Week of March 22 - March 18
Thursday, March 21
Colloquium
Topic: Mathematics of magic angles for twisted bilayer graphene
Maciej Zworski - UC Berkley (Host: Punshon-Smith)
Abstract: Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties such as superconductivity.
Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence be shown for the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two-dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution as well as on the properties of the associated eigenstates.
The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten.
Location: Gibson Hall 126A
Time: 3:30 pm
Wednesday, March 20
AMS/AWM
Topic: Universality, random matrices, and data science through the lens of high-dimensional scale invariance
Gustavo Didier | Tulane University
Abstract: In this talk, we show how the topic of scale invariance (fractality) in high dimensions naturally brings together some major topics of modern mathematical research such as universality, random matrix theory, high-dimensional probability, data science and machine learning. No prior knowledge of these topics will be assumed.
Location: Gibson Hall 126A
Time: 4:00 PM
Tuesday, March 19
Graduate Colloquium
Topic: Geometric Realization: How to add shape to otherwise shapeless data sets
Will Tran - Tulane University
Abstract: We will learn how to add shape to data sets, even when those sets are not necessarily plottable or graphable in n-dimensional real space. Then, we’ll learn what the shape of our data sets could tell us about our data
Location: Gibson 126
Time: 3:30pm
Week of March 15 - March 11
Friday, March 15
Applied and Computational Math Seminar
Topic: Response theory for dissipative SPDEs.
Giulia Carigi
Abstract: A framework suitable to establish response theory for a class of nonlinear stochastic partial differential equations is presented. With response theory we mean in this context the following: one considers a dynamical system whose dynamical law depends on a parameter (here given by an SPDE where the parameter is in the forcing) and we say that one has a response theory if one can show a regularity in the dependence of the invariant measure on the parameter (here differentiability or Hölder continuity in weak topology). The results are applied to the 2D stochastic Navier-Stokes equation and the stochastic two-layer quasi-geostrophic model, an intermediate complexity model popular in the geosciences to study atmosphere and ocean dynamics. This work is jointly with Jochen Bröcker (University of Reading) and Tobias Kuna (University of L’Aquila).
Location: Gibson Hall 126
Time: 3:00pm
Thursday, March 14
Colloquium
Topic: The fractional Yamabe equation on homogeneous groups
Dimiter Vassilev - University of New Mexico (Host: Dr. Can)
Abstract: The general themes of the talk are Dirichlet forms, fractional (non-local) operators and associated Sobolev type spaces on groups of homogeneous type. I will recall some general motivating examples for considering non-local operators and particular equations before focusing on the respective questions in the setting of homogenous groups. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type condition. I will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.
Location: Gibson Hall 126A
Time: 3:30 pm
Wednesday, March 13
Probability and Statistics
Topic: Overview of Master Protocol Trials and Statistical Considerations
Xiaoyun(Nicole) Li – Senior Director at BeiGene
Abstract: Master protocol is a trial structure that evaluates multiple diseases or multiple drugs (or drug combinations) within the same trial. There are three main types of master protocols, i.e., basket trials, umbrella trials and platform trials. Basket trials evaluate the same drug/drug combination in different diseases within the same trial, with the assumption that similar drug activities may seen and data may be borrowed. Various basket trial designs have been proposed over the years and I will give a flavor of it. Umberlla trials evaluate multiple drugs/drug combinations in the same disease within a trial. There is usually a shared control arm for all the different experimental arms to increase the efficiency. I will talk about the statistical consideration in terms of type I error control and other statistical errors in terms of umbrella trials. Platform trials refer to umbrella trials in a perpetual manner. Statistical considerations arise as to whether we could use the non-contemporaneous (non-concurrent) control and if so, how to use it. I will also talk about a phase 3 umbrella trial design as an illustration.
Location: 4:00pm
Time: Zoom with meeting ID: 932 4354 5612
Wednesday, March 13
Algebra and Combinatorics
Topic: Differential operators: simplicity and combinatorial properties of affine semigroup rings
Janet Vassilev - University of New Mexico
Abstract: We will discuss the ring of differential operators of an affine semigroup ring $R$ and how combinatorial properties of the affine semigroup translate into both the simplicity of the ring of differential operators,$ D(R)$, and the simplicity of the ring as a $D(R)$-module. This is joint work with Berkesch, Chan, Matusevich, Page and Traves.
