Topic: Explicit non-Gorenstein R=T via rank bounds
Cathy Hsu | ???
Abstract: In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.
Location: Gibson 126A
Time: 3:00 pm
Topic: Quiver representations over F_1 and Euler characteristics of quiver Grassmannians.
Jaiung Jun - SUNY at New Paltz
Abstract:
A quiver is a directed graph, and a representation of a quiver assigns a vector space to each vertex and a linear map to each arrow. Quiver representations arise naturally in many areas of mathematics. Quiver representations over F_1, ``the field with one element'' can be considered as a (degenerated) combinatorial model of quiver representations over a field, where vector spaces and linear maps are replaced by F_1-vector spaces and F_1-linear maps. I will introduce several aspects of quiver representations over F_1, and its potential applications. This is joint work with Alex Sistko.
Location: Dinwiddie 103
Time: 3:00pm
Topic: The Mathematics of Musical Tuning
Peter Marcus - Tulane University
Abstract: Designers of musical instruments need to decide how they will be tuned, meaning what pitches the instrument can produce. Today the most common tuning system by far is 12-tone equal temperament, but how was this chosen and why is it so popular? I will give a brief history of European tuning systems and dive into the math behind them.
Location: Stanley Thomas 316
Time: 4:00pm Friday THIS IS A DIFFERENT DAY AND TIME
Topic: Using regularity to estimate Lyapunov exponents
Sam Punshon-Smith - Tulane University
Abstract: I will discuss a new method for estimating Lyapunov exponents for hypoelliptic diffusions from below using quantitative local regularity estimates on the stationary measure of the lift to the projective bundle. For damped driven truncations of nonlinear conservative stochastic partial differential equations (SPDE) in fluctuation dissipation scaling, I will outline a general strategy for proving positivity of the top Lyapunov exponent that can be applied to Galerkin truncations of the stochastic Navier-Stokes equations on T^2. The proof in the Navier-Stokes case involves verifying transitivity properties of a certain matrix Lie algebra associated with the linearized Euler equations using tools from algebraic geometry. The techniques are quite general and can be applied more broadly to Galerkin truncations of many other evolution type SPDE. This work is joint with Jacob Bedrossian and Alex Blumenthal.
Location: Gibson 325
Time: 3:00pm
Topic: From integrals to combinatorial formulas of finite type invariants - a case study.
Rafal Komendarczyk | Tulane University
Abstract:
We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the argument presented here and the classical arguments is that the vanishing of integrals over hidden faces does not require the involution trick due to Kontsevich. The integral formula yields the well-known arrow diagram expression for the
invariant, first presented in the work of Polyak and Viro. We also take the next step of extending the arrow diagram expression to multicrossing knot diagrams. The primary motivation is to better understand a connection between the classical configuration space integrals and arrow diagram formulas for finite type invariants. This is a joint work with Robyn Brooks, which builds on her thesis.
Location: Dinwiddie 103
Time: 3:00pm
Topic: From integrals to combinatorial formulas of finite type invariants - a case study.
Rafal Komendarczyk | Tulane University
Abstract:
We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the argument presented here and the classical arguments is that the vanishing of integrals over hidden faces does not require the involution trick due to Kontsevich. The integral formula yields the well-known arrow diagram expression for the
invariant, first presented in the work of Polyak and Viro. We also take the next step of extending the arrow diagram expression to multicrossing knot diagrams. The primary motivation is to better understand a connection between the classical configuration space integrals and arrow diagram formulas for finite type invariants. This is a joint work with Robyn Brooks, which builds on her thesis.
Location: Dinwiddie 103
Time: 3:00pm
Wednesday, September 21
Topic: Wavelet random matrices and fractals in high dimensions
Gustavo Didier | Tulane University
Abstract: Scale invariance, or fractality, is a fundamental feature of the long term behavior of many stochastic systems. The study of fractality in high dimensions is a frontier subject in the field of probability theory, and one which impacts several fields of application. In this talk, we show that wavelet random matrices provide a natural framework for the study of scale invariance in high dimensions. No background on the subject will be assumed.
Location: Gibson Hall 126A
Time: We will be following the Algebra and Combinatorics Seminar
Topic: Weakly Complete Universal Enveloping Algebras of Profinite-Dimensional Lie Algebras
Karl Hofmann | Tulane University
Location: Gibson 126A
Time: 3:00 pm
Topic: Shor's Algorithm and Quantum Supremacy
Rubaiyat Islam | Tulane University
Abstract: Shor's algorithm is a quantum computing algorithm which shows that a programmable quantum computer can factor integers in polynomial time. Currently used important cryptographic algorithms critically depend on the fact that prime factorization of large numbers with classical algorithms takes a long time. A factoring problem can be turned into a period finding problem and the quantum part of Shor's algorithm significantly speeds up this period finding step. In this talk, we will look at the relation between the two problems, what gives quantum advantage and try to see some output from an actual quantum computer.
Location: Stanley Thomas 316
Time: 5:00pm
Week of September 16 - September 12
Topic: Modeling aspects of vesicle electrodeformation and transport
Adnan Morshed | Tulane University
Abstract: Characterization of the mechanical properties of cells and other sub-micron vesicles such as virus and neurotransmitter vesicles are necessary to understand their deformation dynamics. This characterization can be done by submerging the vesicle in a fluid medium and deforming it with controlled electric field exposure known as electrodeformation. Electrodeformation of biological and artificial lipid vesicles is directly influenced by the vesicle and media properties and geometric factors. The problem is compounded when the vesicle is naturally charged which creates electrophoretic forcing on the vesicle membrane. This talk will highlight some of the modeling aspects of electrodeformation and transport of charged vesicles immersed in a fluid media under the influence of a DC electric field. The electric field and fluid-solid interactions are resolved using a hybrid immersed interface-immersed boundary technique. Viability of the width averaged domain conductivity and current as indicators of vesicle deformation and movement will be discussed. Vesicle movement due to electrophoresis is also characterized by the change in local conductivity which can serve as a potential sensor for electrodeformation experiments and in solid-state nanopore applications.
