- 1:00pm - 2:00pm

**Speaker:** Ljudmila Kamenova**Title:** Non-hyperbolicity of hyperkahler manifolds and Kobayashi's conjectures**Abstract:** The Kobayashi pseudometric $d_M$ on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi conjectured that this pseudometric vanishes on Calabi-Yau manifolds, and in particular, Calabi-Yau manifolds have "entire curves". Using ergodicity of complex structures, together with S. Lu and M. Verbitsky we prove this conjecture for all K3 surfaces and for many classes of hyperkahler manifolds. The next step of the project is to generalize these results to singular hyperkahler varieties.

- 7:30pm - 10:10pm
- in online

**Title:** TBA **Speaker:** Bowen, Zhang [Stony Brook University] **Abstract:**

View Details

- 2:30pm - 3:30pm

**Paolo Muratore-Ginanneschi**, University of Helsinki

- 2:30pm - 3:30pm
- in zoom

**Title:** Statistical conservation laws, scaling and renormalisation in the Kraichnan model of passive advection. **Speaker:** Paolo Muratore-Ginanneschi [University of Helsinki] **Abstract:** A complete understanding of the link between the dynamics and the statistics of indicators of a turbulent Newtonian fluid remains a grand open challenge in theoretical and mathematical physics. In particular, a theory predicting in a mathematically controlled fashion experimentally and numerically observed scaling properties of indicators of the flow is still missing.

During the last decade of last century and the first decade of the present one, the study of the Kraichnan model of passive advection made possible to obtain a mathematically controlled derivation of multiscaling for the first time in turbulence theory. The proof of multiscaling highlighted the existence of a general

relation between the statistical conservation laws satisfied by Lagrangian particle shapes contributing to the correlation functions and the occurrence anomalous scaling.

In the inertial range the statistically conserved structures can be also described as composite operators perturbing a scaling fixed point of a Wilsonian renormalization map.

In my talk I will give an overview on these developments with emphasis on how renormalization group ideas have been applied in the context of studying intermittency in the Kraichnan model.

View Details

- 4:30pm - 5:30pm
- in Online

**Title:** Special Colloquium: A Personal Tribute to Louis Nirenberg **Speaker:** Joel Spruck [Johns Hopkins University] **Abstract:** I first met Louis Nirenberg in person in 1972 when I became a Courant Instructor. He was already a celebrated mathematician and a suave sophisticated New Yorker, even though he was born in Hamilton, Canada and grew up in Montreal. In this talk I will describe some of his famous papers, some of our joint work and other work he inspired. For reasons of exposition, I will not follow a strict chronological order and I will concentrate on some of Louis' work inspired by geometric problems beginning around 1974, especially the method of moving planes and implicit fully nonlinear elliptic equations. During the twenty year period 1953-1973 he produced an incredible body of work in many fields of pde including pseudo differential operators and local solvability. Of course I cannot begin to talk about this work. Louis loved to collaborate and I apologize for omitting many other important results of the last twenty five years, a majority of which were in collaboration with his brilliant and devoted student Yanyan Li.

This is the opening talk of the 35th Annual Geometry Festival.

http://www.math.stonybrook.edu/geomfest21/

View Details

- 4:30pm - 5:30pm

35th Annual Geometry Festival

Abstract

- 9:00am - 10:00am
- in Online

**Title:** Deformation Quantization, and Obstructions to the Existence of Closed Star Products **Speaker:** Akito Futaki [Yau Center, Tsinghua University] **Abstract:** A star product is a non-commutative product on the set of formal functions, i.e. formal power series with coefficients in smooth functions. Giving a star product is called deformation quantization. The trace of a star product is an algebra character from the non-commutative algebra of formal functions into the abelian algebra of formal constants. The trace is expressed as an $L^2$ product with a function called the trace density. A star product is said to be closed if the trace density is constant, i.e. the trace is given by the integration. In this talk, we discuss on obstructions to the existence of closed star product as in the similar spirit of Kähler geometry.

