Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
To post a seminar which takes place at the Mathematics department, please email with date, time, room, title and possibly with an abstract.
LOGIC SEMINAR: Sebastien Vasey
Harvard University
Weak factorization systems and stable independence
on Monday, April 22, 2019, at 5:40 pm in Science Center 507
We exhibit a bridge between the theory of weak factorization systems, a categorical concept used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the cofibrantly generated weak factorization systems (those that are, in a precise sense, generated by a set) are exactly those that give rise to stable independence notions. This two way connection yields a powerful new tool to build tame and stable abstract elementary classes. In particular, we generalize a construction of Baldwin-Eklof-Trlifaj to prove that the category of flat modules with flat monomorphisms has a stable independence notion, and explain how this connects to the fact that every module has a flat cover. Joint work with Jiří Rosický and Michael Lieberman.

Quasimap wall-crossing for GIT quotients
on Monday, April 22, 2019, at 12:00 pm - 1:00 pm in CMSA, 20 Garden St, G10
For a large class of GIT quotients X=W//G, Ciocan-Fontanine--Kim--Maulik have developed the theory of epsilon-stable quasimap invariants. They are conjecturally equivalent to the Gromov--Witten invariants of X via explicit wall-crossing formulae, which have been proved in many cases, including targets with good torus action and complete intersections in a product of projective spaces. In this talk, we will give a proof for all targets in all genera. The main ingredient is the construction of some moduli space with C^* action whose fixed-point loci precisely correspond to the terms in the wall-crossing formulae.

Tsinghua University
Quantum many-body computation on a small quantum computer
on Tuesday, April 23, 2019, at 3:00 pm in Jefferson 356
What can we do on a small scale near-term noisy quantum computer with O(10) qubits? We propose a variational scheme to study ground state properties of quantum many-body systems with fewer qubits. Studying ground state of quantum magnets and quantum chemistry problems in this way will be one of few killer applications of a near-term quantum computer.

Harvard University
Systolic inequality for K3 surfaces via stability conditions
on Tuesday, April 23, 2019, at 4:00 pm in Science Center 507
We propose a natural generalization of Loewner's torus systolic inequality from the perspective of Calabi-Yau geometry. We ask whether the square of the minimum volume of special Lagrangians in a Calabi-Yau manifold is bounded above by the total volume of the Calabi-Yau. We introduce the categorical analogues of systole and volume in terms of Bridgeland stability conditions, which enables us to formulate the mirror question under mirror symmetry. Finally, we give an affirmative answer to the mirror question for K3 surfaces of Picard rank one.

NUMBER THEORY SEMINAR : Yiannis Sakellaridis
A new paradigm for the comparison of trace formulas
on Wednesday, April 24, 2019, at 3:00 - 4:00 pm in Science Center 507
Trace formulas are the "mainstream" method for proving Langlands' functoriality, relating the local and automorphic spectra of different groups; in the generalization provided by the relative trace formula, groups can be replaced by appropriate homogeneous spaces. However, the "endoscopic" paradigm of comparisons has nearly reached its limits. In this talk, I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms and L-functions. More precisely, for an affine spherical variety X=H\G of rank one, I will prove that there is an explicit integral "transfer operator" which transforms the orbital integrals of the relative trace formula for H\G/H to the orbital integrals of the Kuznetsov formula for PGL(2) or SL(2), equipped with suitable non-standard test functions. The operator is determined by the L-value associated to the square of the H-period integral, and the proof uses a deep theory of Friedrich Knop on the cotangent bundles of spherical varieties, viewed as Hamiltonian manifolds. If time permits, I will also discuss other instances of such non-standard comparisons, including Venkatesh's thesis that gave a "beyond endoscopy" proof of functoriality from tori to SL(2), and Hankel transforms which encode the functional equations of L-functions.

Virginia Tech
Turbulence Modeling in the Age of Data: From Data Assimilation to Machine Learning
on Wednesday, April 24, 2019, at 3:00 - 4:00 pm in CMSA, 20 Garden St, G10
Many complex systems are characterized by physics at a wide range of scales, for which first-principle-based high-fidelity models resolving all the scales are prohibitively expensive to run. Consequently, practical simulations have primarily relied on low-fidelity models with approximate closure models, which introduce large model-form uncertainties and diminish their predictive capabilities. Turbulent flows are a classical example of such complex physical systems, where numerical solvers with turbulence closure models are widely used in industrial flow simulations. In light of the decades-long stagnation in traditional turbulence modeling, data-driven methods have been proposed as a promising alternative. We present a comprehensive framework for using data to reduce model uncertainties in turbulent flow simulations. For online, continuously streamed monitoring data, we use data assimilation and Bayesian inference to reduce model-form uncertainties; For offline data from a database of flows, we proposed a physics-informed machine learning approach to reduce model discrepancies. In both cases, we emphasized enforcing physical constraints in the data-driven modeling. More information:

Harvard University
Credible Mechanisms
on Wednesday, April 24, 2019, at 4:30 pm- 5:30 pm in CMSA, 20 Garden St, G10
Consider an extensive-form mechanism, run by an auctioneer who communicates sequentially and privately with agents. Suppose the auctioneer can deviate from the rules provided that no single agent detects the deviation. A mechanism is credible if it is incentive-compatible for the auctioneer to follow the rules. We study the optimal auctions in which only winners pay, under symmetric independent private values. The first-price auction is the unique credible static mechanism. The ascending auction is the unique credible strategy-proof mechanism.