Location: Gibson 325
Time: 3:00pm
Topic: Generalized Wilf Conjecture
Naufil Sakran | Tulane University
Abstract: The Wilf Conjecture is a longstanding conjecture regarding the complement finite submonoids of N, the monoid of natural numbers. There have been several attempts to generalize the conjecture for higher dimensions. In this talk, we will extend the conjecture for complement finite submonoids of unipotent group with entries from N. This extension gives a better bound compared to the previous generalizations. Also, we will prove our conjecture for certain subfamilies (thick and thin) of the unipotent groups.
Location: Gibson 126A
Time: 3:00 pm
Topic: Hilbert's 17th Problem
Naufil Sakran | Tulane University
Abstract: Hilbert proposed 23 questions in 1900 at the conference of the International Congress of Mathematics. Answers to these questions have high influence in Modern Mathematics. Hilbert's 17 problem asks about the minimal requirement to a polynomial so that it has representation as sum of squares. The talk will focus on answering the question. Finally, we will discuss open problems regarding it.
Location: Stanley Thomas 316
Time: 5:00pm
Monday, September 12
Topic: we welcome Dr. Samuel Punshon-Smith
Samuel Punshon-Smith | Tulane University
Abstract: We welcome Dr. Samuel Punshon-Smith to discuss his research and highlight a woman in Mathematics of his choice.
Location: Gibson Hall 400D
Time: 3:00pm
Topic: Microswimmer in confinements
Hongfei Chen | Tulane University
Abstract: We consider the active Brownian particle (ABP) model for a two-dimensional microswimmer with fixed speed, whose direction of swimming changes according to a Brownian process. The probability density for the swimmer obeys a Fokker–Planck equation defined on the configuration space, whose structure depends on the swimmer’s shape, center of rotation and domain of swimming. We enforce zero probability flux at the boundaries of configuration space. We consider the dynamics of a swimmer in an infinite channel and a lattice of point obstacles. At first neglecting hydrodynamic interactions, we derive a reduced equation for a swimmer in an infinite channel, in the limit of small rotational diffusivity, and find that the invariant density depends strongly on the swimmer’s precise shape and center of rotation. We also give a formula for the mean reversal time: the expected time taken for a swimmer to completely reverse direction in the channel. Using homogenization theory, we find an expression for the effective longitudinal diffusivity of a swimmer in the channel, and show that it is bounded by the mean reversal time. Finally, we include hydrodynamic interactions with walls, and examine the role of shape. For a swimmer in a periodic lattice of point obstacles, we apply the same framework as in the channel, solve the invariant density and use the homogenization approach to compute effective diffusivity. Moreover, when the swimmer is only a circular or needle diffuser that cannot swim, we solve the cell problem and effective diffusivity asymptotically.
Location: Gibson 325
Time: 3:00pm
Location: Gibson 426
Time: 1:00pm
Topic: Nonlocality and the Pauli Group
Victor Bankston | Tulane University
Abstract: The Pauli group is fundamental in quantum information and is related to certain extraspecial 2-groups. In this talk, I will introduce the Pauli group and pose a novel problem about its structure in terms of partial homomorphisms to products of 2-element groups. Finally, I will motivate the problem by describing a relationship to a novel class of nonlocal games by using the eigenvalue expansion properties of a distance-regular graph.
Location: Gibson 126A
Time: 3:00 pm
Topic: Projection Method and Linear Algebra
Sang-Eun Lee | Tulane University
Abstract: Due to the millennium problem for the global existence and regularity of the 3D Navier-Stokes equations for the viscous and incompressible flow, we cannot obtain an explicit solution to the system of fluid equations. So, we rely on the computational method to describe and understand fluid motion. There are a few numerical methods to find an approximated solution, but due to the massiveness of the matrix size, people have tried to find to reduce the computational cost of the methods. A well-known method to find the numerical solution of fluid motion is called the projection method, due to Alexandre Chorin. This talk briefly describes the fluid-structure and discusses some linear algebraic skills in each scenario.
Location: Stanley Thomas 316
Time: 5:00pm
Topic: Quaternary Lattices, Modular Forms and Elliptic Curves
Daniel Fretwell
Abstract: A common theme in modern Number Theory is to find non-trivial links between objects coming from very different places, by relating their arithmetic. In this talk we will (hopefully) see a surprising example of this, connecting the first and third objects in the title…using the second to bridge the gap. Time permitting, we will sketch the proof, motivated by a hidden 1.5th object (Clifford algebras). (Based on joint work with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight)
Location: DW 103
Time: 3:00pm
Topic: The transportation method in Concentration Inequalities.
Ying Bi | Tulane University
Abstract: The Concentration inequalities are widely used in Probability Theory, Statistical mechanics, information theory and many other fields. Among all the methods in proving such inequalities, the entropy method performs especially well when dealing with suprema of empirical processes, but often faces difficulties when deal with left tail. However, the transportation method provided by Prof Marton in the mid 1990s is more efficient for left tail. We will see the basic idea of proving inequalities based on cost function and the induction spirit in this talk with several examples at the end. Some references will also shown in the appendix for further reading.
Location: Stanley Thomas 316
Time: 5:00pm
