Xujia Chen
May 4, 2021

For symplectic 4-manifolds with a real structure, Welschinger (2003) showed that counts of real rational pseudo-holomorphic curves, with appropriate signs, are well-defined invariants. They are called Welschinger invariants and are analogues of Gromov-Witten invariants in the real setting. In 2007, Solomon proposed two WDVV-type relations for them, which determine these numbers recursively in many good cases. They are real analogues of the usual WDVV relation.

We establish Solomon’s WDVV-type relations by implementing Georgieva’s suggestion to lift homology relations from the Deligne-Mumford moduli spaces of stable real curves. This is accomplished by lifting judiciously chosen cobordisms realizing these relations. Our topological approach provides a general framework for lifting relations via morphisms between not necessarily orientable spaces.

April 23-25, 2021

Stony Brook University, Stony Brook, NY

Nathan Chen
April 16, 2021

A complex projective variety is called rational if there is a Zariski-open subset on which it is isomorphic to a Zariski-open subset of projective space. There has been a huge amount of progress and activity in determining when varieties are rational. One the other hand, one can ask: given a projective variety whose nonrationality is known, can we measure how far it is from being rational?

Measures of irrationality provide an answer to the question above; they are birational invariants that offer an orthogonal viewpoint to questions concerning rationality. They have recently gained interest, in part due to work of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery [BDELU] on hypersurfaces of large degree. In this dissertation, we make advances in the study of measures of irrationality on abelian surfaces and codimension two complete intersections, which answer a few questions posed in [BDELU].

Simons Lectures in Mathematics
Spring 2021
April 12-15, 2021

The last decade has witnessed tremendous progress in algebraic geometry in a p-adic setting: new tools have been introduced, unexpected connections between different areas of mathematics have emerged, and longstanding problems have been solved. In this lecture series, I'll survey some of the developments in this area.

More precisely, the first lecture will be an overview of some of the major results in the area. The second lecture will explain the origins (partially in homotopy theory, partially in the Langlands program) and applications of recently discovered p-adic cohomology theories. The final talk will be dedicated to progress on the p-adic Riemann-Hilbert problem and its implications for birational geometry.

Lecture 1: Overview
Monday, April 12, 2021, 4:30 – 5:30 pm, online

Lecture 2: Prismatic cohomology
Tuesday, April 13, 2021, 4:30 – 5:30 pm, online

Lecture 3: p-adic Riemann-Hilbert Correspondence
Thursday, April 15, 2021, 2:30 – 3:30 pm, online

Aleksandar Milivojevic
April 14, 2021

The homotopy theory of rationalized simply connected spaces was shown by Quillen to be encoded algebraically in differential graded Lie algebras in his seminal work on rational homotopy theory. Motivated by this theory and Whitney's treatment of differential forms on arbitrary complexes, Sullivan later described a theory of computable algebraic models for rational homotopy types in terms of differential graded algebras of differential forms in his "Infinitesimal Computations in Topology". Following a problem posed therein, we give a characterization of the simply connected rational homotopy types, together with a choice of rational Chern classes and fundamental class, that are realized by closed almost complex manifolds in complex dimensions three and greater, with a caveat in complex dimensions congruent to two modulo four depending on the first Chern class. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a closed almost complex manifold, of complex dimension not congruent to two modulo four, depends only on its cohomology ring.

Congratulations to our colleague John Milnor for receiving the 2020 Russian Academy of Sciences Lomonosov Gold Medal.

The Lomonosov Gold Medal named after Russian scientist and polymath Mikhail Lomonosov, is awarded each year since 1959 for outstanding achievements in the natural sciences and the humanities by the USSR Academy of Sciences and later the Russian Academy of Sciences (RAS). Since 1967, two medals are awarded annually: one to a Russian and one to a foreign scientist. It is the Academy's highest accolade.

Ruijie Yang
March 12, 2021

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules.

In this thesis defense, we would like to explain a simpler proof in the case of semisimple local systems, using a more geometric approach adapting de Cataldo- Migliorini. On the one hand, we complement Simpson’s theory of weights for local systems by proving a global invariant cycle theorem in the setting of local systems. On the other hand, we define a notion of polarization via Hermitian forms on pure twistor structures. This is partially based on joint work with Chuanhao Wei.

March 8-12, 2021

The goal of this Workshop is to explore connections between various aspects of Renormalization in Dynamics (unimodal and circle, holomorphic and cocyclic, Henon, KAM, and stochastic renormalizations) and Physics (QFT and statistical mechanics, fluid dynamics, and KPZ), which could help to reveal a unifying theme for all these phenomena.

This workshop is part of the Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory: February 3 – June 5, 2020. There will also be Renormalization retrospective: Feigenbaum Memorial Conference held right before (May 28-29, 2020).