Date: March 12, 3:30 pm.
Place: JMH 405 (RH)
Speaker: Thomas Beck, Wen Li, Han-Bom Moon (Fordham)
Title: Summer Research Opportunity
Abstract: Are you interested in having some research experience this summer? Do you want to explore mathematics beyond classrooms? In this seminar, three math professors will present their summer projects with students. Please come and check the projects, and talk to us if you are interested! We are recruiting students! Here are the titles of the presentations.
Thomas Beck: Nodal Sets of Laplace Eigenfunctions
Wen Li: Physics-Informed Fundamental Kernel Network for Inverse Cauchy Problems
Han-Bom Moon: On Algebraic Space-Filling Curves
Everyone is welcome!
Date: April 2, 3 pm.
Place: JMH 405 (RH)
Speaker: Swarnava Mukhopadhyay (Tata Institute of Fundamental Research)
Title: Graph Potentials
Abstract: We introduce graph potentials, which are Laurent polynomials $L_{\Gamma}$ associated to (colored) trivalent graphs $\Gamma$. An interesting question is to compute the generating series $\sum_{i=0}^{\infty} L_{\Gamma}^{n}_{0}t^n$ of the constant term of the powers of $L_{\Gamma}$. The main goal of this talk is to answer the above question.
We first show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and we use this to define a topological quantum field theory. A similar construction was recently introduced independently by Kontsevich--Odesskii under the name of multiplicative kernels. As an application we give an efficient computational method to compute its partition function answering the main question. We will also discuss the origin of graph potentials as mirror to the Fano variety of the moduli space of rank two bundles on a curve with fixed determinant. This is a joint work with Pieter Belmans and Sergey Galkin.
Date: April 9, 3 pm.
Place: JMH 405
Speaker: Azita Mayeli (CUNY)
Title: What eigenvalues tell us about the time-frequency concentration of a function
Abstract: The Heisenberg uncertainty principle tells us that a function and its Fourier transform cannot both be sharply localized. This gives rise to the concentration problem: determining which functions, when restricted to a short time interval, can still be significantly concentrated in the frequency domain, and vice versa.
In one dimension, this question was extensively studied in the 1960s--1980s by Landau, Slepian, Daubechies, and Polak, who analyzed the eigenfunctions and eigenvalues of certain compact, self-adjoint localization operators. These eigenfunctions, known for their remarkable time-frequency concentration properties, play an important role in applications such as signal interpolation and signal recovery.
In this talk, I will begin by reviewing the one-dimensional theory in an accessible way. I will then explain how these ideas extend to higher dimensions, focusing on the case where the time domain is a cube and the frequency domain is either a cube or a ball. In these settings, we observe a clustering behavior in the eigenvalues of the associated operators, which captures the effective dimension of the corresponding concentrated function spaces. These results are based on two joint works with Arie Israel and Kevin Hughes.
If time permits, I will also discuss related results in the finite-dimensional setting, involving so-called prolate Toeplitz matrices. This part of the work is in collaboration with my undergraduate and master's students at CUNY.
Date: April 30, 3 pm.
Place: JMH 405
Speaker: Sijong Kwak (KAIST)
Title: Symmetric products of a curve, the gonality sequence and syzygies of secant varieties of a curve
Abstract: In this talk, I'd like to introduce Koszul cohomology method via symmetric products of a curve to understand the graded Betti numbers appearing in the minimal free resolution of curves and higher secant varieties of curves. In this direction, there is a well known conjecture, called "gonality conjecture" due to Green-Lazarsfeld. This conjecture has been proved by Aprodu-Voisin, Ein-Lazarsfeld and Farkas-Kemeny etc. We also consider "the generalized gonality conjecture" for higher secant varieties of a curve using the gonality sequence and show that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a joint work with J. Choe and J. Park.
Date: September 11, 3 pm.
Place: JMH 405 (RH)
Speaker: Marco Castronovo (Columbia)
Title: The Fukaya category of Fano manifolds with global symmetries
Abstract: The concept of manifold is one of the most influential in modern mathematics. In both algebraic and differential geometry major efforts have been made to understand the structure of manifolds of Fano type, which are basic building blocks for those defined by polynomials. I will focus on a third and less explored direction, that investigates this class from the point of view of topology. In particular, I will describe a research program aimed at computing generators for the Fukaya category of Fano manifolds with global symmetries, which leads to a rough classification of their Lagrangian submanifolds.
Date: September 25, 3 pm.
