Time & Location: All talks are on Thursdays in Gibson Hall 126A at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.
Organizer: Gustavo Didier
Tuesday, January 14
Title: From Zariski-Nagata to local fundamental groups
Jack Jeffries - Mathematics Research Center, Mexico
Hilbert's Nullstellensatz gives a dictionary between algebra and geometry; e.g., solution sets to polynomial equations over the complex numbers (varieties) translate to (radical) ideals in polynomial rings. A classical theorem of Zariski-Nagata gives a deeper layer to this correspondence: polynomial functions that vanish to certain order along a variety correspond to a natural algebraic notion called symbolic powers.
In this talk, we will explain this theorem, and then pursue a couple of variations on this theme. First, we will consider how the failure of this theorem over ambient spaces with bends and corners allows us to study the geometry of such spaces; in particular, we will give bounds on size of local fundamental groups. Second, we will consider what happens when we replace the complex numbers by the integers; we will show that "arithmetic differential geometry" (in the sense of Buium) allows us to obtain a Zariski-Nagata theorem in this setting. Only a passing familiarity with polynomials and complex numbers is assumed.
This is based on joint projects with Holger Brenner, Alessandro De Stefani, Eloísa Grifo, Luis Núñez-Betancourt, and Ilya Smirnov.
Location: Stanley Thomas 316
Tuesday, January 21
Title: Math in the lab: mass transfer through fluid-structure interactions
Jinzi Mac Huang - University of California San Diego
The advancement of mathematics is closely associated with new discoveries from physical experiments. On one hand, mathematical tools like numerical simulation can help explain observations from experiments. On the other hand, experimental discoveries of physical phenomena, such as Brownian motion, can inspire the development of new mathematical approaches. In this talk, we focus on the interplay between applied math and experiments involving fluid-structure interactions -- a fascinating topic with both physical relevance and mathematical complexity. One such problem, inspired by geophysical fluid dynamics, is the experimental and numerical study of the dissolution of solid bodies in a fluid flow. The results of this study allow us to sketch mathematical answers to some long standing questions like the formation of stone forests in China and Madagascar, and, how many licks it takes to get to the center of a lollipop. We will also talk about experimental math problems at the micro-scale, focusing on the mass transport process of diffusio-phoresis, where colloidal particles are advected by a concentration gradient of salt solution. Exploiting this phenomenon, we see that colloids are able to navigate a micro-maze that has a salt concentration gradient across the exit and entry points. We further demonstrate that their ability to solve the maze is closely associated with the properties of a harmonic function – the salt concentration.
Location: Stanley Thomas 316
Title: Hessenberg varieties and the Stanley--Stembridge conjecture
Martha Precup - Washington U in St Louis (Host: Mahir Can)
Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. In 2015, Brosnan and Chow proved the Shareshian-Wachs conjecture, linking the Stanley-Stembridge conjecture in combinatorics to the geometry of Hessenberg varieties through Tymoczko's permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give an overview of that story and present a new set of linear relations satisfied by the multiplicities of certain permutation representations in Tymoczko's representation. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts. This is joint work with M. Harada.
Friday, January 24
Title: Modeling and simulation of symmetry breaking in cells
Calina Copos - New York University
In order to initiate movement, cells need to form a well-defined "front" and "rear" through the process of cellular polarization. Polarization is a crucial process involved in embryonic development and cell motility and it is not yet well understood. Mathematical models that have been developed to study the onset of polarization have explored either biochemical or mechanical pathways, yet few have proposed a combined mechano-chemical mechanism. However, experimental evidence suggests that most motile cells rely on both biochemical and mechanical components to break symmetry. We have identified one of the simplest quantitative frameworks for a possible mechanism for spontaneous symmetry breaking for initiation of cell movement. The framework relies on local, linear coupling between minimal biochemical stochastic and mechanical deterministic systems; this coupling between mechanics and biochemistry has been speculated biologically, yet through our model, we demonstrate it is a necessary and sufficient condition for a cell to achieve a polarized state.
Location: Stanley Thomas 316
Tuesday, January 28
Title: Quantum and symplectic invariants in low-dimensional topology.
Nathan Dowlin - Columbia University
Khovanov homology and knot Floer homology are two powerful knot invariants developed around two decades ago. These invariants have been applied to problems all over low-dimensional topology, from detecting exotic smooth structures on 4-manifolds to determining whether a given knot diagram is the unknot. Knot Floer homology is defined using symplectic techniques, while Khovanov homology has its roots in the representation theory of quantum groups. Despite these differences, they seem to have many structural similarities. A well-known conjecture of Rasmussen from 2005 states that for any knot K, there is a spectral sequence from the Khovanov homology of K to the knot Floer homology of K. Using a new family of invariants defined using both quantum and symplectic techniques, I will give a proof of this conjecture and describe some topological applications.
Location: Stanley Thomas 316
Wednesday, January 29
Title: Numerical methods for ocean models and venous valve simulations
Sara Calandrini - Florida State University
Location: Stanley Thomas 316
Title: Tropical Algebraic Geometry
Kalina Mincheva - Yale University
Abstract: Tropical geometry provides a new set of purely combinatorial tools to approach classical problems in algebraic geometry. The fundamental objects in tropical geometry are tropical varieties -- combinatorial ``shadows" associated to more traditional geometric objects, algebraic varieties. Until recently, the theory has focused on the geometric aspects of tropical varieties as opposed to the underlying algebra, largely due the lack of tropical analogues to commutative algebra tools. Consequently, there has recently been a lot of effort dedicated to developing such tools using different frameworks -- notably prime congruences, tropical ideals, and tropical schemes. These approaches allow for the exploration of tropical spaces as inherently tropical objects. In this talk, we present a notion of prime congruences and discuss the resulting analogues to classical theorems, such as a Nullstellensatz and aspects of dimension theory. We also demonstrate connections to algebraic geometry via the theory of tropical schemes and ideals.
