CMSA

Geometric Analysis Approach to AI Workshop
2019-01-18 08:30 - 2019-01-21 17:30 (America/New_York) | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Due to inclement weather on Sunday, the second half of the workshop has been moved forward one day. Sunday and Monday’s talks will now take place on Monday and Tuesday.
On January 18-21, 2019 the Center of Mathematical Sciences and Applications will be hosting a workshop on the Geometric Analysis Approach to AI.
This workshop will focus on the theoretic foundations of AI, especially various methods in Deep Learning. The topics will cover the relationship between deep learning and optimal transportation theory, DL and information geometry, DL Learning and information bottle neck and renormalization theory, DL and manifold embedding and so on. Furthermore, the recent advancements, novel methods, and real world applications of Deep Learning will also be reported and discussed.
The workshop will take place from January 18th to January 23rd, 2019. In the first four days, from January 18th to January 21, the speakers will give short courses; On the 22nd and 23rd, the speakers will give conference representations. This workshop is organized by Xianfeng Gu and Shing-Tung Yau.
The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Speakers:

Sarah Adel Bargal, Boston University
Guy Bresler, MIT
Tina Eliassi-Rad, Northeastern
Yun Raymond Fu, Northeastern
Brian Kulis, Boston University
Na Lei, Dalian University of Technology
Yi Ma, UC Berkeley
Minh Hoai Nguyen, Stony Brook
Francesco Orabona, Boston University
Cengiz Pehlevan, Harvard SEAS
Tomaso Poggio, MIT
Zhiwei Qin, DiDi Research America
Kate Saenko, Boston University
Dimitris Samaras, Stony Brook
Johannes Schmidt-Hieber, University of Twente
Steven Skiena, Stony Brook
Vivienne Sze, MIT
Naftali Tishby, ICNC
Jiajun Wu, MIT
Ying Nian Wu, UCLA
Gangqiang Xia, Morgan Stanley
Eric Xing, Carnegie Mellon
Donghui Yan, UMass Dartmouth
Alan Yuille, Johns Hopkins
Juhua Zhu, Argus

1/16/2019 Hodge Seminar
2019-01-16 20:46

1/8/2019 Topology Seminar
2019-01-08 14:00

12/12/2018 Hodge Seminar
2018-12-12 20:43

12/10/2018 Topology Seminar
2018-12-10 20:43

12/10/2018 Mathematical Physics Seminar
2018-12-10 20:42

12/6/2018 Special Mathematical Physics Seminar
2018-12-06 20:45

Displacement convexity of Boltzmann's entropy characterizes positive energy in general relativity
2018-12-05 16:30 - 17:30 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Robert McCann (University of Toronto)
Title: Displacement convexity of Boltzmann’s entropy characterizes positive energy in general relativity
Abstract: Einstein’s theory of gravity is based on assuming that the fluxes of a energy and momentum in a physical system are proportional to a certain variant of the Ricci curvature tensor on a smooth 3+1 dimensional spacetime. The fact that gravity is attractive rather than repulsive is encoded in the positivity properties which this tensor is assumed to satisfy. Hawking and Penrose (1971) used this positivity of energy to give conditions under which smooth spacetimes must develop singularities. By lifting fractional powers of the Lorentz distance between points on a globally hyperbolic spacetime to probability measures on spacetime events, we show that the strong energy condition of Hawking and Penrose is equivalent to convexity of the Boltzmann-Shannon entropy along the resulting geodesics of probability measures. This new characterization of the strong energy condition on globally hyperbolic manifolds also makes sense in (non-smooth) metric measure settings, where it has the potential to provide a framework for developing a theory of gravity which admits certain singularities and can be continued beyond them. It provides a Lorentzian analog of Lott, Villani and Sturm’s metric-measure theory of lower Ricci bounds, and hints at new connections linking gravity to the second law of thermodynamics. Preprint available at http://www.math.toronto.edu/mccann/papers/GRO.pdf

