BRIDGES

Angela Gibney, University of Pennsylvania

Angela is a Presidential Professor at the University of Pennsylvania specializing in algebraic geometry.

Lecture Series: A combinatorial introduction to curves of genus zero and their moduli
Abstract: The moduli space of n-pointed (Deligne-Mumford) stable curves of genus g provides a natural environment in which we may study smooth curves and their degenerations. These spaces, for different values of g and n, are related to each other through systems of tautological maps. Consequently, problems about curves of positive genus can frequently be reduced to the smooth, projective, rational variety $\overline{\mathcal{M}}_{0,n}$ parametrizing stable pointed rational curves.

In my three lectures at the BRIDGES program I will discuss these ideas. In the first lecture, I will introduce moduli spaces and the moduli space of curves. In the second lecture I will give a combinatorial introduction to $\overline{\mathcal{M}}_{0,n}$ and in my third lecture I will describe two open problems about it, one that has been solved (nearly at least), and another that remains stubbornly open.
Lecture notes

Daniel Hernández, University of Kansas

Daniel is an Associate Professor in the Department of Mathematics at the University of Kansas specializing in commutative algebra. Daniel was born and raised in El Paso, Texas, a vibrant city along the Mexican border, in the middle of the Chihuahuan Desert. He is proud of his hometown, and working-class roots. Growing up, he enjoyed every class that he ever took, and as a junior in college, inspired by an introductory course in linear algebra, he left engineering behind to focus on math. He still finds it hard to believe that he is able to support himself as a professional mathematician. He feels lucky to have had amazing mentors at each critical stage of his life, and perhaps because of this, he feels a deep sense of responsibility and desire to help others succeed. To help cope with the frustration that comes with being a mathematician, he regularly runs and participates in group exercise classes. He has a life-long love of basketball, and plays as often as he can.

Lecture Series: Perspectives in algebraic singularity theory
Abstract: The notion of a singular point of an algebraic variety, or manifold, can be regarded as a generalization of the cusps, or sharp corners, that one encounters in calculus. Of course, just as with cusps and sharp corners, the nature of a singular point (i.e., the "badness" of the singularity) can vary quite dramatically, and in this series, we will discuss ways in which one can quantify this type of "badness". Though we will consider the classical case of singular points over the real and complex numbers, our emphasis will be on the setting of positive characteristic.
Lecture 1
Problems

Marissa Loving, Georgia Tech

Marissa is an NSF Postdoctoral Research Fellow and Hale Assistant Professor in the School of Math at Georgia Tech. She graduated with her PhD in mathematics in August 2019 from the University of Illinois at Urbana-Champaign where she was supported by an NSF Graduate Research Fellowship and an Illinois Graduate College Distinguished Fellowship. Marissa was born and raised in Hawai'i where she completed her B.S. in Computer Science and B.A. in Mathematics at the University of Hawai'i at Hilo. She is the first Native Hawaiian woman to earn a PhD in mathematics. She is also Black, Puerto Rican, and Japanese. Her research interests are in low dimensional topology and geometric group theory, especially mapping class groups of surfaces (of both finite and infinite type). Marissa is also deeply invested in making the mathematics community a more equitable place. Some of her work includes mentoring undergraduate research (through programs such as Summer@ICERM, MSRI-UP, and the Georgia Tech School of Math’s REU) and co-organizing initiatives like SUBgroups and paraDIGMS.

Lecture Series: An intro to mapping class groups
Abstract: The mapping class group of a connected, oriented, finite-type surface S, denoted Mod(S), is the group of homotopy classes of orientation-preserving homeomorphisms of S. In this mini-course, we will cover some of the basics of Mod(S) such as finding a (finite) generating set, producing actions of Mod(S) on various metric spaces (the curve complex, Teichmuller space, etc.), and building a (coarse) model for Mod(S). In the final lecture, we will explore "big" mapping class groups of infinite-type surfaces, which is an area of increasing interest and is full of open questions and directions.
Problems

Angela Robinson, National Institute of Standards and Technology

Angela Robinson is a mathematician in the Cryptographic Technology Group of the National Institute of Standards and Cryptography. She completed her Mathematics B.S. at Baylor University and Mathematics PhD at Florida Atlantic University under the advisement of Rainer Steinwandt. Her existing research includes cryptanalysis and quantum-resistant algorithms. Her current research is focused on the decoding performance of moderate-density parity check matrices under iterative decoders.

Lecture Series: Code-based Cryptography
Abstract: Public key cryptography protects the privacy and security of our global digital communication infrastructure. All widely-deployed public key cryptographic systems are based on the difficulty in solving variations of the integer factorization and discrete logarithm problems. In 1991, Peter Shor presented quantum algorithms that could solve these problems significantly faster than classical computers. Consequently, a full-scale quantum computer would upend the security and privacy of our digital world. The National Institute of Standards and Technology (NIST) initiated a process to update current public-key standards to schemes believed to be quantum-resistant. NIST made a worldwide call for quantum-resistant public-key cryptographic algorithms and, in response, received over 80 submissions to be considered for standardization. NIST is currently in the 3rd round of analysis and 3 of the remaining 15 algorithms are based on error-correcting codes.

Error correcting codes were originally designed to improve communication across noisy channels, enabling the correction of errors introduced in transit. Messages are encoded by adding some redundancy in such a way that errors introduced by the channel can be removed from the received information, and then the receiver can decode (remove redundancy) to recover the original message. In the 1970, it was discovered that cryptosystems could be designed based on error-correcting codes if errors were strategically introduced by the sender so that only the intended receiver could decode. Due to the inefficiency of early schemes, these results were not actively pursued by cryptographers until decade later. In this series, we will explore the foundations of code-based cryptography, the history of securing code-based cryptosystems, and role code-based cryptosystems could play in securing our digital world.
Lecture 1
Problems