Title: Reproducible Bootstrap Aggregating
Meimei Liu - Duke University
Heterogeneity between training and testing data degrades reproducibility of a well-trained predictive algorithm. In modern applications, how to deploy a trained algorithm in a different domain is becoming an urgent question raised by many domain scientists. In this paper, we propose a reproducible bootstrap aggregating (Rbagging) method coupled with a new algorithm, the iterative nearest neighbor sampler (INNs), effectively drawing bootstrap samples from training data to mimic the distribution of the test data. Rbagging is a general ensemble framework that can be applied to most classifiers. We further propose Rbagging+ to effectively detect anomalous samples in the testing data. Our theoretical results show that the resamples based on Rbagging have the same distribution as the testing data. Moreover, under suitable assumptions, we further provide a general bound to control the test excess risk of the ensemble classifiers. The proposed method is compared with several other popular domain adaptation methods via extensive simulation studies and real applications including medical diagnosis and imaging classifications.
Location: Hebert Hall 213
Title: How Can Mathematics Save the Honeybees?
Yun Kang - Arizona State University (Host: James Hyman)
The honeybee is crucial in maintaining biodiversity by pollinating 85% of the world’s plant species. This bee is the most economically valuable pollinator of agricultural crops worldwide. Recently the Varroa mite has infected honeybee hives and caused sharp declines in honeybee populations, resulting in a global crisis. I will demonstrate how we develop tractable mathematical models to clarify the principal mechanisms responsible for colony growth dynamics and survival in a dynamic environment. The mathematical analysis of these models can help us understand the crucial feedback mechanisms linking disease, parasitism, nutrition, and foraging behavior. We consider both nonlinear nonautonomous and delayed differential equations. The models are integrated with data to create a metapopulation framework for exploring the contributing factors to the mysterious and dramatic loss of honeybees. We use numerical simulations to identify new strategies for controlling Varroa, reducing colony losses for beekeepers, and maximizing the benefits for land managers.
Title: A probabilistic framework for models of dependent network data, with statistical guarantees
Jonathan Stewart - Rice University
The statistical analysis of network data has attracted considerable attention since the turn of the twenty-first century, fueled by the rise of the internet and social networks and applications in public health (e.g., the spread of infectious diseases through contact networks), national security (e.g., networks of terrorists and cyberterrorists), economics (e.g., networks of financial transactions), and more. While substantial progress has been made on exchangeable random graph models and random graph models with latent structure (e.g., stochastic block models and latent space models), these models make explicit or implicit independence or weak dependence assumptions that may not be satisfied by real-world networks, because network data are dependent data. The question of how to construct models of random graph with dependent edges without sacrificing computational scalability and statistical guarantees is an important question that has received scant attention.
In this talk, I present recent advancements in models, methods, and theory for modeling networks with dependent edges. On the modeling side, I introduce a probabilistic framework for specifying edge dependence that allows dependence to propagate throughout the population graph, with applications to brokerage in social networks. On the statistical side, I obtain the first consistency results in settings where dependence propagates throughout the population graph and the number of parameters increases with the number of population members. Key to my approach lies in establishing a direct link between the convergence rate of maximum likelihood estimators for exponential families and the scaling of the Fisher information matrix. Last, but not least, on the computational side I demonstrate how the conditional independence structure of models can be exploited for local computing on subgraphs, facilitating development of parallel computing algorithms for multi-core computers or computing clusters.
Location: Hebert Hall 212
Title: Symbolic powers
Eloísa Grifo - UC Riverside (Host: Tai Ha)
The main goal of this talk it to introduce symbolic powers and discuss what the main open problems in this area are. Symbolic powers arise naturally from the theory of primary decomposition, an extension of the fundamental theorem of arithmetic. These are algebraic objects that also contain geometric information. Hilbert's Nullstellensatz gives a dictionary between algebra and geometry: solution sets to polynomial equations over the complex numbers (varieties) translate to (radical) ideals in polynomial rings. A classical theorem of Zariski and Nagata gives a deeper layer to this correspondence: polynomial functions that vanish up to a certain order along a variety correspond to symbolic powers.
Title: Group actions and finite free complexes over polynomial rings.
Srikanth B Iyengar - University of Utah (Host: Tai Ha)
This talk will be about various results (some of recent vintage) and conjectures concerning finite free complexes over polynomial rings. Many of these concern numerical invariants associated with such a complex; notably, the length of the complex, and the ranks of the free modules that appear in it. This thread of research can be traced back to Hilbert's Syzygy Theorem (1890) that states that each finitely generated module over a polynomial ring over a field has a finite free resolution. The modern developments in this subject started with the work of Auslander, Buchsbaum, and Serre in the 1950s, and have since then been a centerpiece in commutative algebra. Another impetus for the subject has come from results and conjectures of Adem, Browder, Carlsson, Halperin, and Swan, among others, on obstructions to groups acting freely on spaces.
Title: Confessions of an experimental mathematician: on computer-assisted proofs, moving sofas, and other curiosities
Dan Romik - UC Davis (Host: Victor Moll)
Many of my research projects began with coding a simulation or visualization of a mathematical process or concept and discovering interesting conjectures, which in some cases I ended up proving. In this talk I will show a few such simulations, involving objects such as Young tableaux, random sorting networks, moving sofas, and more. The stories of the rigorous results that resulted from these simulations will provide interesting lessons on the role that experimental mathematics can play in modern math research
Frederi Viens - Michigan State (Host: Glatt-Holtz)
Fabrizio Zanello - Michigan Tech (Host: Tai Ha)