12/5/2018 Random Matrix and Probability Theory Seminar
2018-12-05 15:00 - 16:00

12/5/2018 Hodge Seminar
2018-12-05 13:30

12/05/2018 General Relativity Seminar
2018-12-05 11:00

12/3/2018 Mathematical Physics Seminar
2018-12-03 12:00 - 13:00

13/3/2018 Topology Seminar
2018-12-03 10:00 - 11:30

Morphogenesis: Geometry and Physics
2018-12-03 08:30 - 2018-12-05 14:30 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book – a literary masterpiece – is a visionary synthesis of the geometric biology of form. It also served as a call for mathematical and physical approaches to understanding the evolution and development of shape. In the century since its publication, we have seen a revolution in biology following the discovery of the genetic code, which has uncovered the molecular and cellular basis for life, combined with the ability to probe the chemical, structural, and dynamical nature of molecules, cells, tissues and organs across scales. In parallel, we have seen a blossoming of our understanding of spatiotemporal patterning in physical systems, and a gradual unveiling of the complexity of physical form. So, how far are we from realizing the century-old vision that “Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed ?”
To address this requires an appreciation of the enormous ‘morphospace’ in terms of the potential shapes and sizes that living forms take, using the language of mathematics. In parallel, we need to consider the biological processes that determine form in mathematical terms is based on understanding how instabilities and patterns in physical systems might be harnessed by evolution.
In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems.
The first workshop will focus on the interface between Morphometrics and Mathematics, while the second will focus on the interface between Morphogenesis and Physics.The workshop is organized by L. Mahadevan (Harvard), O. Pourquie (Harvard), A. Srivastava (Florida).
As part of the program on Mathematical Biology a workshop on Morphogenesis: Geometry and Physics will take place on December 3-5, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Videos
Speakers:

Arkhat Abzhanov, Imperial College
Yohanns Bellaiche, Paris
Cheng Ming Chuong, USC
Zev Gartner, UCSF
Thomas Gregor, Princeton
Dagmar Iber, Zurich
Ian Jermyn, Durham University
Raymond Keller, UVA
Allon Klein, HMS
Lisa Manning, Syracuse
Cristina Marchetti, UCSB
Sean Megason, HMS
Elliot Meyerowitz, Caltech
Michel Milinkovitch, Geneva
Leonardo Morsut, USC
Olivier Pourquié, HMS
Eric Siggia, Rockefeller University
Ben Simons, Cambridge
Sebastian Streichan, UCSB
Aryeh Warmflash, Rice

Recent progress on mean curvature flow
2018-11-28 16:30 - 17:30 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Robert Haslhofer (University of Toronto)
Title: Recent progress on mean curvature flow
Abstract: A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution in extrinsic geometry and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In the first half of the talk, I will give an overview of the well developed theory in the mean convex case, i.e. when the mean curvature vector everywhere on the surface points inwards. Mean convex mean curvature flow can be continued through all singularities either via surgery or as level set solution, with a precise structure theory for the singular set. In the second half of the talk, I will report on recent progress in the general case without any curvature assumptions. Namely, I will describe our solution of the mean convex neighborhood conjecture and the nonfattening conjecture, as well as a general classification result for all possible blowup limits near spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.

11/28/2018 Random Matrix and Probability Theory Seminar
2018-11-28 15:00

11/28/2018 Hodge Lecture
2018-11-28 13:30

11/26/2018 Mathematical Physics Seminar
2018-11-26 12:00 - 13:00

11/21/2018 Hodge Seminar
2018-11-21 13:30

Computational Principles of Auditory Cortex
2018-11-19 15:00 - 16:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Xiaoqin Wang (Johns Hopkins University)
Title: Computational Principles of Auditory Cortex
Abstract: Auditory cortex is located at the top of a hierarchical processing pathway in the brain that encodes acoustic information. This brain region is crucial for speech and music perception and vocal production. Auditory cortex has long been considered a difficult brain region to study and remained one of less understood sensory cortices. Studies have shown that neural computation in auditory cortex is highly nonlinear. In contrast to other sensory systems, the auditory system has a longer pathway between sensory receptors and the cerebral cortex. This unique organization reflects the needs of the auditory system to process time-varying and spectrally overlapping acoustic signals entering the ears from all spatial directions at any given time. Unlike visual or somatosensory cortices, auditory cortex must also process and differentiate sounds that are externally generated or self-produced (during speaking). Neural representations of acoustic information in auditory cortex are shaped by auditory feedback and vocal control signals during speaking. Our laboratory has developed a unique and highly vocal non-human primate model (the common marmoset) and quantitative tools to study neural mechanisms underlying audition and vocal communication.

11/19/2018 Mathematical Physics Seminar
2018-11-19 12:00 - 13:00

Current Developments In Mathematics 2018
2018-11-16 08:00 - 2018-11-17 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Current Developments in Mathematics 2018 Conference.
Friday, Nov. 16, 2018 2:15 pm – 6:00 pm
Saturday, Nov. 17, 2018 9:00 am – 5:00 pm
Harvard University Science Center, Hall B
Youtube Playlist

The virtual fundamental class in symplectic geometry
2018-11-14 16:00 - 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Dusa McDuff (Columbia University)
Title: The virtual fundamental class in symplectic geometry
Abstract: Essential to many constructions and applications of symplectic geometry is the ability to count J-holomorphic curves. The moduli spaces of such curves have well understood compactifications, and if cut out transversally are oriented manifolds of dimension equal to the index of the problem, so that they a fundamental class that can be used to count curves. In the general case, when the defining equation is not transverse, there are various different approaches to constructing a representative for this class, We will discuss and compare different approaches to such a construction e.g. using polyfolds or various kinds of finite dimensional reduction. Most of this is joint work with Katrin Wehrheim.