Location: Gibson Hall 126A
Time: 3:00pm
Tuesday, March 12
Graduate Colloquium
Topic: A few non-classical time stepping methods to study fluid flow problems at low Reynolds number
Moslem Uddin - Tulane University
Abstract: Very often the dynamics of a system mimicking real-world phenomena seem to be well modeled by a system of ordinary differential equations. In practice, it's nearly impossible to solve such systems analytically, and this is why lots of efforts have been made to approximate those numerically. Usually, explicit methods(those that require knowledge from previous steps only) are very popular due to less computational effort required. However, those methods tend to return unstable solutions(the solution becomes unbounded in finite time). In this talk, I'll try to review a few non-explicit time integrators with the intention reduce the level of this type of shortcoming considering an example emerging from fluid flow governed by Stokes' equation.
Location: Gibson 126
Time: 3:30pm
Tuesday, March 12
Integrability and beyond!!!
Topic: Umbral calculus, a method for symbolic computation
Christophe Vignat - Tulane University
Abstract:
Umbral calculus is a computation method that represents a sequence of numbers or functions as a sequence of moments. It allows a significant simplification in the computation of some sequences, such as those associated with orthogonal polynomials.
This talk will introduce umbral calculus through some examples such as Hermite or Gegenbauer polynomials, and will show some applications.
Location: Dinwiddie 102
Time: 2:00pm
Week of March 4 - March 8
Friday, March 8
Applied and Computational Math Seminar
Topic: A tractable algorithm, based on optimal transport, for computing adversarial training lower bounds.
Nicolas Garcia Trillos - University of Wisconsin Madison
Abstract: Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust to adversarial perturbations of data. In response to this, a popular paradigm that has been developed to enforce robustness of learning models is adversarial training (AT), but this paradigm introduces many computational and theoretical difficulties. Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem. In this talk, I will leverage the MOT connection to discuss new computationally tractable numerical algorithms for computing universal lower bounds on the optimal adversarial risk. The key insight in the AT setting is that one can harmlessly truncate high order interactions between classes, preventing the combinatorial run times typically encountered in MOT problems. I’ll present a rigorous complexity analysis of the proposed algorithm and validate our theoretical results experimentally on the MNIST and CIFAR-10 datasets, demonstrating the tractability of our approach. This is joint work with Matt Jacobs (UCSB), Jakwang Kim (UBC), and Matt Werenski (Tufts).
Location: Gibson Hall 126
Time: 3:00pm
Wednesday, March 6
Algebra and Combinatorics
Topic: Pick's formula and Castelnuovo polytopes
Takayuki HIbi - Osaka University
Abstract: Pick’s formula and Castelnuovo polytopes Let P ⊂ Rd be a lattice polytope of dimension d. Let b(P) denote the number of lattice points belonging to the boundary of P and c(P) that to the interior of P. It follows from the lower bound theorem of Ehrhart polynomials that, when c(P) > 0, vol(P) ≥ (d · c((1) P) + (d − 1) · b(P) − d2 + 2)/d!, where vol(P) is the (Lebesgue) volume of P. Pick’s formula guarantees that, when d = 2, the inequality (1) is an equality. One calls P Castelnuovo if c(P) > 0 and if the equal sign holds in (1). A quick introduction to Ehrhart theory of lattice polytopes will be presented. Furthermore, a historical background on polarized toric varieties to explain the reason why one calls Castelnuovo will be briefly reviewed.
Location: Gibson Hall 126A
Time: 3:00 pm
Tuesday, March 5
Graduate Colloquium
Topic: Intro to Continued Fractions
Peter Marcus - Tulane University
Abstract: Continued fractions are representations of real numbers that use infinitely nested fractions, in contrast to decimal representations which use infinite sums. They provide excellent rational approximations and don't require choosing a base beforehand, which are benefits over decimal representations. However, there are issues of convergence and uniqueness which need to be addressed. I will discuss this as well as more examples and properties of continued fractions.
Location: Gibson 126
Time: 3:30pm
Monday, March 4
Joint AG & GT seminar
Topic: Knot Invariants, Categorification, and Representation Theory
Arik Wilbert - University of South Alabama
Abstract: I will provide a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry central to my research. I will recall basic facts about the representation theory of the Lie algebra sl2 and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. Time permitting, I will discuss how this picture might generalize to other Lie types beyond sl2.