Place: JMH 405 (RH)
Speaker: Yunping Jiang (CUNY)
Title: An Overview of Holomorphic Motions Theory and Holomorphic Liftings in Teichmuller Spaces
Abstract: The holomorphic lifting, which was used to be called one of the important problems in Teichmuller theory, is equivalent to the full extension of a holomorphic motion of a subset of the Riemann sphere over a hyperbolic Riemann surface. We proved the holomorphic lifting under the trivial monodromy condition and showed that the trivial monodromy condition is necessary and sufficient. I will provide an overview of this work in this talk.
Date: October 16, 3 pm.
Place: JMH 405 (RH)
Speaker: Hyunchul Park (SUNY New Paltz)
Title: Recent Developments of the Spectral Heat Content for Jump Processes
Abstract: In this talk, we explore recent developments on the spectral heat content associated with various jump processes. The spectral heat content quantifies the total heat contained in a domain D with an initial temperature of one with Dirichlet exterior condition. Recently, there have been growing interests on studying the spectral heat content for jump processes as the spectral heat content contains important spectral and geometric information about the underlying processes and domains. We will focus on small time asymptotic behavior of the spectral heat content on bounded C^{1,1} open sets for various jump processes including stable processes and beyond.
This is based on joint works with Kei Kobayashi (Fordham), Jaehun Lee (SAARC, South Korea), and Renming Song (UIUC).
Date: October 23, 3 pm.
Place: JMH 405 (RH)
Speaker: Jaiung Jun (IAS/SUNY New Paltz)
Title: Quiver representations over the field with one element
Abstract: A quiver is a directed graph, and a representation of a quiver involves assigning a vector space to each vertex and a matrix to each arrow. In this talk, I will introduce a combinatorial model of quiver representations known as quiver representations over the field with one element. Additionally, I will discuss some applications of this model. No prior knowledge of the topic is required, and the talk is accessible to undergraduate students.
Date: November 6, 3 pm.
Place: LL 701 (LC)
Speaker: Farhan Azad (U.S. News & World Report)
Title: An isoperimetric inequality for surfaces formed from spherical polygons
Abstract: This talk will focus on the isoperimetric inequality for a family of surfaces with Gaussian curvature equal to 1 anywhere the surface is smooth. These surfaces are formed by gluing two copies of an isometric embedding of a spherical polygon along the boundary, with each vertex of the polygon leading to a non-smooth point on the surface. I will first discuss the proof of the inequality for a surface of revolution formed by gluing two copies of an isometric embedding of a spherical lune. Then I will discuss the proof for the general family of surfaces and an application of this inequality, the Faber-Krahn theorem.
Date: November 13, 3 pm.
Place: JMH 405 (RH)
Speaker: Cormac O'Sullivan (CUNY)
Title: Topographs and some infinite series
Abstract: The Fibonacci numbers are a familiar recursive sequence. Topographs are a kind of two dimensional version conjured up by J.H. Conway in his study of integral binary quadratic forms. These forms are ax^2 + bxy + cy^2 with integer coefficients and their number theoretic study has a long history. We'll review Conway's classification of topographs into 4 types and look at some new discoveries. Applications are to new class number formulas and a simplification of a proof of Gauss related to sums of three squares. We'll also see how certain infinite series over all the numbers in a topograph may be evaluated explicitly. This generalizes and extends results of Hurwitz and more recent authors and requires modular forms.
Date: February 14, 3:00 PM
Speaker: Han-Bom Moon (Fordham University)
Place: JMH 405 (RH)
Title: On roots of complex polynomials
Abstract: This talk is prepared for students. I want to explore the following question -- as a student or a (yet) non-professional mathematician, how can we find an interesting question? I will give some suggestions with a concrete example from the study of the roots of complex polynomials. Only Calculus is required to follow the talk.
Date: February 28, 3:45 PM (Speical Time!)
Speaker: Francesco Iafrate (Sapienza University of Rome)
Place: LL 508 (LC)
Title: Regularized statistical problems for diffusion processes
Abstract: We address the estimation of stochastic differential equations parameters in sparse, high-dimensional settings. In order to recover the true reduced model, we apply adaptive penalized estimation techniques, such as the bridge and the Lasso, under a high frequency sampling scheme. These methods are adaptive in the sense that they build on initial, unpenalized, estimators, such as those based on quasi-likelihood. We present the asymptotic behaviour of these estimators, in particular we argue that they possess the so-called oracle properties. This class of problems has applications in determining causal relations in high-dimensional systems of stochastic differential equations, for example in the context of financial time series. We introduce generalizations and future extensions, including problems with multiple penalties in the mixed-rates asymptotic regime.
