a set of conditions. You can define these things purely algebraically. There's a set of conditions that tell you what point-like vectors are. I'm not going to go through all of that. You can read in Kuznetsov, but these are abstractly, these are things living inside a pseudo-lattice that behave like the structure sheaf of a point behaves if you're living in this universe of work. There's some more things here. So if I take your orthogonal complement of P quotient by P, if I'm working in here, what does that give me? That gives me the narrants of every group of X. And so I call that the narrants of every group of G. Again, if I've got just an abstract pseudo-lattice along with a point-like vector, I can do P perp divided by P. Part of the condition, part of the definition of point-like vectors says that this thing is going to be well-defined. This thing is actually a lattice. So the restriction of the bilinear form to this neuron is very, part of the definition of the point-like vector is that the restriction of the bilinear form to this thing becomes symmetric. So this is actually like a proper lattice and not a pseudo-lattice. I can also define KG. Again, this can be defined purely algebraically in terms of the pseudo-lattices, but we can think of it as the class of the canonical bundle of X. And this is called the canonical class of G. So these are things we do. We can also, we can define a SAIR operator. So the SAIR operator on this category is given by tensor by O of X and then shift by N. This is the SAIR operator, it's got the following property. The following property is if I take a pair of things, V and U, I can sort of commute them if I use the SAIR operator. All fairly standard stuff so far, hopefully. But all of this can be done purely algebraically without resorting to talking about the, without resorting to a specific example. Okay, right. So what can I do? Let's say if I've got C in the anti-canonical linear system, let's say this is a smooth curve. So then I can take I would be the embedding of C into X. Then what I can do is I can, I can take the pullback, the pullback by I, this is a map from G to the numerical growth in the big group of the different, all this stuff. And the derived category of coherent sheets and C. So C is going to be a smooth elliptic curve just by a junction. So this is what we call a spherical homomorphism. It's an example of, again, this is something you can define completely abstractly, it's unilatses, but I'm not going to do so. Part of the definition of that is that it has to have what's called a right adjoint. So the right adjoint in this case is by lower star and the defining property of that is the, if I do I up a star of U with V, that gives me the same, this is just gives me an integer, right? That's the same as doing C. So we can sort of pull classes up and push classes down and that respects the bilinear forms. So everything works in this nice example, but we can also, you know, the sort of the thing I want to get across somehow is that we can also do all of this purely algebraically in terms of abstract structures, unilatses, bilinear forms, this kind of thing. Okay.  And I will say I'm going to get even more vague. We're going to get back to being specific in a minute. A spherical homomorphism, let's say F from G to E. E is isomorphic to this thing of an elliptic curve, brackets. So a spherical homomorphism from a pseudo lattice to the boundary, the numerical growth in the group of the boundary derived category of coherent sheaves on elliptic curve. This thing is called quasi-delpetso, quasi-delpetso, if it looks like this thing, I have a star in this example. So again, this is something that, you know, one can make a purely algebraic definition of in terms of the pseudo lattice, but you know, to give a talk where I give all these definitions, I would spend an hour and not get through the definition. So I'm just kind of saying, I'm giving you an example and saying all of this can be generalized purely in terms of, okay, we're all happy. We're given value of happy. Okay, so right now we're going to get into the, well, we're getting closer to the meat. So let's suppose I've got F from G to E and G from G to E. These are two spherical homomorphisms. I can use this, I can put a pseudo lattice structure on the direct sum, G direct sum H, and how do I do it? I do sort of my bilinear form is given by, make myself to say, block diagonal stuff. So I've got, if I've got two things, U and V, what do I mean here? I've got two elements, U and V that are both in G. I take the intersection in G. If I've got two elements, U and V in H, I take the intersection in H. I take zero down there. And if I'm up here, I take F of U with G of V inside E. So this is what to do if I get something where one of them is in G and one is in H, then this is extended bilinear. So we call this F semi-direct sum G mapping from the definition right, G semi-direct sum H to E. Okay, so this is, you can prove this is also a, this is also a spherical homomorphism. You can prove that certain properties of this can be deduced from certain properties of the factors.  