Departmental Colloquium
Spring 2021
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 19 |
Kristin DeVleming
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A foundational goal in algebraic geometry is the classification of all varieties, which are locally vanishing loci of polynomials. To make this problem tractable, one fixes discrete invariants like dimension or genus of the variety, and then studies continuous families of varieties with those invariants. The parameters for those continuous families are called moduli, and the associated parameter space is called a moduli space. Classically, there are many techniques to construct moduli spaces and compactifications of moduli spaces, from geometric invariant theory to the minimal model program, and recently there has been great progress constructing them using K-stability. I will motivate each of these techniques and ultimately focus on comparing the different results. There will be many pictures and examples! Fall 2020
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January 21 |
Man-Wai Cheung
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Mikhalkin established the correspondence between holomorphic curves and tropical curves on toric surfaces. Tropical curve (disk) counts are then seen as the algebro-geometric analogue holomorphic disk countings in mirror symmetry. The invariants are heavily used in the mirror construction. On the other hand, spaces of quiver representations have been extensively studied in the representation theory world. Together with Travis Mandel, we have linked up these two appearingly independent areas together and developed an expression of tropical disks countings in terms of quiver representations.
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January 26 |
Harold Blum
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Algebraic geometry is focused on the study of algebraic varieties, which are shapes cut out by polynomial equations. A fundamental problem in the field is to construct and study spaces that parametrize algebraic varieties of a given type, referred to as moduli spaces. The quintessential example of such a parameter space is the moduli space of curves, which is a space whose points are in bijection with isomorphism classes of Riemann surfaces. After surveying these ideas, I will discuss the problem of constructing moduli spaces parametrizing Fano varieties, which are a class of complex manifolds that form one of the three main building blocks of varieties in algebraic geometry. A new approach to this problem relies on the notion of K-stability, which is an algebraic definition introduced by differential geometers to characterize when a Fano variety admits a certain canonical metric.
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January 28 |
Junliang Shen
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Topology of Hitchin's integrable systems and character varieties play important roles in many branches of mathematics. In 2010, de Cataldo, Hausel, and Migliorini discovered a surprising phenomenon which relates these two very different geometric objects in an unexpected way. More precisely, they predict that the topology of Hitchin systems is tightly connected to Hodge theory of character varieties, which is now called the "P=W" conjecture. In this talk, we will discuss recent progress of this conjecture. In particular, we focus on general interactions between topology of Lagrangian fibrations and Hodge theory in hyper-Kähler geometries. This hyper-Kähler viewpoint sheds new light on both the P=W conjecture for Hitchin systems and the Lagrangian base conjecture for compact hyper-Kähler manifolds.
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February 2 |
Tony Yue Yu
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Mirror symmetry is one of the most mysterious dualities in mathematics. My research explores a new approach to mirror symmetry, via non-archimedean geometry. I will give an overview of the recent progress in this direction. I will talk about non-archimedean enumerative geometry, the Frobenius structure conjecture, and derived non-archimedean geometry. I will also describe an application towards cluster algebras in representation theory, and another application towards the moduli spaces of Calabi-Yau pairs and their compactifications.
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February 18 |
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The Black Lives Matter movement, and many like it, has garnered widespread support for dismantling the racist structures woven into the fabric of our society at large. The academic discipline of mathematics--alongside many institutions of higher education--has also reached a point of reckoning in its history of institutionalizing racism. We must acknowledge the necessity, not choice, of persistent and active anti-racist work in realizing transformative, long-lasting change. In this talk, I will discuss the ‘Five Ws and How’ for anti-racism as my vision for the mathematical community.
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February 25 |
Frank Calegari
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Choose a random polynomial with integer coefficients, say x^3 + 1 or x^5 + x + 1. If you choose a prime number p, what is the probability that the values of these polynomials will be squares modulo p? Half the time? More than half the time? Less? Some of these elementary questions ultimately lead to difficult problems in the Langlands program. The goal of this colloquium is to show how recent advances in the Langlands program can provide new answers to these questions, and how they are linked to a possible notion of a “Galois group” of systems of polynomial equations in higher dimensions.
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March 4 |
Jinchao Xu
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In this talk, I will first give an elementary introduction to models and algorithms from two different fields: (1) machine learning, including logistic regression and deep neural networks, and (2) numerical PDEs, including finite element and multigrid methods. I will then explore mathematical relationships between these different models and algorithms and demonstrate how such relationships can be used to understand, study and improve different aspects of deep neural networks, finite element and multigrid methods. We will show that ReLU-DNN corresponds exactly to the traditional piecewise linear finite functions and [ReLU]^k-DNN leads to new finite element of piecewise polynomials of degree k with remarkable approximation properties. We will demonstrate how a new convolutional neural network (CNN), known as MgNet, can be derived by making very minor modifications of a classic geometric multigrid method for the Poisson equation and then discuss the theoretical and practical potentials of MgNet.