Date: April 3, 3:00 PM
Speaker: Melkana Brakalova (Fordham University)
Place: JMH 132 (RH)
Title: On one-quasiconformality at a point
Abstract: The theory of plane qusiconformal (q.c.) mappings (originally known as {\bf almost conformal}) was developed in the first half of the 20th century, most notably in the works of Gr\"otzsch, Teichm\"uller, Lavrentieff, Ahlfors, and Bers. However, properties of such maps were used in the early/mid 19th century e.g. by Gauss in his study of isothermal coordinates and by Tissaut in his work on cartography. As of now the theory of quasiconformal mappings is a major area of geometric function theory and an ubiquitous tool used in Nevanlinna theory, Teichm\"uller theory, hyperbolic geometry, complex dynamics, fluid mechanics, computer vision, to mention a few.
Quasiconformal mappings are useful in applications as they are less rigid then conformal maps while distorting shapes and their measurements such as area, angle, length in a bounded fashion.
In this presentation we introduce the notion of the conformal module and geometric and analytic definitions of the q.c. mapping, one--quasiconformality of a q.c. mapping at a point, and present some known, some of our own results and open questions for the case of conformality at a point.
Further results on one--quasiconformality and applications to the study of the integrable Teichm\"uller subspaces of the universal Teichm\"uller space will be discussed in the future.
Date: April 10
Speaker: Tristan Ozuch (MIT)
Place: LL 508 (LC)
Title: Einstein metrics and Ricci flow in dimension 4
Abstract: Einstein manifolds and Ricci flow were instrumental tools in answering long standing topological questions in dimension 3. It is now in dimension 4 that most open questions remain, and geometers hope that those same tools will provide some answers. However, Einstein 4-manifolds are still far from being fully understood and many 4-dimensional specific techniques have yet to be applied to Ricci flow.
I will motivate Einstein metrics and the Ricci flow approach to topology and discuss what are some of the main difficulties and specificities of dimension 4.
Date: April 17
Speaker: Jennifer Li (Princeton)
Place: JMH 132 (RH)
Title: On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds
Abstract: Generally speaking, the three building blocks in the birational classification of algebraic varieties are Fano, Calabi-Yau, and varieties of general type. Convex geometry is a tool that can be used to understand these varieties. More specifically, we may study the cone of curves of a variety (or, dually, the nef cone of the variety). For Fano varieties, the cone of curves (and therefore the nef cone) is always rational polyhedral, meaning the cone has finitely many rational extremal rays. Varieties of general type lie at the other extreme - they form a class that is impossible to classify, and there are examples of the cone of curves of such variety being round. In the Calabi-Yau case, we do not have complete control, but Morrison's cone conjecture (1993) provides hope: given a Calabi-Yau variety Y, there exists a rational polyhedral cone which is a fundamental domain for the action of the automorphism group of Y on the nef cone. In this talk, I will explain a version of Morrison's cone conjecture that holds for certain types of 3-folds, and it turns out that there are many examples of such 3-folds.
Date: April 24
Speaker: Ishan Datt (J.P. Morgan)
Place: LL 508 (LC)
Title: Statistical Methods for trading commodity markets
Abstract: In commodity markets, there are often dislocations within the market caused by discrepancies between supply and demand. Here we explore a few methodologies to identify and capitalize on these dislocations across the market. The statistical methods presented provide a baseline model for quickly identifying and ranking dislocations based on a variety of factors.
Date: May 1
Speaker: Paul Jung (Fordham)
Place: JMH 132 (RH)
Title: Infinite-width limits of Neural Nets and the Neural Tangent Kernel
Abstract: Artificial Neural Networks (ANNs) have played a pivotal role in the AI revolution, contributing to the development of large language models like ChatGPT. In this talk, we explore the concept of taking the infinite-width limit of ANNs using a scaling technique known as neural tangent scaling. Surprisingly, this limit reveals an intriguing link to machine learning Kernel Methods of yore (harking all the way back to the early parts of this millenium!).
Date: May 8
Speaker: Scott Chapman (Sam Houston State University)
Place: LL 701 (LC)
Title: The Mathematics of Chicken McNuggets
Abstract: Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20. The following problem spawned from the use of these latter three numbers.
The Chicken McNugget Problem. What numbers of Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces? In particular, what is the largest number of McNuggets that cannot be ordered?
While the answers to this question are relatively easy, it opens a new line of exploration into the world of numerical monoids. We consider the basic properties of numerical monoids and discuss how elements in such a monoid can be “factored” (much like a positive integer can be factored into a product of prime numbers). The talk is accessible to students with a fundamental background in number theory and abstract algebra.
Date: Sep. 20.
Place: JMH 405 (RH)
Speaker 1: Joshua Morales (Fordham)
Title: Space filling curves over three-dimensional projective space.
Abstract: Masaaki Homma and Seon Jeong Kim in 2013 classified all the algebraic smooth space filling curves over a projective plane over a finite field $F_q$. Furthermore, they proved the minimum degree of these curves was q+2. We look towards three-dimensional projective space to extend these results and attempt to find a minimum degree of space filling curves over this space. This is joint work with Alana Campbell, Flora Dedvukaj, Donald McCormick III, and Han-Bom Moon.
