
Speaker:Title: Summer Introduction to Mathematical ResearchVenue: virtualThe Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description: Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department) In this course, we will start with an introduction to computer programming, algorithms, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, and differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus… 

Speaker:Title: SIMONS COLLABORATION ON HOMOLOGICAL MIRROR SYMMETRYVenue: VirtualThe Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications. This program is funded by the Simons Foundation. Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or LandauGinzburg model). We envision that our goal of proving HMS in full generality can be accomplished by… 

Speaker:Title: Swampland ProgramVenue: VirtualDuring the 2021–2022 academic year, the CMSA will host a program on the socalled “Swampland.” The Swampland program aims to determine which lowenergy effective field theories are consistent with nonperturbative quantum gravity considerations. Not everything is possible in String Theory, and finding out what is and what is not strongly constrains the low energy physics. These constraints are naturally interesting for particle physics and cosmology, which has led to a great deal of activity in the field in the last years. The Swampland is intrinsically interdisciplinary, with ramifications in string compactifications, holography, black hole physics, cosmology, particle physics, and even mathematics. This program will include an extensive group of visitors and a slate of seminars. Additionally, the CMSA will host a school oriented toward… 

Speaker:Title: Nonlinear Equations ProgramVenue: VirtualMost physical phenomena, from the gravitating universe to fluid dynamics, are modeled on nonlinear differential equations. The subject also makes close connections with other branches of mathematics. In particular, some of the deepest results in complex geometry and topology were obtained through solutions of nonlinear equations. The subject underwent rapid developments in the last century and foundational results were established. Compared to linear equations, the difficulty of solving nonlinear equations is of a different order of magnitude and the methods employed in solving them are also much more diversified. To this date, it is an active field with recent exciting discoveries and renewed interests, and several long standing problems seem to be within reach. The special year aims… 

Speaker:Title: MATHPHYSICS PROGRAMVenue: VirtualIn the past thirty years there have been deep interactions between mathematics and theoretical physics which have tremendously enhanced both subjects. The focal points of these interactions include string theory, general relativity, and quantum manybody theory. String theory has been at the center of the ongoing effort to uncover the fundamental principles of nature and in particular to unify Einstein’s geometric theory of gravity with quantum theory. The development of this field has sparked a historically unprecedented synergy between mathematics and physics. Progress at the forefront of theoretical physics has relied crucially on very recent developments in pure mathematics. At the same time insights from physics have led to both new branches of pure mathematics as well as… 

Speaker:Title: GAMES ON HETEROGENEOUS GRAPHSVenue: VirtualA major challenge in evolutionary biology is to understand how spatial population structure affects the evolution of social behaviors such as cooperation. This question can be investigated mathematically by studying evolutionary processes on graphs. Individuals occupy vertices and interact with neighbors according to a matrix game. Births and deaths occur stochastically according to an update rule. Previously, full mathematical results have only been obtained for graphs with strong symmetry properties. Our group is working to extend these results to certain classes of asymmetric graphs, using tools such as random walk theory and harmonic analysis. Here is a list of the scholars participating in this program. Name ShingTung Yau Martin Nowak Ben Adlam Ben Allen YuTing Chen An… 

Speaker:Title: CONDENSED MATTER PROGRAMVenue: VirtualThe methods of topology have been applied to condensed matter physics in the study of topological phases of matter. Topological states of matter are new quantum states that can be characterized by their topological properties. For example, the first topological states of matter discovered were the integer quantum Hall states. The two dimensional integer quantum Hall effect was characterized by an integral number which can be understood as a Chern number of the Berry phase. Chern numbers are topological invariants that play an important role in different areas of mathematics. More recently, new topological states of matter known as topological insulators and topological superconductors have been realized theoretically and experimentally. The characterization of new phases of matter using… 

Speaker:Title: Summer Introduction to Mathematical ResearchVenue: VirtualThe Math Department and Harvard’s Center of Mathematical Sciences and Applications (CMSA) will be running a math program/course for mathematically minded undergraduates this summer. The course will be run by Dr. Yingying Wu from CMSA. Here is a description: Summer Introduction to Mathematical Research (sponsored by CMSA and the Harvard Math Department) In this course, we will start with an introduction to computer programming, algorithm, and scientific computing. Then we will discuss topics in topology, classical geometry, projective geometry, differential geometry, and see how they can be applied to machine learning. We will go on to discuss fundamental concepts of deep learning, different deep neural network models, and mathematical interpretations of why deep neural networks are effective from a calculus viewpoint…. 

Speaker:Title: RANDOM MATRIX PROGRAMVenue: Virtualarge random matrices provide some of the simplest models for large, strongly correlated quantum systems. The statistics of the energy levels of ensembles of such systems are expected to exhibit universality, in the sense that they depend only on the symmetry class of the system. Recent advances have enabled a rigorous understanding of universality in the case of orthogonal, Hermitian, or symplectic matrices with independent entries, resolving a conjecture of WignerDysonMehta dating back 50 years. These new developments have exploited techniques from a wide range of mathematical areas in addition to probability, including combinatorics, partial differential equations, and hydrodynamic limits. It is hoped that these new techniques will be useful in the analysis of universal behaviour in matrix… 

Speaker:Title: TOPOLOGICAL ASPECTS OF CONDENSED MATTERVenue: VirtualDuring Academic year 201819, the CMSA will be hosting a Program on Topological Aspects of Condensed Matter. New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by foster discussion and seeding new collaborations within and across disciplines. As part of the Program, the CMSA will be hosting two workshops: Workshop on Topology and Quantum Phases of Matter (August 2728, 2018) Workshop on Topological Aspects of Condensed Matter (Septmeber 1011, 2019) . Additionally, a weekly Topology… 

Speaker:Title: THE SIMONS COLLABORATION IN HOMOLOGICAL MIRROR SYMMETRYVenue: VirtualThe Simons Collaboration program in Homological Mirror Symmetry at Harvard CMSA and Brandeis University is part of the bigger Simons collaboration program on Homological mirror symmetry (https://schms.math.berkeley.edu) which brings to CMSA experts on algebraic geometry, Symplectic geometry, Arithmetic geometry, Quantum topology and mathematical aspects of high energy physics, specially string theory with the goal of proving the homological mirror symmetry conjecture (HMS) in full generality and explore its applications. Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves… 

Speaker:Title: Mathematical BiologyVenue: VirtualDuring Academic year 201819, the CMSA will be hosting a Program on Mathematical Biology. Just over a century ago, the biologist, mathematician and philologist D’Arcy Thompson wrote “On growth and form”. The book was a visionary synthesis of the geometric biology of form at the time. 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… 

Speaker:Title: SPACETIME AND QUANTUM MECHANICS, TOTAL POSITIVITY AND MOTIVESVenue: VirtualRecent developments have poised this area to make serious advances in 2019, and we feel that bringing together many of the relevant experts for an intensive semester of discussions and collaboration will trigger some great things to happen. To this end, the organizers will host a small workshop during fall 2019, with between 2030 participants. They will also invite 1020 longerterm visitors throughout the semester. Additionally, there will be a seminar held weekly on Thursdays at 2:30pm in CMSA G10. Organizers: Nima ArkaniHamed (IAS) Lauren Williams (Harvard) Alexander Postnikov (MIT) Thomas Lam (Michigan) . Workshops: Spacetime and Quantum Mechanics Workshop, October 2830, 2019 Here is a partial list of the mathematicians and physicists who have indicated that they will attend part or… 