Again, I don't know if they're too far into these, of course. Okay, now I have to figure out how to clean the board. Watch the least professional example of this you're ever gonna see. If I do it without getting myself covered in filthy chalk water, I'm gonna be happy. That's the bare minimum. Oh, look, I missed a bit in the middle. This is not too disgusting. Don't shake it. That's how you get it all over. Okay, right. So now, you know, what's the main kind of construction here? Let's say I've got FI from G, I to E. These are Pozzidale-Petzso spherical homs on pseudulaxis, my line down the middle. I'm gonna make an assumption. I'm gonna assume, remember, if I had these canonical classes, I'm gonna assume that the K of G1 intersected with itself is the same as the negative of K of G2 intersected with itself. And we'll sort of, I'll talk about where this condition comes from in a moment when I give you an example of where this stuff comes up. And then what I can define is I can define F to be the semi-direct sum of F1 with minus F2. This thing works out to be a spherical homomorphism. I'm gonna let R be its right adjoint, F. And then we have the following lemma. This is a purely, purely a statement about pseudulax. This does not use any of the geometry here.  So let K be the kernel of F and E bar be the saturation of R of E inside, R of E, R of E, inside G. Then the following is true. The bilinear form is symmetric on K. Bilinear form is symmetric on K. And E bar as a subset of K is totally degenerate. So what do we do if we've got a totally degenerate sub that's of a lattice? We can quotient it. So what I can then do is I can write down an exact sequence. Zero goes to E bar, goes to K, goes to K mod E bar, goes to zero. I can call that thing M. Okay, that's the kind of algebraic background. Now I'm gonna show where this comes up in John. All right, so everything I've said so far, I sort of, you know, I've waved my hands an awful lot. But, you know, the takeaway message is we can do these things called pseudul lattices. They can have various properties if you don't care too much about if they're sort of particularly nice, we can glue them in pairs. When we glue them in pairs, we get this set of properties. So we get, we take the kernel of this, this gluing map, we take the saturation of the white adjoint, then we get this. Okay, so where is this gonna, where is this gonna show up? Let's look at sure in the generations of capers. We're in the generations of capers. So I'm gonna take curly V over a complex disk. This is gonna be some degeneration. In here, I've got a point zero. Over that, I'm gonna have V zero. I'm also gonna have V and that's just gonna be over some sort of generic point T. I'm gonna assume, this is what's called a Turing degeneration. So what's this gonna be?  If I got my notation right, if I don't get the notation right, it's a C. So this is a degeneration. I'm degenerating my K3 surfaces. I'm taking a very special type of degeneration. I'm degenerating them to unions of two surfaces, glued along a curve C. So my degeneration looks something like this. So X1 and X2 are rational. And C is smooth anti-economically. Yeah, C is smooth anti-economically. So somehow, these things, there's old results of Friedman. Somehow these things can be thought of as primitive type two degenerations of K3 surfaces. You can sort of think of these things as like minimal type two degenerations of K3 surfaces. Other type two degenerations can to some extent be obtained from these by base change. Do I want to say anything else now? Probably, actually, let's put some more on here before I raise things. So as we saw before, I can let GIs be these things. I can let the FIs be the map from GI to, you know, this thing, K naught num VB co of C. And, you know, everything is I'm in the setting that I was in before, right? You know, I've got these things are quasi-delpetsosunilatices. I've got a pair of them. I've got these maps. So what am I going to do? The semi-stability gives me this condition that the intersection of the canonical devices on one has to be negative the intersection of the canonical device on the other. This is condition on such a degeneration for it to be smoothable. So we should, in theory, well, the idea is we will see these suilatices showing up kind of naturally in the situation of these generations, getting better at this thing. So here is the next lemma. So over Q, what do I get if