Speaker 2: Ritik Jain (Fordham), Peter Wu (Fordham)
Title: The number of zeros of a random polynomial over a finite field
Abstract. The distribution of the average number of zeros of polynomials over real and complex fields is a classical topic with many well-known results. In this talk, we present an equivalent finding for multivariable polynomials over finite fields with arbitrary order $q$.
First, we obtain the expected number of zeros for such polynomials. Next, we derive the probability generating function of the distribution in the single-variable case. Notably, as $q$ grows without bound, it asymptotically resembles Poisson distribution with parameter 1. Then, we compute the probability generating function of the number of zeros in the two-variable case. Though it is not explicitly shown, the method of proof extends easily to the general multivariable case via induction, for which the result is stated. Finally, we discuss some related questions. This is a joint work with Han-Bom Moon.
Date: Oct. 11.
Place: JMH 405 (RH)
Speaker: Jinyoung Park (NYU)
Title: Thresholds
Abstract: For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a "threshold." Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.
Date: Oct. 18.
Place: LL 510 (LC)
Speaker: Phillip Kerger (Johns Hopkins)
Title: Quantum Image Denoising: A Framework via Boltzmann Machines, QUBO, and Quantum Annealing
Abstract: I will present recent work on image denoising using quantum annealing via restricted Boltzmann machines (RBMs) and quadratic unconstrained binary optimization (QUBO). First, an overview of quantum annealing and Boltzmann machines will be given. For the denoising problem, we then formulate a denoising objective function by balancing the distribution learned by a trained RBM with a penalty term for derivations from a given noisy image. We derive the statistically optimal choice of the penalty parameter assuming the target distribution has been well-approximated, and further suggest an empirically supported modification to make the method robust to that idealistic assumption. Under additional assumptions, the denoised images attained by our method are, in expectation, strictly closer to the noise-free images than the noisy images are. While we frame the model as an image denoising model, it can be applied to any binary data. As the QUBO formulation is well-suited for implementation on quantum annealers, we test the model on a D-Wave Advantage machine, and also test on data too large for current quantum annealers by approximating QUBO solutions through classical heuristics.
Date: Oct. 24. (Tuesday!) 3:00 pm.
Place: JMH 331 (RH)
Speaker: Arthur Jakot (NYU)
Title: Bottleneck Structure in Deep Neural Networks: Mechanisms of Symmetry Learning
Abstract: Deep Neural Networks (DNNs) have proven to be able to break the curse of dimensionality, and learn complex tasks on high dimensional data, such as images or text. But we still do not fully understand what makes this possible. To answer this question, I will describe the appearance of a Bottleneck structure, where the network learns low-dimensional features in the middle of the network. This allows the network to identify and learn symmetries of the task it is trained on, without any prior knowledge. This could explain the success of deep learning on image and text tasks which feature many `hidden' symmetries.
Date: Nov. 8.
Place: JMH 405 (RH)
Speaker: Kimoi Kemboi (IAS)
Title: Full exceptional collections on linear GIT quotients
Abstract: In algebraic geometry, the derived category of a variety is a crucial algebraic invariant with several profound implications on the geometry of the underlying variety. This talk will focus on a particular structure of derived categories called a full exceptional collection. We will discuss the landscape of full exceptional collections and then explore how to produce them for linear GIT quotients using ideas from "window" categories and equivariant geometry. As an example, we will consider a large class of linear GIT quotients by a reductive group of rank two, where this machinery produces full exceptional collections consisting of tautological vector bundles.
This talk is based on joint work with Daniel Halpern-Leistner.
Date: Nov. 15.
Place: TBA (RH)
Speaker: Wen Li (Fordham)
Title: Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise
Abstract: Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. This is a joint work with Joel Barnett, Elena Resmerita, and Luminita Vese.
Date: Nov. 29.
Place: TBA (RH)
Speaker: Thomas Beck (Fordham)
Title: Pleijel estimates for nodal domains of eigenfunctions
Abstract: The eigenvalues and eigenfunctions of the Laplacian of a plate determine the constant frequencies at which the plate can vibrate. The zero set of the eigenfunction then corresponds to the curves on the plate that are stationary as the plate vibrates at that frequency. This zero set partitions the plate into a finite number of connected components called nodal domains, and a result of Pleijel provides an asymptotic upper bound on their number. We will discuss the key ingredients in the proof of this upper bound, and an extension of this result to regions with less regular boundaries and thin necks. This is joint work with Yaiza Canzani and Jeremy Marzuola at UNC - Chapel Hill.
