Week of February 1, 2026

Mon Feb 2, 2026
12:00pm to 1:00pm - RH 340N - Mathematical Physics
Netanel Levi - (UCI)
Dynamical and Dimensional Properties of Schrödinger Operators Under Finite-Rank Perturbations

Abstract: In this lecture, we will present several dynamical and fractal-dimensional ways of characterizing the spectral measures of Schrödinger operators, such as Rajchman behavior and Hausdorff/packing dimensions, and discuss the extent to which these properties are stable under rank-one perturbations.
We begin with the concrete setting of half-line Schrödinger operators, where a theorem of Gordon shows that generic rank-one perturbations eliminate pure point spectrum, ruling out the most extreme dynamical and dimensional behavior. I will then describe constructions demonstrating that properties only slightly weaker than pure point spectrum can, in fact, be entirely stable: for certain sparse half-line models, both packing-dimension-zero and non-Rajchman behavior persist for every rank-one perturbation.
In the second part, we examine how spectral dimensions behave when passing from the whole line to the half-line. I will present an operator whose spectral measure on the line has Hausdorff dimension one, whereas every half-line restriction - under any boundary condition - has dimension zero, even though the two settings differ only by a finite-rank perturbation.
4:00pm to 5:00pm - RH 340N - Geometry and Topology
Dave Auckly - (Kansas State University)
Smoothly knotted surfaces in small closed 4-manifolds.

It has long been known that homeomorphic 4-manifolds may admit inequivalent smooth structures. Analogous behavior holds for embedded surfaces. Some surfaces are topologically isotopic without being smoothly isotopic. Such surfaces are said to be smoothly knotted. 

 

It turns out that it is easier to construct inequivalent smooth structures on larger 4-manifolds. Similarly, it is easier to construct closed, smoothly knotted surfaces in large 4-manifolds. 

 

In this talk, we will explain how to construct smoothly knotted surfaces in a small 4-manifold. This talk will have many pictures.

 

 

This is joint work with Konno, Mukherjee, Ruberman, and Taniguchi. 
4:00pm to 5:00pm - Virtual (Zoom) - Applied and Computational Mathematics
Elisa Negrini - (UCLA)
A Multimodal PDE Foundation Model for Prediction and Scientific Text Descriptions

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to train approximations to multiple differential equations simultaneously and are thus a general purpose solver that can be adapted to downstream tasks. Current PDE foundation models focus on either learning general solution operators and/or the governing system of equations, and thus only handle numerical or symbolic modalities. However, real-world applications may require more flexible data modalities, e.g. text analysis or descriptive outputs. To address this gap, we propose a novel multimodal deep learning approach that leverages a transformer-based architecture to approximate solution operators for a wide variety of ODEs and PDEs. Our method integrates numerical inputs, such as equation parameters and initial conditions, with text descriptions of physical processes or system dynamics. This enables our model to handle settings where symbolic representations may be incomplete or unavailable. In addition to providing accurate numerical predictions, our approach generates interpretable scientific text descriptions, offering deeper insights into the underlying dynamics and solution properties. The numerical experiments show that our model provides accurate solutions for in-distribution data (with average relative error less than 3.3%) and out-of-distribution data (average relative error less than 7.8%) together with precise text descriptions (with correct descriptions generated 100% of times). In certain tests, the model is also shown to be capable of extrapolating solutions in time.
Wed Feb 4, 2026
2:30pm to 3:30pm - ISEB 1200 - Applied and Computational Mathematics
Lisa Fauci - (Tulane University)
Flexible filaments and swimming cups: just go with the flow

The motion of waving or rotating filaments in a fluid environment is a common element in many biological and engineered systems. Examples at the microscale include chains of diatoms moving in the ocean, flagella of of individual cells comprising multicellular microbial colonies, as well as engineered helical nanorobots designed to deliver drugs to tumors. Complex fluid environments, such as networks of polymers, can have dramatic effects upon the dynamics of microorganisms as they move through mucus or tissues. In this talk we will present mathematical and computational insights into these viscosity-dominated flows. Our modeling approaches will vary from detailed models that capture flagellar material properties and wave geometry to minimal forcedipole models that represent a flagellum by a single point. We will investigate a few intriguing systems, including helical filaments that penetrate, break, and move through a polymeric network, the journey of extremely long insect sperm flagella through tortuous female reproductive tracts, and the hydrodynamic performance of shape-shifting Choanoeca flexa colonies.