I take this suilatice G tensed with Q? So what was G? G was defined just exactly as, oh, I didn't tell you what G is. This is the gluing of these along F1 direct sum, minus F2. So it's that over Q. This is isomorphic by the churn character map to the even cohomology of X1, direct sum even cohomology of X2, where this is Q cohomology. And moreover, KQ by the same thing is isomorphic to H0 of V0 direct sum second weight graded piece of H2 of V0 direct sum H4 of V0. So somehow this is the middle graded piece. I mean, these ones, the mixed hodg structures are not interesting. I mean, they're pure on the top and the bottom. So this is the middle weight graded piece of the mixed hodg structure on the cohomology of the central fiber. So what have we achieved here? What we've achieved here is we've achieved in a sort of natural way coming from the lattice theory of the two components. We've achieved the lattice theory on the appropriate graded piece of the cohomology of the central fiber of the generation. So we've got that, then we can look, we've got this exact sequence. Can we identify what that is? Those of you who study degenerations and mixed hodg structures probably know where I'm going already. So M, this portion up here is isomorphic to the limiting mixed hodg structure of H0, second weight graded piece, limiting mixed hodg structure and H4 limb. Yes, K mod E bar is this thing. This is K mod E bar. Wait for the next statement. So now I haven't missed a cue here. This is a legitimate isomorphism over Z. There's actually a Z isomorphism between these two things. So over here we've got the cues, but this one we do not need the cues. So what does this give us overall? I can take my exact sequence, tensor it over Q, and this is gonna give me zero goes to, I get two classes here, Q eta and Q xi. I'll tell you what those are in a second. This goes to H0, V0, direct sum. The thing that's K, and this goes to H0, limb, V, direct sum, blah, blah, blah goes to zero. What is this? We all recognize this sequence. This is Clemench-Schmidt. So this is the Clemench-Schmidt exact sequence, and what we've done is we've endowed it with a lattice theory from these, coming from these two relax. Yeah, maybe, maybe I need like just second weight graded piece. It's just second weight piece rather than second weight graded piece. Let me check it off. I've got a set of notes and maybe there's, maybe there's, maybe this is not exactly, it's something very much like. It's some weight graded piece in the Clemench-Schmidt, but I'll need to check my notes that aren't these ones because it's sort of technical. I can tell you what xi and eta are. xi is the class of minus C, C in H2X1, H2X1, direct sum H2X2, and eta is the class of minus a point, a point in H4 of X1, direct sum H4 of X2. So these are sort of very nice identifiable things. Now there's results of, so you can ask like, okay, I've got these cues here. The cues are not really ideal. This isomorphism over Z, what goes wrong? There's results of freedom from 1984 that say that actually, this exact sequence at the bottom actually holds over Z. So do we have any hope that this is actually a set of the dead isomorphisms? The answer is no, but it's not very far off. So actually this KQ, there's a failure of isomorphism between the integral part of this and the integral part of that, but that failure is restricted to H4. You can get rid of the H4 by doing the following thing. So if you restrict to Neuron-Severi, what do you get? You get zero goes to, then everything becomes exact over there. Well, I'm gonna write this and it might make Matt angry. Now over Z goes to GRW2, H2-LIM, BZ goes to zero. And this is something which, this limiting host structure is something which you can really identify if you've got a degenerating family of K3s, this is something you can work out lattice theoretically. This is a lattice you can calculate. So here we've actually, here we've actually, yeah, so maybe, yeah, we'll figure it out. So, here we've got an actual sort of proper recognized exact sequence of lattices. Okay, so so far, what have I done? I've done some complicated algebra. Well, not so complicated, linear algebra. I've done some fiddly linear algebra to define these pseudo lattices. And then I've derived using them some results about the generations of K3 surfaces that have been known since before I was born. Big deal. So, what's kind of the point of this and where is it going? The point of this and where it's going is that I can make statements about what the mirrors for all this stuff should be. And that's what's coming. That's sort of the next bit when I have watered the board. So this whole picture shows up in a second way. Elliptic vibrations. So, let's suppose I've got pi from Y to P1. This is a elliptically-fibre