# Calendar

## Today

### Math Colloquium, Neshan Wickramasekera, University of Cambridge, REC 122

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

**Title: Recent advances in the regularity theory of hypersurfaces with prescribed ****mean curvature.**

Abstract: The notion of a generalized solution is a familiar concept in the theory of partial differential equations. For much the same reasons (existence of solutions, compactness of families of solutions etc.), it is of interest to study generalized submanifolds of a Riemannian manifold that satisfy natural geometric constraints.

In a series of works in the past several years (most recent of which in separate joint projects with C. Bellettini and with C. Bellettini and O. Chodosh), a sharp regularity and compactness theory has been developed for uniformly area bounded stable prescribed-mean-curvature hypersurfaces. This work includes minimal (i.e. zero mean curvature) and constant mean curvature (CMC) hypersurfaces as important special cases, and it unifies and extends two celebrated theories for minimal hypersurfaces dating back to the period 1960-1980: one is the regularity theory for locally area minimizing hypersurfaces (due to De Giorgi, Federer and Simons) and the other is the compactness theory for area bounded stable minimal hypersurfaces with small singular sets (due to Schoen--Simon--Yau in low dimensions and Schoen--Simon in general dimensions). Both these theories played a key role in the Almgren--Pitts geometric approach to existence of minimal hypersurfaces in compact Riemannian manifolds, developed in the same period.

Among the applications of the recent work is a considerably simpler PDE alternative to the Almgren--Pitts theory (due to the combined work of Guraco, Hutchinson, Tonegawa and the speaker). We will discuss these old and recent developments and outline some future directions.

## Tomorrow

**Title: Sequentially Cohen-Macaulay modules and graded local cohomology tables.**

Abstract: We consider, along the lines of Boij-Soderberg theory, the problem of writing local cohomology tables as finite sums, with positive rational coefficients, of tables from an explicit list. We show how this can be easily done for sequentially Cohen-Macaulay modules. We then extend previous work of De Stefani and Smirnov by considering modules with large E-depth, which is a weaker condition than being sequentially Cohen-Macaulay. Finally we show how the E-depth can be computed using partial generic initial ideals. This is joint work with Alessandro De Stefani.

**Title: New Directions in Cryptography.**

Abstract: In 1976, Whit Diffie and Martin Hellman published their seminal paper titled New Directions in Cryptography, in which they described a method of exchanging keys over an unsecure channel.** **In this talk we will discuss variations on a Diffie-Hellman Key Exchange melody, including how it was originally proposed, how it is used today using elliptic curves, and how it may be used in the future in a post-quantum world.

The only assumed knowledge to understand this talk is basic group theory and a little bit of ring theory (basically just the definition of an ideal). In particular, no knowledge of cryptography is assumed.

## Thursday

### Mathematical Physics Seminar, Anna Vershynina, University of Houston, UNIV 319

Thursday, Feb 21 1:30 pm - 2:30 pm

**Title: Stability of the recovery map and the quantum quasi relative entropy.**

Abstract: Quantum relative entropy can be viewed as a kind of a measure of a distance between two quantum states. The larger it gets, the easier it is to distinguish two states. One of the most fundamental quantum information inequality is the data processing inequality, which states that the relative entropy can not increase

after states pass through a quantum channel. In 1986 Petz showed that there is no decrease in the relative entropy, if and only if, it is possible to perfectly recover both states with which is now known, as the Petz recovery map. The standing goal in this area is to provide a universal and easily computable quantitative bound in data processing inequality. Based on the joint work with E. A. Carlen, I will provide the most elegant sharpening of the data processing inequality so far. It is evident that if the change in relative entropy between two states after passing through a quantum channel is small, the original Petz recovery map recovers one state perfectly, and the other approximately well. I will also provide related results for various quasi-relative entropies.

Title: Regularity of absolute minimizers for continuous convex Hamiltonians Abstract: For any $n\ge 2$, $\Omega\subset R^n$, and any given convex and coercive Hamiltonian function $H\in C^{0}(R^n)$, we find an optimal sufficient condition on $H$, that is, for any $c\in\mathbb R$, the level set $H^{-1}(c)$ does not contains any line segment, such then any absolute minimizer enjoys the linear approximation property. As consequences, we show that when $n=2$, if any absolute minimizer $u\in C^1$; and if $u$ is an absolute minimizer in $R^2$ satisfies a linear growth at the infinity, then $u$ is a linear function. In particular, if $H$ is a strictly convex Banach norm $R^2$, e.g. the $l_\alpha$-norm for $1<\alpha<1$, then any absolute minimizer is $C^1$. The ideas of proof are, instead of PDE approaches, purely variational and geometric. This is a joint work with Feng Ya, and Yuan Zhou.

### Basic Skills Workshop, Anderson, Harris and Mariano, Purdue, BRNG 1260

Thursday, Feb 21 3:30 pm - 4:20 pm

**Title: Panel Discussion on Postdoctoral Careers.**

Abstract: Stressed out about finding a postdoctoral position after graduation? Our panel will discuss obtaining a postdoctoral job. Our three speakers will open the discussion by talking about their career paths, and then we will open it up to questions from the audience. If you are curious about the process of searching for a postdoc position, how the applications process works at various universities, or even what a day in the life of a postdoc looks like; then Drs. Anderson, Harris, and Mariano encourage you to attend!

Light refreshments will be provided.

## Friday

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 itâ€™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.

### 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.

TBA

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

### 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

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

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

TBA

### 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

### 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