Date: March 8, 3:00 - 4:00 pm.
Place: LL510 (Lincoln Center)
Speaker: Heather Macbeth (Fordham)
Title: Making mathematics computer-checkable
Abstract: In the last thirty years, computer proof verification became a mature technology, with successes including the checking of the Four-Colour Theorem, the Odd Order Theorem, and Hales' proof of the Kepler Conjecture. Recent advances such as the "Liquid Tensor Experiment" verifying a recent theorem of Scholze have provided further momentum, as likewise have promising experiments integrating this technology with machine learning.
I will briefly describe some of these developments. I will then try to describe, more generally, what it *feels* like to carry out research-level computer verifications of mathematics proofs: the level of expression one has access to, the ways one finds oneself interrogating and reorganizing a paper proof, the kinds of arguments which are more tedious (or less tedious!) than on paper.
Date: March 22, 3:00 - 4:00 pm.
Place: LL 510 (Lincoln Center)
Speaker: Ian Morrison (Fordham)
Title: The Poset of $0$--$1$ factorizations of $\frac{1-x*n}(1-x}$ -- small sets of dice with equally likely totals, additive systems and nonnegative digit sets
Abstract: The factorizations in the title form a partially ordered set with partial order given by further factorizations. This point of view lets us simplify and sharpen various known results about them and draw pictures like the one of $\Psi_{36}$ above. I will explain how we describe $\Psi_n$ in terms of posets associated to divisors of $n$, indicate how this leads to combinatorial pictures like the example, and derive various structural corollaries. All the proofs are completely elementary (often quite pretty, and occasionally a bit subtle) so undergraduates are encouraged to attend.
Date: March 29, 3:00 - 4:00 pm.
Place: LL 510 (Lincoln Center)
Speaker: Rolf Ryham (Fordham)
Title: Self-organization by moving domain elliptic PDEs
Abstract: Collaborators and I have recently developed a new mathematical model of self-organization in fluid systems. The framework uses small, rigid granules to represent parcels of suspended media such as elastic membranes. Shape and boundary conditions capture the local material properties, while a phase field function represents the properties of the solvent. The approach falls into the category of work that considers particles as a discretization of a continuous problem. Robust and cutting across many of the challenges suffered by stencil-based discretization methods, the framework enjoys flexibility in terms of morphology and topology, is built around important details like molecular orientations, and exhibits behavior like inter-monolayer slip that must be added post hoc into continuum models. Leveraging boundary integral and fast summation methods, the dynamics can in principle be solved in near-linear complexity in the number of granules, which is equal to or better than the computational complexity of related sharp interface methods.
Date: April 5, 3:00 - 4:00 pm.
Place: JMH 406 (Rose Hill)
Speaker: Kei Kobayashi (Fordham)
Title: Spectral heat content for time-changed Brownian motions
Abstract: The spectral heat content for a Brownian motion in a bounded domain represents the amount of heat that remains in the domain until a given time point. I will first explain the small-time asymptotic behavior of the spectral heat content and then further discuss what happens when the Brownian motion is randomly time-changed. Two types of time changes are considered; one is a subordinator and the other is an inverse subordinator. In the case of a subordinator, depending on the underlying index, the small-time behavior of the spectral heat content changes substantially. In contrast, when the time change is given by an inverse subordinator, there is a single statement that is valid for all values of the index. This is joint work with Hyunchul Park.
Date: April 19, 3:00 - 4:00 pm.
Place: JMH 406 (Rose Hill)
Speaker: Tomasz Nowicki (IBM)
Ttile: An exponential-linear map in the space of cumulative distributions.
Abstract: The problem was first stated as an innocent looking exercise in iterations of a map in two dimensions. The question, posed by David Gamarnik, was for an explicitly given expression: what are the parameters which guarantee there exists a unique globally attracting fixed point? The solution was not obvious, that is to say: we did not find it for this map. Under further investigation it turned out that we are dealing with a special case of a multi-dimensional problem. It did not help. In fact it was technically much more difficult. However moving from a finite dimensional space to the functional space the solution became quite easy, visible and understandable. Some basic knowledge of: probability, exponential function (in dimension 1), monotonicity and integrals would be helpful. Not much more is needed to understand the solution to an open (then) problem in random graphs.
Date: April 26, 3:00 - 4:00 pm.
Place: JMH 406 (Rose Hill)
Speaker: Matthew Romney (Stony Brook)
Title: CAT(k) surfaces and bounded integral curvature
Abstract: The theory of surfaces of bounded integral curvature is a classical topic in geometry, with major contributions by Aleksandrov and Reshetnyak, among others. In this talk, we investigate CAT(k) surfaces: surfaces with curvature locally bounded above by some parameter k. Despite ongoing interest in the topic, the basic properties of CAT(k) surfaces seem to be mathematical folklore. Our goal is to rectify this situation. We provide a complete proof of the fact that CAT(k) surfaces have bounded integral curvature, and thus that the classical theory applies to them. We also consider the problem of smooth approximation by Riemannian surfaces. This research was done as part of Stony Brook University’s Summer 2022 REU program.