K3 surface. I'm gonna let f be a fiber. And I'm not, when I say elliptically-fibre K3 surface, I'm specifically not going to assume there's have to be a section here. I'm just gonna say we've got something like this. I'm gonna have gamma living in P1. This is a loop and using it, I can decompose P1 into a pair of disks or, you know, things topologically equivalent to disks glued along the curve delta, glued along the curve gamma, sorry. And I'm gonna let pi i from Y i to delta i. These are just the restrictions of this vibration to each of these two disks. Let's freshen up the chalk. Now I can define a, so I can look at the second relative homology of these things Y i relative to a fiber. If I've chosen my fiber appropriately to be on the curve gamma, there is a bilinear form on here. So this has a bilinear form. It's called the cipher pairing. And with this, this gives it a pseudo lattice structure. And I can get a spherical homomorphism. What's the spherical homomorphism? I can take the boundary map from this to the first integral homology of an elliptic curve. And this, you can give it symplectic basis. It's got a symplectic intersection form. So this thing is a pseudo lattice. And as a pseudo lattice, it's equivalent to the, it's the same pseudo lattice as the numerical growth and deep group on the right category of boundary coherent sheaves on an elliptic curve. So this thing is isomorphic as a pseudo lattice to what we've been calling E before. It's the same thing. And that's really a manifestation of homological mirror symmetry for elliptic curve. It's just saying that, this is sort of the symplectic focaya category statement. I mean, well, okay. This is numerical growth and deep group of the focaya, appropriate set of adjectives focaya category for elliptic curves. On the other side, you've got the boundary graph categories. They're the same thing. Okay. So you've got that thing. I'm going to make two assumptions on the loop gamma. I'm going to make two assumptions. I'm going to assume there exists a symplectic basis, AB for E such that anticlockwise monodromy around gamma acts as, and when I say anticlockwise monodromy, I mean for either of the discs. I look at each disc and this should be true for each of the two discs. It acts as follows. If this is A and B, A and B, I should get one, one, zero. And the number up here should be E of YI minus 12. What's E of YI? E of YI is the topological oiler number which you can compute for these vibrations over discs by just counting up the contributions from each of the singular fibers living inside each disc. The second condition is if R is the right adjoint, so phi, phi I, think R I, then R I of A is primitive. We think this second condition might not be necessary. We don't know any examples where it fails, but in order to prove the things that come afterwards, we need to know this to be true and it's still a work in progress. So I'm allowed to make assumptions that I might get rid of later. Under these two assumptions, these two maps phi I, these are then quasi-delpezzo homomorphisms of pseudolattices. So I'm in the setup I had before. I've got two quasi-delpezzo homomorphisms of pseudolattices and so using the stuff I was doing before, I can glue them together. So if I glue them together, what do I do? Yeah, I'm doing all right for time actually. I was a bit worried. This was a one hour talk and in the half hour before now, I have cut 20 minutes out of it. So you may notice that there are 20 minutes of this talk missing and they've been excised from the middle and that's why none of it makes any bloody sense. If you want the other 20 minutes, buy me a beer in the pub later. Yep, I'm completely agnostic. We're in Germany, there is no bad beer here. And if you think there is, you haven't been somewhere that's got really bad beer. Okay, so I can again take this direct sum. This is now the direct sum of these, direct sum over I of these two things. What do I get for K this time? K works out to be the compactly supported second homology of Y with F taken out over Z quotiented by the class of a fiber. So this is what my thing K works out to be M works out to be the orthogonal complement of F quotiented by ZF. So both of these both, well, M, this orthogonal complement is taken in H two of the K three surface. So this is sort of subset of H two of Y integral second homology of Y. Y homology, yeah, homology of Y. Make sure I'm on the right place and what does our exact sequence become? And this is exact over Z not exact over Q M, the other thing, right, zero. And this is zero goes to the homology of F goes to a compactly supported homology of Y less F over Z quotiented by ZF. I've made a hash of this, haven't I? And this goes to F over ZF, this goes to zero. That's an isomorphism, that's an