# 2023-2024 Mathematics Colloquium

*Upcoming Talks:*

### TBA

*Recent Talks:*

**April 25** - 3:30pm - SDRICH 207

#### Helge Holden (Norwegian University of Science and Technology)

**Title:**On the Abel Prize – one of the most prestigious prizes in mathematics**Abstract:**The Abel Prize was awarded for the first time in 2003, and has in short time established itself as one of the most prestigious prizes in mathematics. In this talk we will give the background for the prize, and what separates it from other prizes in mathematics. We will present the accomplishments of some of the laureates.

**April 18** - 3:30pm - SDRICH 207

#### Tiago Picon (University of Sao Paulo)

**Title:**Local Lebesgue solvability of elliptic and canceling linear differential equations with measure data**Abstract:**In this talk, we present sufficient conditions for the local Lebesgue solvability of the equation*A*^{∗}(*x*,*D*)*f*= μ with Borel measure data associated to an elliptic linear differential operator*A*(*x*,*D*) of order*m*with smooth variable coefficients. Our method is based on the measures satisfying finite strong (*m*,*p*)−energy in order to obtain local solutions*f*∈*L*for 1 ≤^{p}*p*< ∞. The solvability for the endpoint case*p*= ∞ is studied in the setting of elliptic and canceling operators as a consequence of new local*L*^{1}estimates on measures for a special class of vector fields.

This is joint work with Victor Biliatto (UFSCar - Brazil).

**March 21 **- 3:30pm - SDRICH 207

#### Seick Kim (Yonsei University)

**Title:**On elliptic and parabolic PDEs in double divergence form**Abstract:**We consider an elliptic, double divergence form operator L^{*}, which is the formal adjoint of the elliptic operator in non-divergence form L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L^{*}u=0 and show that Schauder type estimates are available when the coefficients are of Dini mean oscillation and belong to certain function spaces. We will also discuss some applications and parabolic counterparts.

**March 14** - 3:30pm - SDRICH 207

#### Joe Conlon (University of Michigan)

**Title:**Some simple models of Ostwald Ripening**Abstract:**Ostwald Ripening is a phenomenon observed in solid solutions and liquid gels that involves the evolution of inhomogeneous structures over time. Small crystals or sol particles first dissolve and then redeposit onto larger crystals or sol particles. A quantitative theory of Ostwald ripening was proposed in 1961 by Lifschitz-Slyozov and independently by Wagner (LSW theory). In 1935 Becker-Doring (BD) proposed a model to describe a variety of phenomena in the kinetics of phase transition, including metastability, nucleation and coarsening. In 1997 Oliver Penrose argued that the large time behavior of the BD model is well described by the LSW model. This talk will be concerned with the mathematical theory of the BD and LSW models, how they are related and connections with the Fokker-Planck equation.

**February 1** - 3:30pm - SDRICH 207

#### Dmitry Ryabogin (Kent State University)

**Title:**Some problems related to floating bodies**Abstract:**Assume that all sections of an origin-symmetric convex body K in R^{n}, n≥3, have a symmetry of the cube (of the square for n=3). Does it follow that K is a Euclidean ball? We will discuss this and other problems of uniqueness related to symmetries of sections and projections of convex bodies and to floating bodies.

**January 25** - 3:30pm - SDRICH 344 (Baylor Lecture Series)

#### Gigliola Staffilani (MIT)

**Title:**The Schrödinger equations as inspiration of beautiful mathematics**Abstract:**In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on to the concept of energy transfer and its connection to dynamical systems, and I will end with some results following from viewing the periodic nonlinear Schrödinger equation as an infinite dimensional Hamiltonian system.

**November 2** - 3:30pm - SDRICH 207

#### Jussi Behrndt (Technical University of Graz)

**Title:**The Landau Hamiltonian with delta-potentials supported on curves**Abstract:**The spectral properties of a singularly perturbed self-adjoint Landau Hamiltonian in the plane with a delta-potential supported on a finite curve are studied. After a general discussion of the qualitative spectral properties of the perturbed Landau Hamiltonian and its resolvent, one of our main objectives is a local spectral analysis near the Landau levels. This talk is based on joint works with P. Exner, M. Holzmann, V. Lotoreichik, and G. Raikov.

**October 19 - **3:30 PM

#### Edmund Chiang (The Hong Kong University of Science and Technology)

**Title:**Recent development of complex function theory with respect to difference operators**Abstract:**The search for discrete integrable systems (Ablowitz, Halburd & Herbst) prompted researchers to discover previously unknown complex analytic structures for holomorphic functions. Indeed, not only several Little Picard theorems with respect to difference operators have been discovered as consequences, but full-fledged difference versions of Nevanlinna theory have been established. We plan to give an overview on this recent development that connects the areas of integrable systems, special functions and holomorphic functions. We also discuss some fundamental obstacles in understanding nature of holomorphic functions with respect to difference operators may lie in appropriate algebraic interpretations.

Click here to see a list of past colloquium talks.