# Calendar

## Today

Title: Teaching Discussion -- Abstract: This weekly teaching discussion is a place to discuss our practice in the classroom. All are welcome and encouraged to participate, from experienced instructors who have seen it all to those who are just starting.

## Next Week

### Spectral and Scattering Theory Seminar, Antonio Sa Barreto, Purdue, MATH 175

Monday, Feb 25 1:15 pm - 2:15 pm

**Title: Singularities Generated by the Interaction of Semilinear Waves**

Abstract: We will discuss the local propagation of conormal singularities for solutions of semilinear wave equations P u= f(y,u), where P is a second order strictly hyperbolic differential operator. This topic was intensively studied in the 1980s and 1990s, beginning with the celebrated work of J.M. Bony. The subject became too difficult and technical, and sort of faded by mid 2000s. Recently, Kurylev, Lassas and Uhlmann, and Lassas, Uhlmann and Wang, and Uhlmann and Wang, have found a relationship between this subject and non-linear inverse problems.

We will discuss what happens when three three classical conormal plane waves (in two space dimensions) intersect transversally at point q. We show that they will produce singularities on the characteristic cone for P with vertex q (these singularities would not be present if f(u) were a linear function u). The higher dimensional case can be reduced to the two dimensional one, modulo some parameters that really play no role in the problem.

Melrose & Ritter and Bony, working independently, had shown that the singularities over the cone are the only ones that could possibly be generated by the non-linearity. Moreover, they had shown that, away from the original waves, the solution u(y) is a Lagrangian distribution of suitable class with respect to Q (but it could be smooth there). Here we compute its principal symbol, and we show it’s not equal to zero, provided the third derivative of f(y,u) with respect to u at the point (q, u(q)) is not equal to zero. In the mid 1990s, M. Beals had proved this result for f(y,u)=a(y)u^3, in which case the derivative in question is equal to a(q).

We will sketch two proofs of the result. One is seemingly elementary and just uses the Fourier transform (but it hides the use of spaces of distributions commonly used in the field) and is joint work with Yiran Wang. The second proof uses more sophisticated microlocal analysis methods and can (hopefully) be applied in other settings, more precisely when two waves which are tangent to finite order along a line intersect a third wave transversally, and the appearance of caustics.

### Quantum Information Science Seminar, Ryan Mishmash, UC Berkeley, PHYS 242

Tuesday, Feb 26 11:00 am - 12:00 pm

**Unraveling and harnessing non-Abelian topological quantum matter. **

Developing a scalable quantum computing architecture that can withstand decoherence to the extent necessary for real-world applications poses an enormous scientific and technological undertaking. One route forward involves stabilizing exotic topological phases of quantum matter harboring emergent particles known as non-Abelian anyons. With these degrees of freedom in hand, one can in principle construct physical qubits that by themselves function as good logical qubits, thus obviating the need for large-scale error correction. Within this topological approach to quantum computation, the requisite topological phenomena could either be *intrinsic* to a material's internal dynamics or *engineered* in judiciously designed heterostructures. In this talk, I will first describe the current status of the 5/2 fractional quantum Hall plateau, a classic example of a (potentially) intrinsic non-Abelian topological phase. In light of recent thermal Hall measurements indicating a particularly surprising non-Abelian state, I will show that this many-body condensed matter problem is as enigmatic as ever from both a theoretical and numerical perspective [1]. Next, I will turn to recent efforts toward engineering a prototype topological qubit in the Majorana nanowire platform. Within this context, I will describe our recent proposals aimed at verifying topological protection in these devices via time-domain measurements designed to demonstrate (or falsify) the system's exponential insensitivity to all sources of local noise [2,3]. Finally, several directions of future research in these and related areas will be discussed.