isomorphism, that's an isomorphism. And moreover, you know, this is not just an isomorphism of Z module, this is an isomorphism of lattices where this thing gets the usual lattice structure from H two of Y, right? So I can induce a lattice structure on this from H two of Y and this thing is getting its lattice structure off of these unilattices. So what have I done? I've sort of identified this picture and what is this sequence? This maybe doesn't look terribly familiar. This is what Chuck and I in a paper last year called the Mira-Klemmenschmitt sequence. When I say last year, I really mean two years ago. So you might ask, you know, okay, but on the other side, really the interesting information was contained in the Neuron-Severi part, right? I mean, you know, if you take the Neuron-Severi part, it cuts off the H naught and the H four and then what you're left with is sort of a standard exact sequence that we recognize for type 2 degenerations of K3 surfaces. If you do the same thing on this side, I mean, you can do it, you can take these Neuron-Severi lattices, but what the picture you get is not, does not seem to have an obvious geometric interpretation, right? You get some sort of, you know, complicated sub lattices of these things, but you know, we don't, there's not some nice clean way that we could think of to write these down as, you know, nice comologies. So there is, you know, this sort of, we take this as kind of an indication that, you know, working on the Neuron-Severi's, if you want to do mirror symmetry, it's a little bit of a red herring, the Neuron-Severi don't work so much, don't work so well over here. Really, if you want to do sort of some lattice theory here, you've got to be working with the pseudo lattices. Okay, so I've said mirror symmetry a lot. So in the last closing minutes, let's say something about mirror symmetry. So what's the sort of motivation behind all of this and where we're going to, this conjecture was never really written down, but you know, sort of comes out of a circle of ideas by Chuck and Andrew and me. So in this sort of situation, it says something along the lines of, if X and X prime are a mirror pair of K3s, then we should have a mirror correspondence between type two degenerations of X being mirror to, let's say, elliptic vibrations on what? Elliptic vibrations on X check. Now, if I've got a type two degeneration, these are not necessarily unique. If I go to a type two degeneration, you know, there might be multiple different models for the same degeneration, multiple ways of completing the family. So we have something like projective Kulikov models should correspond to these splittings of P1. So taking a projective, yes, I don't know. That, yeah, I don't, yeah, I don't know the answer. It's not that's not what you really need to do. But maybe we can check about, maybe you know something right. Okay, projective Kulikov models should correspond to splittings of P1. And moreover, if you take these splittings of P1 and cut them apart, the components of the degeneration here should correspond to LG models. So you can think of an LG model as like an elliptic vibration over a disc or more generally elliptic vibration over an A1. So the components here are rational surfaces. No, in nice situations, there'll be del petso surfaces. Those are mirror to LG models. You can think of a sort of elliptic vibration, you kind of think. But the point is that like, the pseudo-laxis picture that I very hastily presented is the same on both sides. So over here, we've got these pseudo-laxis G1 and G2. On the other side, we've got the same two pseudo-laxis. These correspond in this picture. So what we're hoping to do is we're hoping now, well, okay, all of this is kind of the ambient space to work in, right? So when you're doing mirror symmetry of K3 surfaces, you're talking about lattices and these sort of orthogonal complements of lattices happening inside. Yeah, you talk about lattice polarizations, orthogonal complements, lattice polarizations. We're hoping that using this, we can lift that whole theory from lattice polarizations on X and X check to pseudo-lattice polarizations on the type two degenerations or the elliptic vibrations so the whole picture will be compact. So that's where it's going. Okay, thank you very much. Well, thank you. So we have time for questions or comments or discussion. Yes. Yeah, I'm just having general questions just out of curiosity. Do you know kind of the string theory literature in this direction? Because this is something which has been studied quite extensively where these type two degenerations are like stable degeneration limits and then you have a dual heterodic model there and you have to respect it. I know little bits of this, but probably not as much as I do. And