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March 11 |
Gilad Lerman
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The problem of group synchronization asks to recover states of objects associated with group elements given possibly corrupted relative state measurements (or group ratios) between pairs of objects. This problem arises in important data-related tasks, such as structure from motion, simultaneous localization and mapping, Cryo-EM, community detection and sensor network localization. Two common groups in these problems are the rotation and symmetric groups. We propose a general framework for group synchronization with compact groups. The main part of the talk discusses a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios. Under our mathematical model of adversarial corruption, it can be used to infer the corrupted group ratios and thus to solve the synchronization problem. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish competitive theoretical results under a uniform corruption model. Finally, we discuss the MPLS (Message Passing Least Squares) or Minneapolis framework for solving real scenarios with high levels of corruption and noise and with nontrivial scenarios of corruption. We demonstrate state-of-the-art results for two different problems that occur in structure from motion and involve the rotation and permutation groups.
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March 16 |
Ila Varma
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Malle's conjecture can be thought of as a generalization of the inverse Galois problem, which asks for every finite group G, is there a number field K such that their Galois group over Q is isomorphic to G? Although open, this question is widely believed to be true, and Malle went further to predict the asymptotics of how many number fields there are with a given Galois group that only depended on the group structure of G and the degree of the number field. In this talk, we will discuss the history as well as recent results and techniques surrounding these conjectures.
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April 1 |
Tatiana Toro
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In this talk we will present an area of analysis that is concerned with the relationship between differential operators, the properties of their solutions, and the geometry of the domain on which they are considered. The goal is to highlight how analytic properties of solutions to PDEs determine the geometry of the domain where they are considered. The tools used in this area come from analysis of partial differential equations, harmonic analysis and geometric measure theory.
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April 6 |
Ryan Hynd
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A curve of constant width has the property that any two parallel supporting lines are the same distance apart in all directions. A fundamental problem involving these curves is to find one which encloses the smallest amount of area for a given width. This problem was resolved long ago and has a few relatively simple solutions. The volume minimizing shapes in three-dimensions have yet to be determined. We’ll discuss the two shapes which are conjectured to be volume minimizing and an approach to this outstanding problem.
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April 8 |
Aaron Pollack
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Classical holomorphic modular forms are intensely studied objects. They are generalized to automorphic forms, which are associated to reductive groups G. When G is an exceptional group, such as E_8, one can ask if there are special automorphic forms on G, which can in some ways take the place of the classical holomorphic modular forms. Gan, Gross, Savin, and Wallach singled out a class of such automorphic forms, called the quaternionic modular forms. I will explain what these objects are, and what is known about them.
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April 22 |
Greg Forest
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Insights into the mechanisms and dynamics of human respiratory tract (HRT) infections from the SARS-CoV-2 virus can inform public awareness as well as guide medical prevention and treatment for COVID-19 disease. Yet, the complex physiology of the human lung and the inability to sample diverse regions of the HRT pose fundamental roadblocks, both to discern among potential mechanisms for infection and disease and to monitor progression of infection. My group has explored lung biology and disease for over 2 decades in an effort called the UNC Virtual Lung Project, spanning many disciplines. We further explored how viruses “traffic” in mucosal barriers coating human organs, including the upper and lower respiratory tract, for the last decade, focusing on natural and synthetic antibody protection. Then along came the novel coronavirus SARS-CoV-2, for which we have no immune protection, requiring a step back to a pre-immunity scenario. We developed a computational model that incorporates: detailed physiology of the HRT, and best current knowledge about the mobility of SARS-CoV-2 virions in airway surface liquids (ASL) as well as epithelial cell infectability and replication of infectious virions throughout the HRT. The model simulates outcomes from any dynamic deposition profile of SARS-CoV-2 throughout the HRT, and tracks the propagation of infectious virions in the ASL and infected epithelial cells. We focus this lecture on two clinical observations, their respective likelihoods, and open questions raised: an upper respiratory tract infection following inhaled exposure to SARS-CoV-2; and, progression to alveolar pneumonia. Our baseline modeling platform is poised to superimpose interventions, from adaptive immune responses to any form of medical or drug treatment, at any point from pre-exposure to disease progression, with several new collaborations to do so. The results presented highlight the urgency to understand the underlying physical and physiological conditions that facilitate transmission, including self-transmission, which we absolutely do not yet understand.
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