**MATH-IMS Joint Colloquium Series (Pure Mathematics)**

**Organizers:**Yi-Jen Lee and Martin Li**Hosts:**Department of Mathematics & Institute of Mathematical Sciences, CUHK**Contact:**Please email to yjlee@math.cuhk.edu.hk or martinli@math.cuhk.edu.hk for any inquiries. The recording of the talks are available upon request.**Remark:**Due to current COVID-19 situation, we will be hosting the colloquium talks online via ZOOM until safe travels are permitted. You can access the talk via the ZOOM link below.

Upcoming talk

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2021-22 Spring schedule

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Jan 21, 2022 (Fri) 4:00PM Zoom link |
Speaker: Cedric Bonnafe (Université de Montpellier) Title: Singular K3 surfaces and complex reflection groups Abstract:
Joint work with A. Sarti. Singular K3 surfaces are the K3 surfaces with maximal Picard number, namely 20. I will explain how to construct families of K3 surfaces with big Picard number using invariants of finite complex reflection groups of rank 4, most family containing some singular ones. This extends earlier work of Barth-Sarti for two reasons: firstly, we obtain much more examples by considering all reflection groups of rank 4 and, secondly, our proofs involve more theory of complex reflection groups and avoids as much as possible (but not completely) a case-by-case analysis. |

Jan 28, 2022 (Fri) 10:00AM Zoom link |
Speaker: Jennifer Hom (Georgia Tech) Title: Unknotting number and satellites Abstract:
The unknotting number of a knot is the minimum number of crossing changes needed to untie the knot. It is one of the simplest knot invariants to define, yet remains notoriously difficult to compute. We will survey some basic knot theory invariants and constructions, including the satellite knot construction, a straightforward way to build new families of knots. We will give a lower bound on the unknotting number of certain satellites using knot Floer homology. This is joint work in progress with Tye Lidman and JungHwan Park. |

Feb 11, 2022 (Fri) 4:00PM Zoom link |
Speaker: Melanie Rupflin (University of Oxford) Title: Lojasiewicz estimates and applications to geometric flows Abstract:
Many interesting geometric objects are characterised as minimisers or critical points of natural geometric energies. To deform a given geometric object towards such an optimal state, it is hence natural to consider the corresponding gradient flow. In this talk we consider one of the most powerful tools that can be used to analyse gradient flows of analytic energies, so called Lojasiewicz estimates, and discuss in particular how such estimates can be obtained also in settings where singularities form and where the classical results on Lojasiewicz-Simon estimates hence do not apply. |

Feb 18, 2022 (Fri) 4:00PM Zoom link |
*** NOTE: This talk is a joint pure & applied mathematics colloquium series. *** Speaker: Geordie Williamson (University of Sydney) Title: Combinatorial invariance conjecture and machine learning Abstract:
The combinatorial invariance conjecture is a fascinating open problem on the border between combinatorics and representation theory. It predicts that certain quantities of central importance in representation theory (Kahzdan-Lusztig polynomials) can be read off a much more elementary object (a Bruhat interval, which is a directed graph). I have been fascinated by this problem since I learned about it, but it has always seemed out of reach. One major difficulty is that Bruhat intervals become very complicated very quickly, so the problem is difficult (for me) to visualize. I will discuss work with the London based AI lab DeepMind, where we used graph neural nets to try to predict the answer. Applying standard techniques ("saliency analysis") in machine learning led to the understanding that certain edges in the Bruhat graph are more important than others, which eventually led me to a conjecture which I hope will solve the conjecture for symmetric groups. |

Feb 25, 2022 (Fri) 11:30AM Zoom link |
Speaker: Pramod Achar (Louisiana State University) Title: The geometry of nilpotent orbits in representation theory Abstract:
An elementary fact from linear algebra is that the set of n x n nilpotent matrices (with entries in some field k) has finitely many conjugacy classes. The geometry of these conjugacy classes (also called nilpotent orbits) has played a major role in numerous results in representation theory for at least the past 50 years. I will briefly mention a few of these results, but I will focus primarily on new work relating nilpotent orbits to a special class of representations of GL_n(k) known as tilting modules. This talk is based on joint work with W. Hardesty and S. Riche. |

Mar 11, 2022 (Fri) 10:00AM Zoom link |
Speaker: Mohammed Abouzaid (Columbia University) Title: Homotopical methods in Floer theory Abstract:
The free loop space of a symplectic manifold is equipped with a canonical (multivalued) functional, which assigns to a 1-parameter family of loops the area of the cylinder that they sweep. Floer's insight that one can assign a homology group to this context by an appropriate reformulation of Morse theory, which led to a revolution in symplectic topology. Applied to toy examples, Floer's homology groups agree with ordinary homology. I will discuss an extension of Floer's idea to generalised cohomology theories. This was first envisioned by Floer himself, but the area of symplectic topology has finally reached the stage where we have concrete applications, which I will describe. |

Mar 25, 2022 (Fri) 4:00PM Zoom link |
Speaker: Penka Georgieva (Sorbonne University) Title: Real Gromov-Witten theory Abstract:
For a symplectic manifold with an anti-symplectic involution one can consider J-holomorphic maps invariant under the involution. These maps give rise to real Gromov-Witten invariants and are related to real enumerative geometry in the same spirit as their more classical counterparts; in physics they are related to orientifold theories. In this talk I will give an overview of the developments in real Gromov-Witten theory and discuss some properties of the invariants. |

Apr 8, 2022 (Fri) 10:00AM Zoom link |
Speaker: Zoltan Szabo (Princeton University) Title: Knot Floer homology constructions and the Pong Algebra Abstract:
In a joint work with Peter Ozsvath we have developed algebraic invariants for knots using a family of bordered knot algebras. The goal of this lecture is to review these constructions and discuss some of the latest developments. |

Jun 2, 2022 (Thur) 4:00PM Zoom link |
Speaker: Thomas Kragh (Uppsala University) Title: Twisted generating functions Abstract:
The nearby Lagrangian conjecture concerns the space of embedded smooth Lagrangians $C^0$-close to another given Lagrangian in a symplectic manifold. Indeed, the conjecture states that the space is connected (a stronger version states that it is locally contractible). The most concise and general statement we know about this conjecture is that if L is $C^0$ close enough to another Lagrangian K then they are simple homotopy equivalent. Generating functions has previously been employed to understand these, and in this talk I will outline the ideas in an extension of this method called twisted generating functions. I will also explain a bit about how this can be used to prove new results regarding the two tangent spaces of the Lagrangians. |

**Past Colloquium**