Microsoft Station Q
Quantum cellular automata in higher dimensions
on Wednesday, April 24, 2019, at 10:30 am in CMSA Building, 20 Garden St, G10
I'll discuss joint work with Matt Hastings on local endomorphisms of the operator algebra. We found these have a cohomological invariant similar to that of an incompressible flow.

Universität Tübingen
Asymptotically flat extensions with charge
on Wednesday, April 24, 2019, at 10:30 am - 11:30 am in CMSA, 20 Garden Street, G02
Inspired by the Mantoulidis and Schoen construction, we obtain time-symmetric black hole initial data sets for the Einstein--Maxwell equations satisfying the dominant energy condition, such that their horizon boundary geometry is prescribed, and their total masses and total charges are controlled. We also formulate a notion of boundary Bartnik mass in this context and compute its value for minimal Bartnik data. This talk is based on a joint work with A. Alaee and C. Cederbaum.

Oxford University
Pathwise well-posedness of nonlinear diffusion equations with nonlinear, conservative noise
on Thursday, April 25, 2019, at 4:30 pm- 5:30 pm in CMSA, 20 Garden St, G10
We present a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, approximate the Dean-Kawasaki equation in fluctuating fluid dynamics, describe the fluctuating hydrodynamics of the zero range process, and model the evolution of a thin film in the regime of negligible surface tension. Motivated by the theory of stochastic viscosity solutions, we pass to the equation’s kinetic formulation, where the noise enters linearly and can be inverted using the theory of rough paths. The talk is based on joint work with Benjamin Gess.

University of Michigan
Perturbations of Null Hypersurfaces and Null Penrose Inequality
on Thursday, April 25, 2019, at 10:30 am - 11:30 am in CMSA, 20 Garden Street, G02
The Penrose inequality in general relativity is a conjectured inequality between the area of the horizon and the mass of a black-hole spacetime. The null Penrose inequality is the case where it concerns the area of the horizon and the Bondi mass at null infinity on a null hypersurface. An effective method to prove Penrose-type inequalities is to exploit the monotonicity of the Hawking mass along certain foliations. The constant mass aspect function foliation is such a desired foliation, but the behavior of the foliation at past null infinity is an obstacle for the proof. An idea to overcome this difficulty is to vary the null hypersurface to achieve the desired behavior of the foliation at null infinity, leading to a spacetime version of the Penrose inequality. To formalise this idea, one need to study perturbations of null hypersurfaces. I will talk about my work on the study of perturbations of null hypersurfaces and its application to the null Penrose inequality.

Georgia Tech
Heegaard Floer and homology cobordism
on Friday, April 26, 2019, at 3:30 pm in Science Center 507
We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong. Future schedule is found here:

UC Davis
Adiabatic limits of Yang-Mills connections on collapsing K3 surfaces
on Tuesday, April 30, 2019, at 4:00 pm in Science Center 507
In this talk I will discuss the vector bundle analogue of the degeneration problem for Ricci flat K3 surfaces considered by Gross-WIlson (and later Gross-Tosatti-Zhang). Namely, given an elliptically fibered K3 surface equipped with complex vector bundle, what are the convergence properties of a family of SU(n) ASD Yang-Mills connections as the elliptic fibers collapse? Under certain geometric assumptions, I will demonstrate W^{1,p} convergence away from a finite number of fibers, and show how the limit is uniquely determined by the sequence of holomorphic structures. This is joint work with Ved Datar and Yuguang Zhang.

Iwasawa modules in higher codimension
on Wednesday, May 01, 2019, at 3:00 - 4:00 pm in Science Center 507
Classically speaking, Iwasawa theory concerns the growth of p-parts of class groups in towers of number fields of p-power degree. One considers an inverse limit of such groups as a finitely generated, torsion module over a completed pro-p group ring of the tower. Often, this growth can be slow enough that the support of this unramified module lies in codimension two and higher, while invariants like characteristic ideals which one hopes might be described through L-values fail to measure anything beyond codimension one. The typical way around this is to allow ramification at enough primes over p to make the module larger. Yet, the original modules are certainly of arithmetic interest. I'll discuss how, over CM fields, these unramified Iwasawa modules and others satisfy nice reflection principles and how p-adic L-functions tell us something about them. More generally, I'll explain how p-adic L-functions can be used to describe quotients of exterior powers of Iwasawa modules of restricted ramification. This is joint work with F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, and M. Taylor, building on prior joint work of these authors and G. Pappas.