Location: Gibson 308
Time: 2:00pm
Week of March 1 - February 26
Wednesday, March 1
Algebra and Combinatorics
Topic: Algebraic Matroids, Monodromy, and the Heron Variety
Barbara Prinari - University at Buffalo
Abstract: We present the inverse scattering transform to solve the Maxwell-Bloch system of equations that describes two-level systems with inhomogeneous broadening, in the case of optical pulses that do not vanish at infinity in the future. The inverse problem is formulated in terms of a suitable matrix Riemann-Hilbert problem, and the formulation of the direct scattering problem combines features of the methods with decaying as well as non-decaying fields. We also discuss the asymptotic state of the medium and of the optical pulse.
Location: Gibson Hall 126
Time: 3:00 pm
Thursday, February 29
Colloquium
Topic: A broad conjectural framework for the parity of eta-quotients
Fabrizio Zanello - Michigan Tech (Host: Dr. Ha)
Abstract:
One of the classical and most fascinating problems at the intersection between combinatorics
and number theory is the study of the parity of the partition function. Even though p(n) is widely
believed to be equidistributed modulo 2, progress in this area has always proven exceptionally hard. The
best results we have today, obtained incrementally over several decades by Serre, Soundararajan, Ono
and many others, do not even guarantee that, asymptotically, p(n) is odd for √x values of n ≤ x.
In this colloquium talk, we present a new, general conjectural framework that naturally places the
parity of p(n) into the much broader, number-theoretic context of eta-quotients. We discuss the history of
this problem as well as recent progress on our “master conjecture,” which includes novel results on multiand
regular partitions. We then show how seemingly unrelated classes of eta-quotients carry surprising
(and surprisingly deep) connections modulo 2. One instance is the following striking result: If any tmultipartition
function, with t ̸≡ 0 (mod 3), is odd with positive density, then so is p(n). (Note that
proving either fact unconditionally seems entirely out of reach with current methods.)
Throughout our talk, we will also try to give a sense of the many interesting mathematical techniques
that come into play in this area. They include a variety of algebraic and combinatorial ideas, as well as
tools from modular forms and number theory.
Much of this work is in collaboration with my former Ph.D. student S. Judge or with W.J. Keith (see
my papers in the J. Number Theory, 2015, 2018, 2021, 2022, and 2023; Annals of Comb., 2018; Int. J.
Number Theory, 2021 and 2023).
Location: Gibson Hall 126A
Time: 3:30 pm
Tuesday, February 27
Graduate Colloquium
Topic: Inverse PDE Problem: Comparison Between Numerical Method and Physics-informed Neural Network
Lan Trinh | Tulane University
Abstract: In our problem, we’re interested in the number of particles inside biological cells, which is governed by a Poisson spatial process with the intensity measure u(x). This u(x) is shown to satisfy a PDE with two unknown parameters z (source location) and lambda (nondimensional quantity) constructed from the diffusivity constant, emerging rate, and size of the cell. We also let u(x) equal 0 on the domain's boundary U, assuming that the particles are absorbed once hitting it. In this talk, I will discuss a simple version of this model in the 1D case using two methods: finite difference technique and Physics-informed Neural Network, then discuss the advantages, disadvantages as well as a potential combination of these methods for the full model in the 2D case.
Location: Gibson 126A
Time: 3:30pm
Week of February 25 - February 19
Clifford Lectures, February 22-25
Information: Here
Registration: Here
Wednesday, February 21
AMS/AWM
Topic: Mathematical Crossroads: some connections between very different areas of mathematics
Ken McLaughlin - Tulane University
Abstract: I will try to create a snapshot of the research interests of our small group by taking examples from combinatorics, complex analysis, probability theory, and other areas. There will be pictures and there will be mad, mad limits.
Location: Gibson Hall 126A
Time: 4:00 PM
Wednesday, February 21
Algebra and Combinatorics
Topic: Algebraic Matroids, Monodromy, and the Heron Variety
Taylor Brysiewicz - University of Western Ontario
Abstract: Heron's formula gives the area of a triangle in terms of the lengths of its sides. More generally, the volume of any simplex is determined by its edge-lengths via a Cayley-Menger determinant. In this talk, I will discuss which sets of volumes of faces of an n-simplex determine other volumes. The answer to this question is encoded in the algebraic matroid of the Heron variety. Whether this determination is in terms of a formula in terms of radicals is controlled by the monodromy groups of certain branched covers. We answer these questions for n<5 by combining techniques in computational group theory, computer algebra, field theory, and numerical algebraic geometry. Of particular focus is recovering the 10 edge lengths of a 4-simplex from its 10 triangular face areas, a problem motivated by applications in theoretical physics.