[1] Mishmash, Mross, Alicea, and Motrunich, PRB 98, 081107(R) (2018)

[2] Aasen, Hell, Mishmash, Higginbotham et al., PRX 6, 031016 (2016)

[3] Mishmash, Bauer, von Oppen, and Alicea (in preparation)

### Colloquium, L. Ridgway Scott, Louis Block Emeritus Professor, University of Chicago, MATH 175

Tuesday, Feb 26 3:30 pm - 4:30 pm

**Title: Automating PDE Teaching and Research**

Abstract: Automated PDE software systems are transforming both teaching and research in partial differential equations (PDE). The changes in education will be briefly illustrated, and then its role in recent research will be described. We will present examples where it has been used to answer questions that would have required months of programming using traditional techniques. This will include fundamental techniques for numerical solution of PDEs, as well as examples related to recent research on theoretical PDE models for non-Newtonian fluids.

### Commutative Algebra Seminar, Linquan Ma, Purdue University, LWSN B151

Wednesday, Feb 27 1:30 pm - 2:30 pm

**Title: Homological conjectures, perfectoid spaces, and singularities in mixed characteristic**

Abstract: The homological conjectures have been a focus of research in commutative algebra since the 1960s. They are interconnected conjectures that relate homological properties of commutative rings to their internal ring structure. These conjectures had largely been resolved for rings that contain a field, but several remained open in mixed characteristic---until recently Yves Andre made a breakthrough by proving Hochster’s direct summand conjecture and the existence of big Cohen-Macaulay algebras, which lie at the heart of these homological conjectures. The main new ingredient in the solution is to systematically use the theory of perfectoid spaces, which leads to further developments in the study of mixed characteristic singularities. For example, using integral perfectoid big Cohen-Macaulay algebras, one can define the mixed characteristic analog of rational/F-rational and log terminal/F-regular singularities, and they have applications to the study of singularities when the characteristic varies (based on joint work with Karl Schwede). In this talk, we will give a survey of these results and methods.

### Algebraic Geometry Seminar, Nathan Grieve, Michigan State University, MATH 731

Wednesday, Feb 27 3:30 pm - 4:30 pm

Title : Stability, complexity of rational points and arithmetic of linear series -- Abstract: I will survey concepts that are near to K-stability and which have origins in toric geometry. A main goal will be to explain their role in measuring arithmetic complexity of rational points, for example questions in Diophantine approximation for projective varieties. There are also important connections to measures of growth and positivity of line bundles. These arithmetic results build on a number of earlier related works including those of Ru-Vojta, McKinnon-Roth, Evertse-Ferretti and Fujita.

### Quantum Information Science Seminar, Ching-Kai Chiu, Kavli Institute for Theoretical Sciences, PHYS 242

Thursday, Feb 28 11:00 am - 12:00 pm

**Majorana vortex modes in iron-based superconductors.**

Recent scanning tunneling spectroscopy measurement observed the zero-bias conductance peaks at the vortex cores of the topological superconductor candidates, being consistent with Majorana tunneling interpretation. Hence, the observation of the zero bias peak is a strong clue of multiple Majorana zero modes trapped in vortices on the surface of the iron-based superconductor (FeTexSe1-x), which is a potential ideal platform for scalable quantum computing. In the current experiment, the most controllable parameter is magnetic field strength adjusting the intervortex distance. The change of the magnetic field strongly affects Majorana physics near the vortex cores. In this talk, we theoretically study the Majorana vortex lattice with and without disorder as a function of the magnetic field and further propose the readout of Majorana qubits in this platform toward quantum computing processing.

Refreshments served at 10:45 am.

**Title: A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants.**

Abstract: We discuss a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. We find that a tangency requirement along the phase boundary for competitors in the conjectured Γ-limit becomes a mechanism for development of singularities. We discuss several examples involving analysis and numerical experiments. This is joint work with Peter Sternberg, Dmitry Golovaty, and Raghav Venkatraman.

Title: Teaching Discussion -- Abstract: This weekly teaching discussion is a place to discuss our practice in the classroom. All are welcome and encouraged to participate, from experienced instructors who have seen it all to those who are just starting.

