I am a regular organizer of seminars, workshops, conferences, and other mathematical gatherings. Below you can find information on the events, past, current, and future.
A joint research seminar of the Algebraic Geometry, Commutative Algebra, and Number Theory groups at the University of South Carolina featuring both external and internal speakers. If you are interested in speaking, please contact any of us. Held at the University of South Carolina.
As part of the 2020 Canadian Mathematical Society Winter Meeting in Montreal, there will be a special session on derived categories and (non)commutative algebraic geometry. This event will connect a researchers in Canada with the broader world community to exchange ideas and drive progress. The event has moved to a virtual format and scheduling is still in progress.
Birational geometry is a classical mathematic subject dating back to the late 1800s. It aims to classify geometric shapes by looking at the majority of their points and asks "if I remove some points from shape A and some other points from shape B, do they become the same shape?". This turns out to be a surprisingly difficult and fundamental mathematical question. Perhaps more surprising is that the classification of geometric shapes from birational geometry is related to the classification of abstract mathematical gadgets called derived categories, a formal language developed by prominent mathematicians in France in the 1960s. However, the relationship between derived categories and birational geometry remains unproven.
One avenue to pursue this connection is through a modern mathematical subject known as "derived algebraic geometry". This workshop brings together experts in birational geometry, derived categories, and derived algebraic geometry in an effort to connect the disciplines and more thoroughly understand this deep mathematical phenomenon.
Do not despair. The in-person workshop scheduled still exists! The new dates will be set soon.
This special session, to the upcoming Mathematical Congress of the Americas, will bring together leading experts in the Americas and their collaborators to discuss the state of the art of tropical and toric geometry from the perspective of mathematical physics (integrable systems, string theory, mirror symmetry, and related areas).
Determining whether a given field extension is purely transcendental, also known as the Lüroth problem, is one of the most basic questions in mathematics. In the geometric context, the rationality of an algebraic variety is a central question in algebraic geometry. In recent decades, the derived category of coherent sheaves has been developed into a robust tool for subtly encoding geometric information. Once viewed as a book-keeping tool for homological invariants, the study of derived categories has connected algebraic geometry to far-flung areas, including noncommutative algebra, symplectic geometry, string theory, higher category theory, and homotopy theory, in deep and interesting ways. The structure of the derived category of coherent sheaves on a variety X is intimately connected with questions of rationality of X. The momentum behind this topic and the remaining unexplored territory make it a wonderful time to bring together experts in both areas to share results and pollinate future advances.
Organized with Emanuele Macrì (Paris-Saclay) and Patrick McFaddin (Fordam). Held at Université de Grenoble-Alpes, Grenoble, France during a joint meeting of the American Mathematical Society, Société Mathématique de France, and European Mathematical Society.
A colloquium-style seminar series I created in 2014 to provide an opportunity for graduate students to their current ideas with their colleagues. Interspersed within are talks and panels focused on career development. Topics have included: picking a major advisor, applying for jobs, and navigating the graduate program. While I am no longer the faculty advisor, the Graduate Colloquium is alive and strong. Held at the University of South Carolina.
Mirror Symmetry is a phenomenon first observed to occur in High-energy Theoretical Physics. Since the early 90s, mirror symmetry has been used to solve difficult mathematical problems known as curve-counting or enumerative questions in Algebraic Geometry. In his 1994 ICM address, Kontsevich proposed that mediating the rich geometry of Mirror Symmetry was abstract foundational mathematics which amounts to a categorical equivalence. From this sprang the conjecture of Homological Mirror Symmetry (HMS). HMS views two very different areas of mathematics, Symplectic Geometry and Algebraic Geometry, as opposite sides of the same categorical coin. Today, HMS is the cornerstone of an extremely active research field, reaching in influence far beyond its original formulation as a duality between Calabi-Yau manifolds, to such subjects as representation theory, singularity theory, and knot theory. At the core of the HMS research community are three different groups of researchers: algebraic geometers, symplectic geometers, and mathematical physicists. This thematic period will focus on teaching young researchers all aspects of the field. This 6-month thematic period will consist of algebraic geometers, symplectic geometers, and mathematical physicists from around the world focusing on the most modern aspects of HMS with an emphasis on engagement with students and young researchers.
There were six workshops and conferences
Organized with Mohammed Abouzaid (Columbia), Kevin Costello (Perimeter), David Favero (Alberta), Ludmil Katzarkov (Miami/Vienna), Ailsa Keating (Cambridge), Tony Pantev (Penn), Johannes Walcher (Heidelberg), and Ursula Whitcher (AMS/Michigan). Held at the Fields Institute. Partial funding provided by the Fields Insitute and the National Science Foundation.