Date: April 28 (Friday), 3:00 - 4:00 pm.
Place: JMH 405 (Rose Hill)
Speaker: Kim Klinger-Logan (Rutgers/Kansas State University)
Title: How to fail to prove the Riemann Hypothesis
Abstract: In this talk, we will begin by telling explaining the Riemann Hypothesis. We will then a story about how to (interestingly) fail to prove the Riemann Hypothesis. (Yes, we all fail to prove the Riemann Hypothesis every day but don't worry, I will not make you just stare at an empty blackboard.) The story begins with a computational error that leads to lots of fun math including complex analysis, automorphic forms, PDEs, functional analysis, and more! Time permitting, we will allude to the connection that this problem has to particle physics.
Date: October 5
Place: JMH 405
Speaker: Harry Hyungryul Baik (KAIST, CUNY Queens College)
Title: Surface and 3-manifolds
Abstract: How should we understand surfaces or 3-manifolds? One reasonable thing to do is to use even lower-dimensional objects to probe the inside of the space you want to understand. We will discuss a few explicit examples of this approach and see what we can do with them.
Date: October 26
Place: JMH 406
Speaker: Han-Bom Moon (Fordham)
Title: Derived category of moduli space of vector bundles on a curve
Abstract: The derived category of a smooth projective variety is an invariant expected to encode birational geometric information of the variety. One way to study its structure is to divide it into smaller building blocks, and the semiorthogonal decomposition provides a systematic approach. In this talk, I will explain the current status of the study of the derived category of moduli spaces of vector bundles on a curve in the framework of semiorthogonal decomposition. In particular, I will describe how one can embed the derived category of the symmetric product of the base curve into the derived category of the moduli space and an implication in the Fano visitor problem.
Date: November 2
Place: JMH 406 (at 3:30)
Speaker: Debanjana Kundu (University of British Columbia)
Title: Iwasawa Theory and Arithmetic Statistics
Abstract: In this talk I will give a brief overview of my current research on Iwasawa theory. The first part of this talk will be expository in nature, where I will give an introduction to Iwasawa theory and explain the main questions that drive a lot of the research in this area. In the second part of the talk, I will discuss some of the progress made towards questions on the average behaviour of Iwasawa invariants - both the classical Iwasawa invariants and Iwasawa invariants of Selmer groups of elliptic curves.
Date: November 9
Place: JMH 406
Speaker: Wen Li (Fordham)
Title: Simulation and analysis of groundwater flow captured by the horizontal reactive media well using CS-RPIM
Abstract: The Horizontal Reactive Media Treatment (HRX) well is a novel technology for in situ treatment of contaminated groundwater. The most significant technical performance risk associated with field-scale implementation of the HRX well is the potential for water to bypass the well untreated. Therefore, the groundwater capture simulation plays a very important role in the design of the HRX well. To help engineers design the HRX well, I build a model based on a cell-based smoothed radial point interpolation method (CS-RPIM) for simulating groundwater flow captured by the HRX well in both 3D test pit pilot scale and field scale.
Date: November 16
Place: JMH 406
Speaker: Manami Roy (Fordham)
Title: Dimensions of spaces of modular forms and Siegel modular forms
Abstract: Modular forms are complex differentiable functions with rare symmetry. The study of classical modular forms spans many fields of mathematics, including numerous applications in complex analysis, number theory, algebraic geometry, and physics, to name a few. Modular forms have been instrumental in resolving many classical problems in number theory, from ``calculating the number of ways to represent a positive integer using four integer squares" to being at the center of ``the proof of Fermat’s Last Theorem." In this talk, we will briefly introduce modular forms and look at the dimensions of spaces of modular forms. Lastly, we will discuss some new dimension formulas of Siegel modular forms (modular forms in more than one variable).
