feldman (at) math.utah.edu),
Victoria Kala (victoria.kala (at) utah.edu),
or Akil Narayan (akil (at) sci.utah.edu)Monday, February 9 at 4pm. In-person LCB 222
Speaker: Héctor Chang-Lara
Área de Matemáticas Básicas, Centro de Investigación en Matemáticas
Title: Regularity Estimates for Zeroth Order Operators
Abstract: A very interesting limit case of the fractional Laplacian in R^d is given by
Lu(x) := \int_{B_1(x)} (u(x) - u(y))|y - x|^{-d} dy,
which serves as a principal example of a zeroth-order integro-differential operator. This operator arises naturally as the leading term of the logarithmic Laplacian which has been studied in recent years. In contrast with the fractional Laplacian, the scaling properties in this scenario are very delicate; in particular, the dilation of the kernel leads to a non-integrable tail, which represents a challenge for the regularity theory of solutions of equations governed by L. In this talk, I will present interior continuity estimates for solutions to a family of operators comparable to the one above, obtained in collaboration with Alberto Saldaña and Sven Jarosh.
Monday, March 2 at 4pm. In-person LCB 222
Speaker: Bogdan Raita
Department of Mathematics and Statistics, Georgetown University
Title: Solving Linear PDE by Machine Learning and Commutative Algebra
Abstract: We use the theory of linear pde systems with constant coefficients (Malgrange, Palamodov, Pommaret, Sturmfels) to implement a machine learning algorithm which generates solutions to arbitrary linear pdes. Since we preprocess the equations with computer algebra, our methods are applicable to arbitrary pde systems, irrespective of type (elliptic, hyperbolic, etc.) or order. We test our method for classical equations (wave, heat, Laplace) and discuss future applications to equations describing wave-related phenomena, for example direct and inverse problems involving Maxwell’s system and the elasticity equations.
Monday, April 6 at 4pm. In-person LCB 222
Speaker: Muamer Kadic
SUPMICROTECH
Title: Topological and active mechanical metamaterials
Abstract: Nonlinear mechanical metamaterials can exhibit emergent transport phenomena that mimic topological protection without relying on linear band topology. In this talk, we will discuss how we realize a bifurcation-induced nonreciprocal lattice that supports robust propagation of elastic kink waves [1]. Each unit is a prestrained, hinged-beam circulator that develops angular-momentum bias during snap-through transitions between buckling states, producing an effective breaking of time-reversal symmetry. Coupling such units into a hexagonal array yields a mechanically chiral network where localized soliton-like excitations propagate unidirectionally along interfaces and edges, immune to sharp bends. We demonstrate non-dispersive kink transport governed by a Sine-Gordon type field whose effective bias encodes mechanical chirality. This frame work bridges bifurcation dynamics and nonreciprocal transport, establishing a nonlinear route toward topological-like mechanical functionality without magnetic or gyroscopic bias. Finally, we will discuss how active mechanics can be controlled by optical waves and open a fully controllable field of space-time metamaterials [2].
[1] https://arxiv.org/abs/2602.04591
[2] https://journals.aps.org/prapplied/abstract/10.1103/ynnm-619n
Monday, April 13 at 4pm. In-person LCB 222
Speaker: Sebastian Munoz
Department of Mathematics, UCLA
Title: Singularities and regularity in the supercooled Stefan problem
Abstract: The supercooled Stefan problem is a classical model for the solidification of
water below its freezing temperature. Unlike the melting problem, the supercooled problem
is ill-posed: solutions may develop discontinuous freezing fronts, nucleation (the spontaneous
appearance of ice), and fractal freezing sets. In this talk, I will discuss recent results on the
structure and regularity of solutions in arbitrary dimensions. On one hand, for general
initial data, weak solutions can exhibit genuinely pathological free boundaries: the freezing
set may contain arbitrary prescribed fractals, and nucleation can occur even for smooth data.
On the other hand, maximal solutions - which delay solidification as much as possible - are
remarkably well-behaved: the free boundary is the graph of a C1 freezing time, the singular
set (where the front moves with infinite speed) satisfies sharp Hausdorff dimension bounds,
and in the subcritical regime, fractal freezing does not occur. I will discuss some of the ideas
behind these results. Based on joint work with Raymond Chu, Max Engelstein, and Inwon
Kim
[Special Seminar] Tuesday, April 14 at 4pm. In-person LCB 323
Speaker: Zhiyuan Geng
Department of Mathematics, Purdue
Title: Asymptotics for 2D vector-valued Allen-Cahn minimizers
Abstract: For the scalar two-phase Allen-Cahn equation, the celebrated De Giorgi conjecture has inspired a rich literature, revealing deep connections between diffuse interfaces and minimal surfaces. In the presence of three or more equally preferred phases, one must instead consider a vector-valued order parameter, and the diffuse interfaces are expected to resemble weighted minimal partitions.
In this talk, I will present recent results on minimizers of a 2D Allen-Cahn system with a multi-well potential. We describe the asymptotic behavior near the junction of three phases by analyzing the blow-up limit, which is a global minimizing solution converging at infinity to a Y-shaped minimal cone. A key ingredient in our approach is the derivation of sharp upper and lower energy bounds via a slicing argument, which implies the localization of the diffuse interface within a small neighborhood of the sharp interface. Consequently, we obtain a complete classification of global two-dimensional minimizers in terms of their blow-down limits at infinity.
feldman (at) math.utah.edu),
Victoria Kala (victoria.kala (at) utah.edu),
and
Akil Narayan (akil (at) sci.utah.edu).
Past lectures: Fall 2025, Spring 2025, Fall 2024, Spring 2024, Fall 2023, Spring 2023, Fall 2022, Spring 2022, Fall 2021, Spring 2021, Fall 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.