http://www.math.stonybrook.edu/geomfest/

View Details

- 9:00am - 10:00am

35th Annual Geometry Festival

Abstract

- 11:00am - 12:00pm
- in Online

**Title:** Holomorphic Morse inequalities, old and new **Speaker:** Jean-Pierre Demailly [Institut Fourier, Universite Grenoble Alpes] **Abstract:** We will review some recent developments around holomorphic Morse inequalities, as well as geometric applications and unsolved problems.

http://www.math.stonybrook.edu/geomfest/

View Details

- 11:00am - 12:00pm

35th Annual Geometry Festival

Abstract

- 1:00pm - 2:00pm
- in Online

**Title:** SYZ Mirror Symmetry for del Pezzo Surfaces **Speaker:** Tristan Collins [MIT] **Abstract:** If X is a del Pezzo surface and D is a smooth anti-canonical divisor, we can regard the complement X\D as a non-compact Calabi-Yau surface. I will discuss a proof of a strong form of the Strominger-Yau-Zaslow mirror symmetry conjecture for these non-compact surfaces. It turns out the mirror Calabi-Yau is a rational elliptic surface (in particular, it has an elliptic fibration onto $P^1$) with a singular fiber which is a chain of nodal spheres. I will discuss how we can construct special Lagrangian fibrations on these manifolds, as well as moduli of complex and symplectic structures and how hyper-Kahler rotation allows us to construct an identification of these moduli spaces. This is joint work with A. Jacob and Y.-S. Lin.

http://www.math.stonybrook.edu/geomfest/

View Details

- 1:00pm - 2:00pm

35th Annual Geometry Festival

Abstract

- 3:00pm - 4:00pm
- in Online

**Title:** Some Recent Results on Ricci Flow **Speaker:** Jim Isenberg [University of Oregon] **Abstract:** We discuss the results of two recent collaborative works on Ricci Flow. The first of these results, done with Eric Bahuaud and Chris Guenther, shows that 'convergence stability' holds for Ricci Flow solutions converging to the flat metric on the torus as well as for Ricci Flow solutions converging to the hyperbolic metric. Convergence stability tells us that if the Ricci flow starting at a metric h converges to a metric g, then it follows that the Ricci Flow starting at metrics sufficiently close to h (relative to a specified topology) must also converge to g. Convergence stability is a consequence of stability at g combined with long-time continuous dependence for the class of geometries including h and g. Our verification that convergence stability holds for the hyperbolic metric depends on geometric analysis results for asymptotically hyperbolic metrics contained in a specified weighted Holder space. The second of our results, done with Tim Carson, Dan Knopf and Natasa Sessum, involves the study of singularity formation at spatial infinity for Ricci Flow of certain multi-warped complete geometries on non-compact manifolds. We use this analysis to show that there is unexpected behavior of blowup sequences for Ricci Flows developing Type I singularities at spatial infinity. In particular, we find that some blowup sequences form gradient solitons, while others form ancient solutions which are not solitons.

http://www.math.stonybrook.edu/geomfest/

View Details

- 3:00pm - 4:00pm

35th Annual Geometry Festival

Abstract

- 5:00pm - 6:00pm
- in Online

**Title:** Topological Recursion and Crepant Transformation Conjecture **Speaker:** Chiu-Chu Melissa Liu [Columbia University] **Abstract:** The Crepant Transformation Conjecture (CTC), first proposed by Yongbin Ruan and later refined/generalized by others, relates Gromov-Witten (GW) invariants of K-equivalent smooth varieties/orbifolds. The Remodeling Conjecture (proposed by Bouchard-Klemm-Marino-Pasquetti and proved in full generality by Fang, Zong and the speaker) relates open and closed GW invariants of a symplectic toric Calabi-Yau 3-orbifold to invariants of its mirror curve defined by Chekhov-Eynard-Orantin Topological Recursion. We will explain how to use the Remodeling Conjecture to derive all genus open and closed CTC for symplectic toric Calabi-Yau 3-orbifolds. This is based on joint work with Bohan Fang, Song Yu, and Zhengyu Zong.