Location: Gibson Hall 126A
Time: 3:00 pm
Tuesday, February 20
Graduate Colloquium
Topic: Statistical Phylogenetic Approach to Characterize the Evolutionary Impact of Interlocus Gene Conversion (IGC)
Yufei Zou | Tulane University
Abstract: The Interlocus Gene Conversion (IGC) is a type of mutation that homogenizes repeated DNA sequences. Although substantial progress has been made with regard to inferring nucleotide substitutions that result from point mutations, IGC has typically been ignored when the genomes of related species are studied. This can potentially lead to misleading inferences about evolutionary history and process. Here we apply a composite likelihood approach to IGC inference. By applying this approach to data sets from segmentally-duplicated regions of primates, our results show that evolutionary changes from IGC occur at substantially different rates in different segmentally-duplicated regions.
Location: Gibson 126A
Time: 3:30pm
Week of February 16 - February 12
Friday, February 16
Applied and Computational Math Seminar
Topic: Self-Similar Blow up Profiles for Fluids via Physics-Informed Neural Network
Javier Gomez Serrano - Brown University
Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution (or asymptotically self-similar solution) for different equations in fluid dynamics, such as Euler or Boussinesq. The new numerical framework is shown to be both robust and readily adaptable to several situations. Joint work with Tristan Buckmaster, Gonzalo Cao-Labora, Ching-Yao Lai and Yongji Wang.
Location: Gibson Hall 126
Time: 3:00pm
Thursday, February 15
Colloquium
Topic: DeLTA: Changing Teaching Evaluation through Departmental Action
Paula P Lemons - University of Georgia (Host: TBA)
Abstract: The University of Georgia DeLTA project works toward new core commitments in undergraduate STEM education: collaboration about teaching, basing educational decisions on evidence, and continuously improving our teaching. Modernizing our teaching evaluation is a primary way to achieve these commitments. In the DeLTA project we have achieved change in teaching evaluation by working at the departmental level. A leadership action team of department chairs convenes several times per year to learn about national models for effective teaching evaluation and to exchange ideas about the teaching evaluation practices in their units. Department chairs recruit faculty members who collaborate with faculty from other units to understand, revise, and implement new teaching evaluation practices, such as peer observation by trained peers and instructor self-reflection. The change we have achieved at the department level has been facilitated by changes in policy at the university level. DeLTA research shows that change takes place in departments at different rates and suggests factors that may contribute to departmental outcomes. This seminar will present the UGA DeLTA model, including principles and details about implementation, and will explain our research findings.
Location: Gibson Hall 126A
Time: 3:30 pm
Week of February 9 - February 5
Wednesday, February 7
AMS/AWM
Topic: Tropical Geometry
Kalina Mincheva - Tulane University
Abstract: In this talk I will give a brief overview of tropical geometry and the philosophy behind it. I will introduce algebraic varieties and their tropical counter parts. I will give some applications and open problems associated to them related to toric degenerations and dual curves.
Location: Gibson Hall 126A
Time: 4:00 PM
Wednesday, February 7
Algebra and Combinatorics
Topic: On partial trace ideals of one-dimensional local rings
Souvik Dey - Charles University, Czech Republic
Abstract: In this talk, based on joint work with S. Kumashiro, we define and study a slight generalization of the notion of partial trace ideals and h-invariant of S. Maitra. We show that for one-dimensional local rings, h-invariant of a module is finite if and only if the co-length of its trace is so. For ideals in nice enough local domains of dimension one, we give an explicit tangible formula for the h-invariant. We also discuss some characterizations of rings, including three-generated numerical semigroup rings, whose canonical ideal have low h-invariant, and how the h-invariant of the canonical module changes with respect to forming fiber products and gluing of numerical semigroup rings.