I don't know what the necessarily, what the interpretation of this is going to actually be connected to this. I know, I mean, most of the things, so the challenge here, you know, there's a lot of things I can get into and sort of the challenge here is understanding the structure of the pseudolattices. So the challenge of what we've done here so far is understanding the pseudolattice structure on these where these YIs, the fibers are not necessarily all I want. So the fibers don't necessarily satisfy this thing where they're like simple pinch tori. And I haven't, that's something that I haven't really seen much in the literature anywhere. In most cases, people seem to, it's sort of, you know, a lot of the literature, people seem to consider vibrations where they say, okay, you know, we assume that all of our singular fibers are like simple, more singularities. And then we do some protective geometry kind of style stuff. Yeah, yeah, yeah.  So we started off with this with, we started off looking at string junctions and there's the junction pairing in the string theory literature. And the junction pairing works out to be precisely the symmetrization or it's either the symmetrization or the anti-symmetrization of the pairing that we've been looking. So this, there is a relationship there, yeah. So, and I'm now trying to, we ran into some problems with that but it was about eight years ago and so I've forgotten what the problem's done. Yeah, I mean, we have, there have been sort of, there have been relationships that we've observed here. So, you know, there's sort of some old work by, what am I saying, old, like 10 year old or so worked by sort of Grassy and Halverson and Shanison, where they study these kinds of vibrations and they get, and also like the Wolframs, why that? And they get these kinds of decompositions and we show that the same decompositions hold on the pseudolattice level, which is sort of, you know, you can deduce their results from ours but the other way, as best I know, doesn't hold. We've got, you know, they've got these sort of symmetrized pairings and ours has got this non-symmetry which makes it more, but I guess ours is our picture is more natural from a homological mirror symmetry standpoint. So this is all kind of, you know, pairings on numerical growth and deep groups of, you know, derived castings of Coherent sheaves or focaya castings depending on the size of it. But if you do want to see a tie-in to the heterotic type two and F3, Mohsen, my postdoc is going to be talking about that in this lecture, that we're turning it around using this to perform the other dualities. Yes. In the situation when you don't have maximally unipotent monodromes, is there any version of SYZ mirrors, symmetric conjecture or anything like that? I'm not sure. I mean, there is some maximally unipotent degeneracy hiding here. Okay. And the place that it's hiding is that, you know, there's some technical assumptions on your type two degeneration has to be, has to occur along a locus, which is incident to a point of type green, like maximum degeneracy. So you can, I mean, the idea is to try and sort of interpret, make some mirror symmetry interpretation statements, which work sort of away from, you know, allow you to move away from smooth lobes, talk about the singular things.  So, I mean, these are to some extent an SYZ type object. And I think the predictions of SYZ effectively boil down to kind of, I think there's some statements you can deduce from SYZ, which give you kind of some mirror symmetry between these two exact sequences I've got. But we're hoping to go further and do something really with the lattice polarization and that's sort of where this is heading. But I'm not sure SYZ has a lot to say in that sort of, but yeah, you can, you can to some extent see this as a duality between, I think if you're looking at sort of generic K-free surfaces with a sort of generic factor degeneration, you can see this as some sort of duality between, you know, like a symplectic vibration and now it's a great vibration. I don't see any more questions. Oh, you're fine. Okay, fine, thank you. I was going to check it, but I won't now. Ha, ha, ha. You're done. That was it. Okay, thank you. So let's think, I'll speak to you one more time. Maybe I'll go through to stop the video and start.