In 1992 during his Fields medal address to the International Congress of Mathematicians, Maxim Kontsevich proposed that derived categories in Algebraic Geometry could take on a role as boundary conditions for vibrating strings (D-branes) - the smallest units of space and time in String Theory. This is now the ansatz for Kontsevich's Noncommutative Geometry, a prominent field of study in modern Algebraic Geometry and Mathematical-Physics. We brought together 17 early career researchers (graduate and postdoctoral) along with 1 mentor, Eric Sharpe (VaTech) for a Pacific Insitute of Mathematics summer school in Edmonton which studies derived categories both from the perspective of Algebraic Geometry and as D-branes in String Theory. Our aim was to generate bi-lingual researchers working at the interface of high-energy physics and algebraic geometry.
The notes from the Super School are available as a book. Organized with Charles Doran (Alberta & Harvard) and David Favero (Alberta). Held at the University of Alberta. Partial funding provided by the Pacific Insitute for the Mathematics, the National Science Foundation, and the University of South Carolina.
Mirror symmetry, as originally conceived by physicists in the 80's, relates pairs of distinct (Calabi-Yau) manifolds by identifying the seemingly disparate features of the algebraic geometry of one with the symplectic geometry of the other. Since their introduction, conjectures stemming from mirror symmetry have been a continual source of fascination for mathematicians due to the indirect nature of the relationship: the precise underlying mathematical structures indentified under mirror symmetry are often of very different nature, supplying a deep, and often mysterious, duality.
Birational geometry, on the other hand, is one of the most classical areas of algebraic geometry, with foundations dating back at least to the late 1800's. Birational geometry aims to classify the fundamental objects of algebraic geometry by understanding their "generic" behavior. Guided over the decades by luminaries such as Zariski, Kodaira, and Hironaka, it has evolved into one of the richest theories in modern algebraic geometry. In recent years, efforts by several research groups have elucidated several concrete ways in which the birational theory of varieties can be interpreted in terms of mirror symmetry constructions, yielding intriguing new perspectives on classical geometric themes.
This workshop capitalized on the developing momentum which lies at the intersection of these two subjects. Organized with Colin Diemer (IHES) and David Favero (Alberta). Held at the Banff International Research Station.
This event was a Special Session at the American Mathematical Society Southeastern regional meeting. The focus was on Mirror Symmetry broadly. We had 15 speakers from across the USA.
Organized with David Favero (Alberta). Held at UNC-Greensboro.
Seven Commutative Algebraists and Algebraic Geometers have moved to the Southeast since 2011. The conference is designed to welcome these new hires, to foster collaborations between researchers at different institutions, and to encourage and nurture interaction between the Commutative Algebra and Algebraic Geometry communities.
The conferences involved 66 participants from across the Southeast and as far away as Japan. There were nine plenary lecture plus a poster session for early career researchers.
Over the last several years, a variety of new categorical structures have been discovered by physicists. Furthermore, it has become transparently evident that the higher categorical language is beautifully suited to describing cornerstone concepts in modern theoretical physics. The goal of this project was to develop these structures even further. As we head into the second decade of the 21st century, modern geometry and theoretical physics are more intertwined than ever before. The convergence of ideas from mathematics and physics is accelerating at the same time as elementary particle physics is on the cusp of a profound revolution to be brought about by the new experimental results coming out of the Large Hadron Collider (LHC). These will serve to identify among the multitude of theoretical possibilities currently open, which ones best address quantum field theory at the high energy scale. At the same time, a lot of mathematical work remains to be done to provide a suitable framework for the new physical theories that are being proposed. The geometric objects which we investigate today are the foundations for such a framework: homological mirror symmetry is the mathematical realization of dualities among sypersymmetric theories and higher categories are the mathematical foundation for quantum field theories. These new flavors of geometry, in which categorical structures play a primordial role, will certainly continue to play a fundamental role in the 21st century theoretical physics. Developing these new categorical structures was the main goal of the ESI activity Geometry of Topological D-Branes, Categories, and Applications.
plus a series of lectures by experts occurred during the thematic period. The event saw 77 invited scientists visit.
Organized with David Favero (Alberta), Sergei Gukov (Caltech), Anton Kapustin (Caltech), Ludmil Katzarkov (Vienna), Yan Soibelman (Kansas State). Held at the Erwin Schrodinger Institute. Partial funding provided by the ESI.