Calendar

TITLE: Hodge structures and the topology of algebraic varieties

ABSTRACT: We review the major progress made since the 50’s in our understanding of the topology of complex algebraic varieties. Most of the results  we will discuss  rely on Hodge theory, which  has some analytic aspects giving the Hodge and Lefschetz decompositions, and the Hodge-Riemann relations. We will see that a crucial ingredient, the existence of a polarization,  is missing in the general Kaehler context.
We will also discuss some results and problems related to algebraic cycles and motives.

Talk chair: Joe Harris

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

We discover a wide range of new nonperturbative effects in quantum gravity, namely moduli spaces of constrained instantons of the Einstein-Hilbert action.  We find these instantons in all spacetime dimensions, for AdS and dS.  Many can be written in closed form and are quadratically stable.  In 3D AdS, where the full gravitational path integral is more tractable, we study constrained instantons corresponding to Euclidean wormholes.  We show that they encode the energy level statistics of microstates of BTZ black holes, which precisely agrees with a quantitative prediction from random matrix theory.

Zoom: https://harvard.zoom.us/j/977347126

TITLE: Immersions of manifolds and homotopy theory

ABSTRACT: The interface between the study of the topology of differentiable manifolds and algebraic topology has been one of the richest areas of work in topology since the 1950’s. In this talk I will focus on one aspect of that interface: the problem of studying embeddings and immersions of manifolds using homotopy theoretic techniques. I will discuss the history of this problem, going back to the pioneering work of Whitney, Thom, Pontrjagin, Wu, Smale, Hirsch, and others. I will discuss the historical applications of this homotopy theoretic perspective, going back to Smale’s eversion of the 2-sphere in 3-space. I will then focus on the problems of finding the smallest dimension Euclidean space into which every n-manifold embeds or immerses. The embedding question is still very much unsolved, and the immersion question was solved in the 1980’s. I will discuss the homotopy theoretic techniques involved in the solution of this problem, and contributions in the 60’s, 70’s and 80’s of Massey, Brown, Peterson, and myself. I will also discuss questions regarding the best embedding and immersion dimensions of specific manifolds, such has projective spaces. Finally, I will end by discussing more modern approaches to studying spaces of embeddings due to Goodwillie, Weiss, and others. This talk will be geared toward a general mathematical audience.

Talk chair: Michael Hopkins

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

$\ell$-torsion conjecture states that $\ell$-torsion of the class group $|\text{Cl}_K[\ell]|$ for every number field $K$ is bounded by $\text{Disc}(K)^{\epsilon}$. It follows from a classical result of Brauer-Siegel, or even earlier result of Minkowski that the class number $|\text{Cl}_K|$ of a number field $K$ are always bounded by $\text{Disc}(K)^{1/2+\epsilon}$, therefore we obtain a trivial bound $\text{Disc}(K)^{1/2+\epsilon}$ on $|\text{Cl}_K[\ell]|$. We will talk about results on this conjecture, and recent works on breaking the trivial bound for $\ell$-torsion of class groups in some cases based on a work of Ellenberg-Venkatesh.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

The geometry of rhombic tilings and tessellations like the Penrose tiling have captivated mathematicians and artists alike.  Hidden in the geometry of certain rhombic tilings of certain polygons, though, is an unexpected combinatorial structure that not only lends itself to some combinatorial objects, but also is often rather useful for their enumeration.  In this talk, we will highlight the connection between one type of rhombic tiling and the world of plane partitions, monotone discrete functions, and stacks of cubes; and another type of rhombic tiling to the world of permutations, Coxeter groups, and reduced words.

Come learn some neat mathematics connecting permutations, polygons, geometry, and groups.

Zoom link is posted here:
https://calendar.college.harvard.edu/event/math_table

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. I will describe results establishing the existence of intrinsic sign problems in a broad class of topologically ordered phases in 2+1 dimensions. Within this class, these results exclude the possibility of ‘stoquastic’ Hamiltonians for bosons, and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The talk is based on arxiv: 2005.05566 and 2005.05343.

Zoom: https://harvard.zoom.us/j/977347126

TITLE: Birational geometry

ABSTRACT: About main achievements in birational geometry during the last fifty years.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.