11/14/2018 RM & PT Seminar
2018-11-14 15:00 - 16:00

11/14/2018 Hodge Seminar
2018-11-14 13:30

11/14/2018 General Relativity Seminar
2018-11-14 11:00

Hodge and Noether-Lefschetz Loci Seminar
2018-11-07 15:06
In the Fall 2018 Semester the CMSA will be hosting a seminar on Hodge and Noether-Lefschetz loci, with lectures given by Hossein Movasati (IMPA). The seminar will occur weekly on Wednesday at 1:30 in room G10 of the CMSA.

The schedule below will be updated as talks are confirmed.

Date
Title/Abstract

11/7/2018
Video
Title: Hodge and Noether-Lefschetz loci
Abstract: Hodge cycles are topological cycles which are conjecturally (the millennium Hodge conjecture) supported in algebraic cycles of a given smooth projective complex manifold. Their study in families leads to the notion of Hodge locus, which is also known as Noether-Lefschetz locus in the case of surfaces. The main aim of this mini course is to introduce a computational approach to the study of Hodge loci for hypersurfaces and near the Fermat hypersurface. This will ultimately lead to the verification of the variational Hodge conjecture for explicit examples of algebraic cycles inside hypersurfaces and also the verification of integral Hodge conjecture for examples of Fermat hypersurfaces. Both applications highly depend on computer calculations of rank of huge matrices. We also aim to review some classical results on this topic, such as Cattani-Deligne-Kaplan theorem on the algebraicity of the components of the hodge loci, Deligne’s absolute Hodge cycle theorem for abelian varieties etc.
In the theoretical side another aim is to use the available tools in algebraic geometry and construct the moduli space of projective varieties enhanced with elements in their algebraic de Rham cohomology ring. These kind of moduli spaces have been useful in mathematical physics in order to describe the generating function of higher genus Gromov-Witten invariants, and it turns out that the Hodge loci in such moduli spaces are well-behaved, for instance, they are algebraic leaves of certain holomorphic foliations. Such foliations are constructed from the underlying Gauss-Manin connection. This lectures series involves many reading activities on related topics, and contributions by participants are most welcome.

11/14/2018
Video
Title: Integral Hodge conjecture for Fermat varieties
Abstract: We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same. This is a joint work with E. Aljovin and R. Villaflor.

11/21/2018
Video
Title: Periods of algebraic cycles
Abstract: The tangent space of the Hodge locus at a point can be described by the so called infinitesimal variation of Hodge structures and the cohomology class of Hodge cycles. For hypersurfaces of dimension $n$ and degree $d$ it turns out that one can describe it without any knowledge of cohomology theories and in a fashion which E. Picard in 1900’s wanted to study integrals/periods. The data of cohomology class is replaced with periods of Hodge cycles, and explicit computations of these periods, will give us a computer implementable description of the tangent space. As an application of this we show that for examples of $n$ and $d$, the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space.

11/28/2018
Video
Title: Periods of Complete Intersection Algebraic Cycles
Speaker: Roberto Villaflor
Abstract: In order to compute periods of algebraic cycles inside even dimensional smooth degree d hypersurfaces of the projective space, we restrict ourselves to cycles supported in a complete intersection subvariety. When the description of the complete intersection is explicit, we can compute its periods, and furthermore its cohomological class. As an application, we can use this data to describe the Zariski tangent space of the corresponding Hodge locus, as the degree d part of some Artinian Gorenstein ideal of the homogeneous coordinate ring of the projective space. Using this description, we can show that for d>5, the locus of hypersurfaces containing two linear cycles, is a reduced component of the Hodge locus in the underlying parameter space.

12/05/2018
Room G02
Title: Some explicit Hodge cycles
Abstract: Explicit examples of Hodge cycles are due to D. Mumford and A. Weil in the case of CM abelian varieties. In this talk, I will describe few other examples for the Fermat variety. Effective verification of the Hodge conjecture for these cycles is not known.

12/12/2018
Video
Title: A conjectural Hodge locus for cubic tenfold
Abstract: In this talk we will consider the difference of two linear algebraic cycles of dimension 5 inside a smooth cubic tenfold and such that the dimension of their intersection is 3. We will show some computer assisted evidences to the fact that the corresponding Hodge locus is bigger than the expected locus of algebraic deformations of the cubic tenfold together with its linear cycles. A similar discussion will be also presented for cubic six and eightfold, for which we will prove that the corresponding second and third order infinitesimal Hodge loci are smooth. The main ingredient is a computer implementation of power series of periods of hypersurfaces.