Location: Gibson Hall 126A
Time: 3:00 pm
Tuesday, February 6
Graduate Colloquium
Topic: Rigid Microspheres in a Stokes Fluid: Motion Due to White Noise
Irene Erazo Estrada | Tulane University
Abstract: This talk will center around the dynamic behavior of small spherical particles subjected to externally applied random forces while immersed in a viscous fluid. In contrast to the stochastic immersed boundary method which averages fluctuating random forces within the particle location, here, these forces are in the surrounding fluid, external to the particle surfaces.
Location: Gibson 126A
Time: 3:30pm
Week of February 2 - January 29
Friday, February 2
Applied and Computational Math Seminar
Topic: The planar Coulomb gas on a Jordan curve
Klara Courteaut - NYU Courant
Abstract: The eigenvalues of a uniformly distributed unitary matrix (CUE) have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle and at inverse temperature 2. In this talk, I will present a more general model in which the unit circle is replaced by a sufficiently regular Jordan curve, at any positive temperature. In a paper with Johansson, we obtained the asymptotic partition function and the Laplace transform of linear statistics at any positive temperature. These can be expressed using either the exterior conformal mapping of the curve or its associated Grunsky operator.
Location: Gibson Hall 126
Time: 3:00pm
Friday, February 2
Applied Math / Probability and Statistics
Topic: Stochastics in medicine: Delaying menopause and missing drug doses
Sean Lawley - University of Utah
Abstract: Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation. The first problem is based on joint work with Joshua Johnson (University of Colorado School of Medicine), John Emerson (Yale University), and Kutluk Oktay (Yale School of Medicine).
Location: 12:00 pm
Time: Stanley Thomas 316
Thursday, February 1
Colloquium
Topic: Exponential generating functions and their congruences in enumerative combinatorics
Ira Gessel - Brandeis University (Host: Amdeberhan)
Abstract: Enumerative combinatorialists study sequences of integers that count things, and some of us like to find congruences for these integers. I will talk about sequences that have nice exponential generating functions, which are power series in which the numbers of interest are the coefficients of x^n/n!. An important example is the exponential generating function exp(exp(x) -1) for the Bell numbers, which count partitions of a set. I will first discuss how exponential generating functions are used in enumeration. Then I will discuss three methods for finding congruences for coefficients of exponential generating functions. The first method (which does not actually use the generating function) is the combinatorial method: Suppose that we have a finite group acting on a set S. If every element of S is in an orbit of size divisible by m, then the size of S is divisible by m. The second method, the umbral method, works with recurrences that are not so easily derived directly from generating functions. The third method uses the algebra of exponential generating functions modulo a prime, and differential operators on this algebra.
Location: Gibson Hall 126A
Time: 3:30 pm
Tuesday, January 30
Graduate Colloquium
Topic: Student Activities in Mathematics at Tulane
Sang-Eun Lee - Tulane University
Abstract: We will wrap up the activity we did last semester and propose this semester's events.
Location: TBA
Time: 3:30pm
Week of January 19 - January 15
Thursday, January 18
Math Club
Maggie Lai, Tulane Math Club President
Topic: Floer homology and algebraic geometry
Nikolai Saveliev - University of Miami (Host: Komendarczyk)
Abstract: Machine learning is quickly becoming embedded in everyday applications. It’s becoming essential for
students and educators to adopt this technology to solve complex real-world problems. MATLAB and
Simulink provide a flexible and powerful platform to develop and automate data analysis, deep learning,
AI, and simulation workflows in a wide range of domains and industries. In this workshop we will
introduce machine learning with MATLAB. We will utilize a previously trained network and modify it,
using the MATLAB Deep Network Designer. The Deep Network Designer allows you to interactively
build, visualize, and train neural networks. Individuals can generate the code for the neural network and
fine-tune parameters. Users can use popular pre-trained networks or construct their own. We will also
look at the MATLAB Classification Learner to run several models on a single data set. These visual
approaches create a more efficient workflow.
Jon Loftin is a Customer Success Engineer at MathWorks. Jon’s background is in mathematics. More
specifically, implementing mathematics in a computer. He holds degrees in mathematics: a BS from
Southern Arkansas University, a MS from the University of Arkansas, and a Ph.D. from Texas Tech
University. He has had years of teaching experience, from teaching at the Naval Nuclear Power School to
teaching as an Assistant Professor. Jon’s research focus is building efficient integration techniques in
finite element methods.
Location: Newcomb Institute 300, Diboll Gallery (3rd floor of Commons)
Time: 5:00-6:30PM