## Two Weeks

**Title: On the structure of Ricci shrinkers**

Abstract: Ricci shrinkers are self-similar models of the short time singularities of the Ricci flow. In this talk, I will establish some heat kernel estimates and prove the optimal log-Sobolev inequality, Perelman’s pseudolocality and other rigidity results for Ricci shrinkers without any curvature assumption. This is joint work with Bing Wang.

### Probability Seminar, Tai Melcher, University of Virginia, REC 313

Wednesday, Mar 6 1:30 pm - 2:20 pm

Title: Teaching Discussion -- Abstract: This weekly teaching discussion is a place to discuss our practice in the classroom. All are welcome and encouraged to participate, from experienced instructors who have seen it all to those who are just starting.

## Three Weeks

## March

### Math Colloquium, Magda Peligrad, University of Cincinnati, REC 122

Tuesday, Mar 19 3:30 pm - 4:30 pm

TBA

### Probability Seminar, Gennady Samorodnitsky, Cornel University, REC 313

Wednesday, Mar 27 1:30 pm - 2:00 pm

## April

Title: A-Hypergeometric Systems & D-Module Functors

### CCAM Lunch Seminar, Prof. Olivier Goubet, Universite de Picardie Jules Verne, REC 307

Friday, Apr 5 11:30 am - 12:30 pm

TBA

Direct & Inverse Problems in Elastic Scattering

### CCAM Seminar, Prof. Daniele Venturi, University of California, Santa Cruz, REC 113

Monday, Apr 8 4:30 pm - 5:30 pm

### Math Colloquium, Juncheng Wei, University of British Columbia, REC 122

Tuesday, Apr 9 3:30 pm - 4:30 pm

TBA

TBA

**Title: Numerical Approximations of Fractional Operators using Dunford-Taylor Representations****Abstract:** We review numerical algorithms based on Dunford-Taylor representations of fractional diffusion problems with a particular emphasis on their analysis and implementations.

In the case of spectral fractional powers of an elliptic operator, a representation of the solution is obtained in term of an improper integral involving solutions to auxiliary, parameter dependent, reaction-diffusion problems.

The improper integral is approximated using an exponentially convergent SINC quadrature method.

At each quadrature point, a standard finite element method is advocated to approximates the independent auxiliary problems.

The method is easily parallelizable and consists of a straightforward modification of standard finite element methods for reaction-diffusion problems.

For the integral fractional laplacian, the Dunford-Taylor integral representation is instrumental to derive novel variational formulations.

As in the spectral case, SINC quadrature formulas coupled with finite element discretizations on parameter dependent truncated domains are put in place.

This yields a non-conforming method where the action of the stiffness matrix on a vector is approximated (sometimes referred to as a matrix free approach). The efficiency of the method is illustrated in three dimensions.

We then focus on several applications involving either fractional operators and demonstrate how their proposed approximations yield tractable numerical methods even for the approximation of complex systems.

### IDSI Distinguished Lecture, Subbarao Kambhampati (Rao), Arizona State University, LWSN 1142

Thursday, Apr 18 1:30 pm - 2:30 pm

**Rise of AI & The Challenges of Human-Aware AI Systems**

I will start with a perspective on the status and recent progress in AI, and the heightened public expectations surrounding it, with the aim of separating hype from technical reality, and explicating the complementary strengths of data-based and model-based approaches to AI. I will then focus on our ongoing research on designing AI systems that can interact and collaborate fluidly with humans, including modeling the mental states of humans in the loop, recognizing their desires and intentions, providing proactive support, exhibiting explicable behavior, giving cogent explanations on demand, and engendering trust. I will summarize the progress we have made so far on tackling the challenges raised by such human-aware AI systems.

### PDE Seminar, Professor Xiaochun Li, University of Illinois, Urbana-Champaign, REC 122

Thursday, Apr 18 3:30 pm - 4:20 pm

TBA

### Math Colloquium, Fabrice Baudoin, University of Connecticut, REC 122

Tuesday, Apr 23 3:30 pm - 4:30 pm

TBA