Bio: Dr. Lisa Fauci received her PhD from the Courant Institute of Mathematical Sciences at New York University, and directly after that joined the Department of Mathematics at Tulane University in New Orleans, Louisiana, USA. Her research focuses on biological fluid dynamics, with an emphasis on using modeling and simulation to study the basic biophysics of organismal locomotion and reproductive mechanics. Lisa served as president of the Society for Industrial and Applied Mathematics (SIAM) in 2019-2020. She is a fellow of SIAM, the American Mathematical Society, the Association for Women in Mathematics, and the American Physical Society. In 2023, she was elected to the US National Academy of Sciences.

Note: for Fred and Julia Wan Distinguished Lecture in Mathematical Biology.
Thu Feb 5, 2026
3:00pm to 4:00pm - RH 306 - Number Theory
Katy Woo - (Stanford)
The distribution of prime values of random polynomials

The Bateman--Horn Conjecture predicts how often an irreducible polynomial  assumes prime values. We will discuss how with sufficient averaging in the coefficients of the polynomial (exponential in the size of the inputs), one can not only prove Bateman--Horn results on average but also pin down precise information about the distribution of prime values at finite but growing scales. We will prove that 100% of polynomials satisfy the appropriate analogue of the Poisson Tail Conjecture, in the sense that the distribution of the gaps between consecutive prime values around the average spacing is Poisson.

 

We will also study the frequencies of sign patterns of the Liouville function evaluated at the consecutive outputs of f; viewing f as a random variable, we establish the limiting distribution for every sign pattern. 
A key input behind all of our arguments is Leng's recent quantitative work on the higher-order Fourier uniformity of the von Mangoldt and M\"obius functions (in turn relying on Leng, Sah, and Sawhney's quantitative inverse theorem for the Gowers norms).

 

This talk is based on joint work with Noah Kravitz and Max Xu. 
Fri Feb 6, 2026
4:00pm to 4:50pm - NS 1201 - Distinguished Lectures
Menachem Magidor - (Hebrew University of Jerusalem)
Regularity Properties of Subsets of the Real Line and Other Polish Spaces

Using the axiom of choice one can construct sets of reals which are pathological in some sense. Similar constructions can be produce such ”pathological” subsets of any non trivial Polish space (= a complete separable metric space). Typical examples of ”pathology” is non mea- surability, lacking the property of Baire (=not equivalent to an open set modulo meager set), contradicting an infinitary versions of Ramsey theorem , etc.
Another important example is the non determinacy of infinitary games. A subset of the Baire space NN is ”pathological” if in the associated game no player has a winning strategy.

A prevailing paradigm in Descriptive Set Theory is that sets that has a ”simple description” should not be pathological. Evidence for this maxim is the fact that Borel sets are not pathological in any of the senses described above.In this talk we shall present a notion of ”super regularity” for subsets of a Polish space, the family of universally Baire sets.

The universally Baire sets typically do not show the ”pathologies” we listed above, especially if one assumes the existence of large cardi- nals.Also the large cardinals imply that the family of Universally Baire sets is much richer that the class of Borel sets.

The intuitive principle ”try to minimize the family of pathological set of reals ” is fruitful guiding principle for extending the set of axioms for Set Theory . It is connected to the guiding principle of strong axioms of infinity. We shall make some remarks how these principles can shed some light on the Continuum problem.

The talk should be accessible to a wide mathematical audience .