Calendar
- 19September 19, 2021No events
- 20September 20, 2021No events
- 21September 21, 2021
CMSA Algebraic Geometry in String Theory Seminar: What do bounding chains look like, and why are they related to linking numbers?
Gromov-Witten invariants count pseudo-holomorphic curves on a symplectic manifold passing through some fixed points and submanifolds. Similarly, open Gromov-Witten invariants are supposed to count disks with boundary on a Lagrangian, but in most cases such counts are not independent of some choices as we would wish. Motivated by Fukaya’11, J. Solomon and S. Tukachinsky constructed open Gromov-Witten invariants in their 2016 papers from an algebraic perspective of $A_{\infty}$-algebras of differential forms, utilizing the idea of bounding chains in Fukaya-Oh-Ohta-Ono’06. On the other hand, Welschinger defined open invariants on sixfolds in 2012 that count multi-disks weighted by the linking numbers between their boundaries. We present a geometric translation of Solomon-Tukachinsky’s construction. From this geometric perspective, their invariants readily reduce to Welschinger’s.
https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09
Tautological classes of matroids
1 Oxford Street, Cambridge, MA 02138 USAAlgebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng. - 22September 22, 2021
CMSA Quantum Matter in Mathematics and Physics Seminar: Symmetry types in QFT and the CRT theorem
I will discuss ideas around symmetry and Wick rotation contained in joint work with Mike Hopkins (https://arxiv.org/abs/1604.06527). This includes general symmetry types for relativistic field
theories and their Wick rotation. I will then indicate how thebasic CRT theorem works for general symmetry types, focusing on the case of the pin groups. In particular, I expand on a subtlety first flagged by Greaves-Thomas.Galois action on the pro-algebraic completion of the fundamental group
1 Oxford Street, Cambridge, MA 02138 USAGiven a variety over a number field, its geometric etale fundamental group comes equipped with an action of the Galois group. This induces a Galois action on the pro-algebraic completion of the etale fundamental group and hence the ring of functions on that pro-algebraic
completion provides a supply of Galois representations.It turns out that any finite-dimensional p-adic Galois representation contained in the ring of functions on the pro-algebraic completion of the fundamental group of a smooth variety satisfies the assumptions of the Fontaine-Mazur conjecture: it is de Rham at places above p and is a. e. unramified.
Conversely, we will show that every semi-simple representation of the Galois group of a number field coming from algebraic geometry (that is, appearing as a subquotient of the etale cohomology of an algebraic variety) can be established as a subquotient of the ring of functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.
Projective vs. abelian geometry
1 Oxford Street, Cambridge, MA 02138 USAProjective space and complex tori are two of the simplest types of manifolds we encounter, and in many ways they seem very different from each other. I will try to convince you however that, at least if we consider a special (but at the same time very common) class of tori called principally polarized abelian varieties, then the geometry of their subvarieties exhibits surprising, and to date mostly unexplained, similarities to the geometry of subvarieties in projective space. - 23September 23, 2021
CMSA Interdisciplinary Science Seminar: The number of n-queens configurations
The n-queens problem is to determine Q(n), the number of ways to place n mutually non-threatening queens on an n x n board. The problem has a storied history and was studied by such eminent mathematicians as Gauss and Polya. The problem has also found applications in fields such as algorithm design and circuit development.Despite much study, until recently very little was known regarding the asymptotics of Q(n). We apply modern methods from probabilistic combinatorics to reduce understanding Q(n) to the study of a particular infinite-dimensional convex optimization problem. The chief implication is that (in an appropriate sense) for a~1.94, Q(n) is approximately (ne^(-a))^n. Furthermore, our methods allow us to study the typical “shape” of n-queens configurations.Zoom ID: 950 2372 5230 (Password: cmsa)CMSA Quantum Matter in Mathematics & Physics Seminar: Applications of instantons, sphalerons and instanton-dyons in QCD
I start with a general map of gauge topology, including monopoles, instantons and instanton-dyons. Then comes reminder of the “topological landscape”, the minimal energy gauge field configurations, as a function of Chern-Simons number Ncs and r.m.s. size. It includes “valleys” at integer Ncs separated by mountain ridges. The meaning of instantons,
instanton-antiinstanton “streamlines” or thimbles, and sphalerons are reminded, together with some proposal to produce sphalerons at LHC and RHIC.Applications of instanton ensembles, as a model of QCD vacuum, are mostly related to their fermionic zero modes and t’Hooft effective Lagrangian, which explains explicit and spontaneous breaking of chiral symmetries. Recent applications are related with hadronic wave
functions, at rest and in the light front (LFWFs). Two application would be spin-dependent forces and the so called “flavor asymmetry of antiquark sea” of the nucleons. At temperatures comparable to deconfinement transition, instantons get split into constituents called instanton-dyons. Studies of their ensemble explains both deconfinement and chiral transitions, in ordinary and deformed QCD.https://harvard.zoom.us/j/977347126
Password: cmsaCMSA Active Matter Seminar: The many phases of a cell
I will begin by introducing an emerging paradigm of cellular organization – the dynamic compartmentalization of biochemical pathways and molecules by phase separation into distinct and multi-phase condensates. Motivated by this, I will discuss two largely orthogonal problems, united by the theme of phase separation in multi-component and chemically active fluid mixtures.1. I will propose a theoretical model based on Random-Matrix Theory, validated by phase-field simulations, to characterizes the rich emergent dynamics, compositions, and steady-state properties that underlie multi-phase coexistence in fluid mixtures with many randomly interacting components.2. Motivated by puzzles in gene-regulation and nuclear organization, I will propose a role for how liquid-like nuclear condensates can be organized and regulated by the active process of RNA synthesis (transcription) and RNA-protein coacervation. Here, I will describe theory and simulations based on a Landau formalism and recent experimental results from collaborators.Zoom link: https://harvard.zoom.us/j/96657833341*rescheduled from 9/16/21Quantitative Logarithmic Equidistribution and S-Integrality
Given a rational map defined over a number field, the Galois orbits of points with canonical height tending to zero will equidistribute to a measure supported on the Julia set. If one is able to extend the space of test functions to include those with certain logarithmic poles, then it is possible to obtain finiteness results on S-integral points. In this talk, we will study quantitative versions of logarithmic equidistribution in some special situations and their implications.
- 24September 24, 2021No events
- 25September 25, 2021No events