http://www.math.stonybrook.edu/geomfest/

View Details

- 5:00pm - 6:00pm

35th Annual Geometry Festival

Abstract

- 9:00am - 10:00am
- in Online

**Title:** Local entropy along the Ricci flow **Speaker:** Bing Wang [University of Science and Technology of China] **Abstract:** Inspired by the Li-Yau estimates, we localize the entropy functionals of G. Perelman, and generalize his no-local-collapsing theorem and pseudo-locality theorems. The improved no-local-collapsing theorem can be used to study the general Kahler-Ricci flow. The improved pseudo-locality theorem can be used to show the continuous dependence, with respect to the initial metric in the Gromov-Hausdorff topology, of the Ricci flow on manifolds satisfying a lower Ricci-curvature bound; and to prove the compactness for the moduli of Kahler manifolds with bounded scalar curvature that satisfy a rough locally-almost-Euclidean condition.

http://www.math.stonybrook.edu/geomfest/

View Details

- 9:00am - 10:00am

35th Annual Geometry Festival

Abstract

- 11:00am - 12:00pm
- in Online

**Title:** Some boundary value and mapping problems for differential forms **Speaker:** Simon Donaldson [SCGP, Stony Brook University, and Imperial College, London] **Abstract:** Hitchin formulated the equations for $G_{2}$ holonomy in seven dimensions and Calabi-Yau structures in six dimensions in terms of variational problems for closed 3-forms. We will discuss versions of these ideas for manifolds with boundary. In the second case this leads to a mapping problem for maps from a five dimensional manifold to ${\bf C}^{3}$ which is related to CR geometry, contact geometry and four-dimensional Riemannian geometry and has a dimension reduction to the classical Minkowski problem for convex surfaces in 3-space.

http://www.math.stonybrook.edu/geomfest/

View Details

- 11:00am - 12:00pm

35th Annual Geometry Festival

Abstract

- 4:30pm - 5:30pm
- in Online

**Title:** Singularity formation in black Hole interiors **Speaker:** Spyros Alexakis [University of Toronto] **Abstract:** Starting from classical examples of singularity formation inside

black holes, I will recall the strong cosmic censorship conjecture of

Penrose regarding question. I will also review some further predictions

and known results on the generic behavior of the space-time metric as it

terminates at a singularity; these results will be compared with the

complementary picture on initial, big-bang type singularities. The main

new result we will present is a recent proof of the perturbative stability

of the Schwarzschild singularity in vacuum, under polarized perturbations

of the initial data. The singularity that then forms is again of

space-like character, and the solution displays asymptotically-velocity-term-dominated

behavior upon approach to the singularity. Joint with G. Fournodavlos.

View Details

- 3:00pm - 4:15pm
- in Online

**Title:** TBA **Speaker:** Qizheng Yin [Beijing International Center for Mathematical Research] **Abstract:** TBA

View Details

- 1:00pm - 2:00pm

**Speaker:** Yu Li

- 4:30pm - 5:30pm
- in Online

**Title:** Unipotent flows on hyperbolic manifolds, \'a la Ratner **Speaker:** Hee Oh [Yale University] **Abstract:** The celebrated Ratner's orbit closure theorem proved around 1990 says that in a homogeneous space of finite volume, the closure of an orbit of any subgroup generated by unipotent flows is homogeneous. A special case of Ratner's theorem (also proved by Shah independently) implies that the closure of a geodesic plane in a hyperbolic manifold of finite volume is always a properly immersed submanifold. Searching for analogs of Ratner's theorem in the infinite volume setting is a major challenge. We present a continuous family of hyperbolic 3-manifolds, and a countable family of higher dimensional hyperbolic manifolds of infinite volume, for which we have an analogue of Ratner's theorem.

(Based on joint work with McMullen, Mohammadi, Benoist and Lee in different parts.)

View Details