1/16/2019
Title: Algebraic BCOV anomaly equation
Abstract: We introduce the moduli space T of non-rigid compact Calabi-Yau threefolds enhanced with differential forms and a Lie algebra of vector fields in T. This will be used in order to give a purely algebraic interpretation of topological string partition functions and the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation (joint work with M. Alim, E. Scheidegger, S.-T. Yau). We will also define similar moduli spaces for even dimensional Calabi-Yau varieties, where we have the notion of Hodge locus.

1/23/2019
Video
Title: A new model for modular curves
Abstract: One of the non-trivial examples of a Hodge locus is the modular curve X_0(N), which is due to isogeny of elliptic curves (a Hodge/algebraic cycle in the product of two elliptic curves). After introducing the notion of enhanced moduli of elliptic curves, I will describe a new model for X_0(N) in the weighted projective space of dimension 4 and with weights (2,3,2,3,1). I will also introduce some elements in the defining ideal of such a model.
The talk is based on the article arXiv:1808.01689.

1/30/2019
Video
Title: Constant Yukawa couplings
Abstract: In this talk I will first introduce algebraic Yukawa couplings for any moduli of enhanced Calabi-Yau n-folds. Then I will list many examples in support of the following conjecture. A moduli of Calabi-Yau n-folds is a quotient of a Hermitian symmetric domain (constructed from periods) by an arithmetic group if and only if the corresponding Yukawa couplings are constants.

2/6/2019
Video
Title: Integrality properties of CY modular forms
Abstract: The integrality of the coefficients of the mirror map is a central problem in the arithmetic of Calabi-Yau varieties and it has been investigated by Lian-Yau (1996, 1998), Hosono-Lian-Yau (1996), Zudilin (2002), Kontsevich-Schwarz-Vologodsky (2006) Krattenthaler-Rivoal (2010). The central tool in most of these works has been the so called Dwork method. In this talk we use this method and classify all hypergeometric differential equations with a maximal unipotent monodromy whose mirror map has integral coefficients.
We also give a computable condition on the parameters of a hypergeometric function which conjecturally computes all the primes which appear in the denominators of the coefficients of the mirror map. This is a joint work with Kh. Shokri.

2/13/2019
Title: Foliations and Hodge loci
Abstract: In this talk I will introduce a holomorphic foliation in a larger parameter space attached to families of enhanced projective varieties. Irreducible components of the Hodge locus with constant periods are algebraic leaves of such a foliation. Under the hypothesis that these are all the algebraic leaves, we get the fact that such algebraic leaves are defined over the algebraic closure of the base field and that Hodge classes are weak absolute in the sense of C. Voisin.

References:

M. Alim, H. Movasati, E. Scheidegger, S.-T. Yau. Gauss-Manin connection in disguise: Calabi-Yau threefolds, Comm. Math. Phys. 344, (2016), no. 3, 889-914.
E. H. Cattani, P. Deligne, and A. G. Kaplan. On the locus of Hodge classes. Amer. Math. Soc., 8(2):483–506, 1995.
B. Haghighat H. Movasati, S.-T. Yau. Calabi-Yau modular forms in limit: Elliptic fibrations, Communications in Number Theory and Physics, Vol. 11, Number 4, 879-912, 2017.
H. Movasati, Modular and automorphic forms & beyond, Book under preparation, 2019.
H. Movasati. A Course in Hodge Theory: with Emphasis on Multiple Integrals.Book submitted,2018.
H. Movasati, On elliptic modular foliation, II, 2018
H. Movasati, R. Villaflor Loyola, Periods of linear algebraic cycles,, 2018.
H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms, Surveys in Modern Mathematics, Vol 13, International Press, Boston.
H. Movasati, Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci, Asian Journal of Mathematics, Vol.21, No. 3, pp. 463-482, 2017.
C. Voisin. Hodge loci and absolute Hodge classes. Compos. Math., 143(4):945–958, 2007.
C. Voisin. Hodge loci. Handbook of moduli. Vol. III, volume 26 of Adv. Lect. Math. (ALM)}, pages 507–546. Int. Press, Somerville, MA, 2013.

11/7/2018 Random Matrix and Probability Theory Seminar
2018-11-07 15:00 - 16:00

11/7/2018 General Relativity Seminar
2018-11-07 11:00

11/5/2018 Math Physics Seminar
2018-11-05 12:00 - 13:00