Date: November 30
Place: JMH 406 (at 3:30)
Speaker: A. Raghuram (Fordham)
Title: Motives and Automorphic Forms
Abstract: One of the central themes in the Langlands Program is a correspondence between motives and automorphic forms. The idea of an L-function acts as a bridge in this correspondence. A motive is an algebro-geometric concept and one aspect of a motive concerns the action of the Galois group of the field of rational numbers on a vector space from which a classical construction of Artin gives the motivic L-function. An automorphic form is an object in the realms of harmonic analysis, and period integrals of automorphic forms give rise to automorphic L-functions. The first part of the talk will be an introduction via examples to such objects and their L-functions. The special values of an L-function encode structural information about motives and/or automorphic forms to which the L-function is attached. An overarching conjecture of Deligne on the special values of motivic L-functions predicts - via the conjectural correspondence in the Langlands program - the nature of results one may expect to prove with automorphic L-functions. In the second part of my talk, I will discuss some results in an ongoing project with Deligne involving period relations for motives that conjecturally explains other results of mine, obtained partly in independent collaborations with Günter Harder and Chandrasheel Bhagwat, on the special values of certain automorphic L-functions.
Date: February 11
Speaker: David Swinarski (video)
Title: Modelling elevator traffic with social distancing
Abstract: Social distancing standards implemented during the COVID-19 pandemic typically decrease both a building's population and the capacity of each elevator car. Elevator traffic may be worsened when the decrease in the elevator car capacity is proportionally greater than the decrease in the building's population. We build a model and use it to predict the elevator traffic under social distancing in a university classroom building, and study the effects of four low-cost interventions aimed at improving this traffic.
Date: March 4
Speaker: Shu Gu
Title: Large-scale Regularity of Nearly Incompressible Elasticity in Stochastic Homogenization.
Abstract: In this talk, we will systematically study the regularity theory of the linear system of nearly incompressible elasticity. In the setting of stochastic homogenization, we develop new techniques to establish the large-scale estimates of displacement and pressure, which are uniform in both the scale parameter and the incompressibility parameter.
Date: March 18
Speaker: Benjamin Davidson and Andrew Souther (video, slides)
Title: An Introduction to the Lean theorem prover
The Lean theorem prover is a software tool growing in popularity among mathematicians for digitizing and verifying math proofs. After an introduction to Lean and its expanding library of computer-verified math, we will present a few demo proofs. We will also present a contribution we made to this library, a test case for several of its recent developments in measure theory and calculus: proving the area of the disc. No prior knowledge assumed beyond Calculus and Discrete Math.
Date: April 1
Speaker: Patrick McFaddin (video)
Title: Categories, functors, and categories of functors
Abstract: Explicitly formalized in the 1940's, category theory has become an indispensable tool for tackling problems in, e.g., algebra, algebraic geometry, and topology. The study of categories and functors has also seen a wide variety of applications to fields outside of pure mathematics, including logic, programming, and data analysis. This talk will be a gentle introduction to category theory, drawing on examples from (primarily) algebra and topology. Time permitting, we will discuss a framework for utilizing homotopical methods in algebraic geometry.
Date: April 22
Speaker: Han-Bom Moon
Title: Can we recover the history of life with mathematics?
Abstract: Can we recover the tree of evolution from a given collection of DNA? This seemingly unrelated question to pure mathematics has a deep connection with algebraic and tropical geometry. In this talk, I will explain classical results, new challenges, and recent joint work with Alessio Caminata, Noah Giansiracusa, and Luca Schaffler. One of the goals is to provide an example of the interaction between pure mathematics and biological science.
Date: October 22
Speaker: Thomas Beck
Title: Eigenfunctions, vibrations, and nodal partitions
Abstract: When a plate vibrates at a constant frequency, the lines on the plate where no vibration occurs are described by the zero set of a Laplace eigenfunction. These lines form a partition of the plate, and the classical Courant nodal domain theorem places a restriction on the number of components of the partition. We will compute some examples where the eigenfunctions and partitions can be explicitly found, and discuss some properties and open questions concerning these partitions.
Date: November 5
Speaker: Manami Roy video
Title: On some equidistribution theorems.
Abstract: In this talk, we will look at various equidistribution results in mathematics. We will start with some examples of equidistribution of sequences of numbers. Then we look at some classical equidistribution results in number theory, namely, Sato-Tate conjectures for elliptic curves and eigenvalues of modular forms. Finally, we will state an equidistribution theorem for certain automorphic representations.
Date: November 12
Speaker: Han-Bom Moon video
Title: Let's count points!
Abstract: A fascinating fact on mathematics is that there are many interesting connections between seemingly different mathematical disciplines. In this talk, I will present a surprising formula counting integral points on polygons and sketch its proof. We will see a delightful interaction between algebra, combinatorics, and geometry. This talk aims primarily for undergraduate students. No prerequisite is assumed beyond calculus.
Date: December 3
Speaker: Farhan Azad, Zixi Chen video
Title: Relations of the Generalized Skein Algebras of Punctured Spheres
Abstract: I will talk about the presentation of skein algebra, which is a topological invariant of a surface constructed by using curves on the surface. The presentation of the generalized skein algebra of a punctured sphere has been computed recently. I will explain the basic relations that are used to calculate various arcs and loops on an n-punctured sphere and provide a visual understanding. Then, I will describe relations which generate the ideal for the presentation of an n-punctured sphere.
