Lectures in Knot Theory: An Exploration of Contemporary Topics

Jozef H. Przytycki (1953-) is Professor of Mathematics at the George Washington University and summer visiting Professor at Gdansk University.

A distinguished mathematician, he received the Kazimierz Kuratowski Prize (1982), the Trachenberg prize (2010) and OVPR Distinguished Researcher Award (GWU). Przytycki received his Ph.D. from Columbia University in 1981, and wrote his dissertation on incompressible surfaces in 3-manifoldsunder Joan S. Birman. He received his Master degree in Mathematics in 1977 from Warsaw University, in his native Poland. He was a postdoctoral fellow with Dale Rolfsen at the University of British Columbia, Kunio Murasugi at the University of Toronto and Vaughan Jones at University of California, Berkeley. He also spent a semester as a member at Institute for Advanced Study, 1990. In 1995 he started working at George Washington University. He obtained his Habilitation at Warsaw University, in December 1994 and Presidential Professorship in Poland in 2013. Over the years he has traveled throughout the word and spent time as a visiting professor in many fine universities (e.g. Berkeley, Toronto, Vancouver).

His research includes, classical knot theory, topology and geometry of 3- manifolds, algebraic topology based on knots, skein modules and algebras, homology theories motivated by knot theory, to name several. Przytycki has been invited to international conferences, and has served in a variety of capacities: as a speaker, co-organizer, and member of scientific committees. He is the co-organizer of a series of conferences: Knots in Washington (started in 1995) as well as Knots in Poland (I, II, III) and Knots in Hellas conferences (I and II). He has served as editor of several journals such as Fundamenta Mathematicae, journal of Knot Theory and Its Ramification, and Involve. Every December he organizes an intensive workshop for his students, "Mathathon", where some open problem is presented and usually at least partially solved.

Till date he has supervised sixteen PhD students.

Rhea Palak Bakshi was born and brought up in India. She did most of her schooling at Welham Girls' School in Dehradun. She received her B.Sc. Honors in Mathematics in 2014 from Lady Sri Ram College for Women at the University of Delhi and her M.Sc. in Mathematics in 2016 from the University of Mumbai. Thereafter, she moved to the United States and got a Ph.D. in Mathematics in 2021 from the George Washington University. Her research interests lie at the confluence of low-dimensional topology, quantum topology, and knot theory. She is particularly interested in the theory of skein modules and algebras, various related conjectures such as the volume conjecture and the AJ conjecture, TQFTs, Khovanov homology, and categorification. She is also the coauthor of two chapters on skein modules and algebras in the Encyclopedia of Knot Theory. She currently works as a Research Fellow at the Institute for Theoretical Studies at ETH Zürich in Switzerland.

Dionne Ibarra is a Research Fellow at Monash University Clayton campus, Australia. She was born in Fresno, California USA and obtained her BA in 2010 and MA in 2012 from California State University, Fresno under the supervision of Professor Carmen Caprau.

In the winter semester of 2020 she was a Junior fellow at Institut Mittag-Leffler in Djursholm, Sweden. In 2022, she received a PhD in Mathematics from the George Washington University and wrote her dissertation on framed links in 3-manifolds and algebraic approaches to knot theory under the supervision of Professor Jozef H. Przytycki. Her main research area is low-dimensional topology, knot theory, diagrammatic algebras, and quantum invariants of links and 3-manifolds. She has also conducted research in interpolation methods using unit quaternions through the National Science Foundation (NSF) Mathematical Sciences Graduate Internship (MSGI) Program.

Gabriel Montoya-Vega was born in Barranquilla, Colombia where in 2015 he obtained a BS in Mathematics from the Universidad del Atlántico. In 2017 he completed a MS in Mathematics at the University of Puerto Rico-Mayaguez. He received a PhD in Mathematics in 2022 from the George Washington University. His main research area is knot theory, including invariants of links such as Fox colorings and Khovanov homology, and the history of knot theory.

After completing his PhD, he was awarded a National Science Foundation Mathematical and Physical Sciences Ascend Postdoctoral Research Fellowship affiliated to the City University of New York Graduate Center and to the University of Puerto Rico at Río Piedras.

Jozef H. Przytycki（1953年-）是喬治華盛頓大學的數學教授，也是格但斯克大學的夏季訪問教授。作為一位傑出的數學家，他獲得了卡齊米日·庫拉托夫斯基獎（1982年）、特拉琴伯格獎（2010年）和OVPR傑出研究者獎（GWU）。Przytycki於1981年從哥倫比亞大學獲得博士學位，並在Joan S. Birman的指導下撰寫了關於三維流形中不可壓縮曲面的論文。他於1977年在他的祖國波蘭的華沙大學獲得數學碩士學位。他曾在不列顛哥倫比亞大學的Dale Rolfsen，多倫多大學的Kunio Murasugi和加利福尼亞大學伯克利分校的Vaughan Jones擔任博士後研究員。他還在1990年作為高等研究院的成員度過了一個學期。1995年，他開始在喬治華盛頓大學工作。他於1994年12月在華沙大學獲得了他的博士後資格，並於2013年在波蘭獲得了總統教授職位。多年來，他遊歷世界各地，並在許多優秀的大學擔任訪問教授（例如伯克利，多倫多，溫哥華）。

他的研究包括古典結疤理論，三維流形的拓撲和幾何學，基於結疤、結疤模塊和代數的代數拓撲，以結疤理論為基礎的同調理論等等。Przytycki受邀參加國際會議，並擔任各種職務：演講者，共同組織者和科學委員會成員。他是一系列會議的共同組織者：華盛頓結疤（始於1995年），以及波蘭結疤（I，II，III）和希臘結疤（I和II）會議。他曾擔任多個期刊的編輯，如《基礎數學》、《結疤理論及其分支》和《參與》。每年12月，他都會為他的學生組織一個密集的研討會“Mathathon”，在這個研討會上，會提出一些開放問題，通常至少部分解決。

迄今為止，他指導了十六位博士生。

Rhea Palak Bakshi在印度出生和長大。她在德拉敦的韋爾漢女子學校完成了大部分學業。她於2014年從德里大學的女士斯里蘭姆學院獲得數學榮譽學士學位，並於2016年從孟買大學獲得數學碩士學位。之後，她移居美國，並於2021年從喬治華盛頓大學獲得數學博士學位。她的研究興趣涉及低維拓撲學、量子拓撲學和結疤理論的交叉領域。她對結疤模塊和代數的理論，以及與之相關的猜想（如體積猜想和AJ猜想）、TQFT、Khovanov同調和分類化感興趣。她還是《結疤理論百科全書》中關於結疤模塊和代數的兩章的合著者。她目前在瑞士蘇黎世聯邦理工學院的理論研究所擔任研究員。

Dionne Ibarra是澳大利亞莫納什大學克萊頓校區的研究員。她出生於美國加利福尼亞州的弗雷斯諾，於2010年獲得學士學位，並於2012年獲得加利福尼亞州立大學弗雷斯諾分校的碩士學位，指導教授為Carmen Caprau。

在2020年冬季學期，她在瑞典杜爾舒姆的米塔格-勒夫勒研究所擔任初級研究員。2022年，她從喬治華盛頓大學獲得數學博士學位，並在Jozef H. Przytycki教授的指導下撰寫了關於三維流形中的框架鏈和結疤理論的代數方法的論文。她的主要研究領域是低維拓撲學、結疤理論、圖解代數和結疤和三維流形的量子不變量。她還通過國家科學基金會（NSF）數學科學研究生實習（MSGI）計劃進行了使用單位四元數的插值方法的研究。

Gabriel M