Blood sugar, also known as blood glucose, is the universal go-to fuel for the cells throughout our bodies. Our brain burns through a quarter pound of sugar a day. It's preferred metabolic fuel. Our body can break down proteins and make glucose from scratch, but most comes from our diet in the form of sugars and starches. If we stop eating carbohydrates or stop eating altogether, most of our cells switch over to burning fat. Fat has difficulty getting through the blood-brain barrier, but our brain has this constant mass of need for fuel, one organ accounting for up to half of our energy needs. Without it, the lights go out permanently. To make that much sugar from scratch, our body would need to break down about a half pound of protein a day. That means we'd cannibalize ourselves to death within two weeks, and people can fast for months. The answer to the puzzle was discovered in 1967. Harvard researchers famously stuck catheters into the brains of obese subjects who had been fasting for over a month, and discovered that ketones had replaced glucose as the preferred fuel for the brain.  Switching fuels has such an effect on brain activity that it has been used to treat epilepsy since antiquity. The prescription of fasting for the treatment of epileptic seizures dates back to Hippocrates. In the Bible, Jesus seems to have concurred. To this day, it's unclear why switching from blood sugar to ketones as a primary fuel source has such a dampening effect on brain overactivity. How long can you fast, though, to prolong the fasting therapy in 1921? A distinguished physician scientist at the Mayo Clinic suggested trying what he called a ketogenic diet, a high-fat diet designed to be so deficient in carbohydrates it could effectively mimic the fasting state. Remarkable improvement was noted the first time it was put to the test, efficacy that was later confirmed in randomized, controlled trials. Ketogenic diets started to fall out of favor in 1938 with the discovery of the anti-seizure drug, which would become known as Dylantin, but ketogenic diets are still in use today as a third or fourth-line treatment for drug refractory epilepsy in children. Oddly, the success of ketogenic diets against pediatric epilepsy seems to get conflated by keto diet proponents into suggesting a ketogenic diet is beneficial for everyone. But you know what else sometimes works for intractable epilepsy? Brain surgery. But I don't hear people at the gym clamoring to get their skulls sought open, since when do medical therapies translate into healthy lifestyle choices? Scrambling brain activity with electroshock therapy can be helpful in some cases of major depression. So what passed the electrodes? Ketogenic diets are also being tested to see if they can slow the growth of certain brain tumors. Even if it works, you know what else can help slow cancer growth? Chemotherapy. So why go keto when you can just go chemo? The use of ketogenic diets for cancer paid for by so-called ketone technology companies that will send you salted caramel bone broth powder for 100 bucks a pound, or companies that market ketogenic meals report extraordinary anecdotal responses in some cancer patients, but more concrete evidence is simply lacking. Even the theoretical underpinnings may be questionable. You know, a common refrain is that cancer feeds on sugar. But all cells feed on sugar. Advocating ketogenic diets for cancer is like saying Hitler breathed air, so let's boycott oxygen. Cancer can feed on ketones, too. Ketones have been found to fuel human breast cancer growth and drive metastases in an experimental model more than doubling tumor growth. Some have even speculated that may be why breast cancer often metastasizes to the liver, the main site of ketone production.  Researchers are even considering designing ketone-blocking drugs to prevent further cancer growth by halting ketone production. And think about what eating a ketogenic diet might entail— high animal fat intake. May increase the mortality risk among breast cancer survivors and potentially play a role in its development in the first place, through oxidative stress, hormone disruption, or inflammation. Men, too. A strong association has been found between saturated fat intake and prostate cancer progression. Those in the top third of consumption of these kinds of fat-rich animal foods appeared to triple their risk of dying from prostate cancer. Not necessarily fat in general.  There's a reason the official American Cancer Society and American Society of Clinical Oncology Breast Cancer Survivorship Care Guidelines recommend a dietary pattern for breast cancer patients that's essentially the opposite of a ketogenic diet. High in vegetables, fruits, whole grains, and legumes, meaning in bean, split peas, chickpeas, and lentils, and low in saturated fats. So far, not a single clinical study has shown a measurable benefit from a ketogenic diet for any human cancer. There are currently at least a dozen trials underway, however, and the hope is that at least some cancer types will respond. Still, even then, that wouldn't serve as a basis for recommending ketogenic diets for the general population, any more than recommending everyone go out and get radiation surgery and chemo for kicks.