Date: January 23rd, 2020
Speaker: Patrick McFaddin
Title: Algebraic groups, torsors, and twisted forms
Abstract: Algebraic groups are geometric objects defined by polynomial equations and which also carry a group structure. An example of such an object is the automorphism group of a (possibly non-commutative) algebra. In this setting, twisted forms of the algebra can be encoded using torsors. In this talk, I will outline this framework by producing a correspondence between torsors and twisted forms, focusing on central simple algebras (twisted forms of matrix algebras). We will then relate these ideas to recent work which utilizes torsors to study questions of rationality.
Date: January 30th, 2020
Speaker: Sixian Jin
Title: A brief introduction to Malliavin calculus with some financial applications
Abstract: Malliavin calculus extends the field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. In this talk, I will start with a brief introduction to Malliavin calculus for Brownian motion. Then I will establish two types of series representations for the conditional expectation of functionals of Brownian motions. Some financial applications on bond pricing problems will be mentioned at the end of the talk.
Date: February 20th, 2020
Speaker: Hans-Joachim Hein
Title: Harmonic functions and hyper-Kähler geometry
Abstract: In this talk I will explain a classical method of creating special solutions to the 4-dimensional Einstein vacuum equations from positive harmonic functions on domains in R^3. Under this correspondence, simple properties of harmonic functions on R^3 (for example, the fact that the net flux of their gradient vector field through any closed surface is zero) translate into much deeper geometric and topological properties of the associated 4-dimensional spaces. Towards the end of the talk I will mention some recent work where this correspondence was used to construct good approximations to Ricci-flat Kähler metrics on K3 surfaces near certain boundary points of the K3 moduli space.
Date: March 5th, 2020
Speaker: Manami Roy
Title: Local representations attached to rational elliptic curves with non-trivial torsion subgroups
Abstract: There is a classification of rational elliptic curves with a non-trivial torsion subgroup given by two-parameter families. By Mazur’s Torsion theorem there are fourteen such families of elliptic curves. For each of these fourteen families, our goal is to find the associated local representations of GL(2, Q_p) at each prime p. In this talk, we will look at a few of the cases.
All of the remaining talks were canceled due to COVID-19.
Date: September 12th, 2019
Speaker: Heather Macbeth
Title: The Yamabe problem
Abstract: I will give an introduction to conformal differential geometry, to scalar curvature, and to the relationship between them, culminating in a discussion of the Yamabe problem, solved by Richard Schoen in 1984. There will be digressions into complex analysis and general relativity.
Date: October 3rd, 2019
Speaker: Xumin Jiang
Title: Isometric embedding
Abstract: I will give an introduction about isometric embedding of Riemannian manifolds in Euclidean spaces and talk about the Weyl problem, completely solved by Nirenberg and Pogorelov independently in the early 1950s. I will also talk about my result when the Gauss curvature is nonnegative.
Date: October 17th, 2019
Speaker: Patrick McFaddin
Title: The arithmetic and geometry of algebraic cycles on twisted Grassmannians
Abstract: Chow groups provide a satisfying algebro-geometric analogue of singular (co)homology, which can also reflect interesting arithmetic properties of a given algebraic variety if one allows for varying coefficient systems. The theory of Chow groups (with coefficients) of homogeneous varieties has seen many useful applications to the study of central simple algebras, quadratic forms, and Galois cohomology. Significant results include the Merkurjev-Suslin Theorem and the Voevodsky-Rost-Weibel Theorem (also known as the Bloch-Kato Conjecture). Despite these successes, a general description of these groups remains elusive and computations are done in various cases. In this talk, I will discuss work on computing zero-cycles with coefficients for some twisted forms of (symplectic) Grassmannians.
Date: October 31st, 2019
Speaker: Han-Bom Moon
Title: Phylogenetic trees, tropical geometry, and point configurations
Abstract: Can we find a nice algebraic curve passing through given n points in a d-dimensional space? Surprisingly, this elementary question has a connection (via tropical geometry) with computational biology. I will explain the connection, known results, and open questions. This talk will be accessible to students, and some of the open questions might be good projects for students who are interested in research experience.
Date: November 21st, 2019
Speaker: David Swinarski
Title: Vector partition functions for conformal blocks
Abstract: A vector partition function is a function that counts the number of lattice points in a polytope defined by the function's arguments. I will define vector partition functions and give examples. One application is in algebraic geometry, where it is conjectured that the ranks and intersection numbers of conformal blocks are given by vector partition functions. I will discuss evidence for these conjectures and the techniques and software I am developing that may help me prove them.