The Mathematics Department holds regular seminars on a variety of topics. Please see below for further details.

Seminars

Seminar Meeting Details Title & Abstract
Algebra Seminar
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place
MSB 110
TBA
Speaker: Yuchen Liu, Northwestern University
Differential Equations Seminar
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place
MSB 111
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Nonuniqueness for continuous solutions to 1D conservation laws

In this talk, we will show that a geometrical condition on \(2 \times 2\) systems of conservation laws leads to nonuniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone—which ensures the uniqueness of small BV solutions—is insufficient to guarantee uniqueness in the continuous setting, even within the mono-dimensional frame. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process. Notably, this result represents the first application of convex integration to construct non-unique continuous solutions in one spatial dimension. This is a joint work with Alexis Vasseur and Cheng Yu.
 

Speaker: Ming Chen (University of Pittsburgh)
Differential Equations Seminar
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place
MSB 111
group
On the precise cusped behavior of extreme solutions to Whitham-type equations

We prove exact leading-order asymptotic behaviour at the origin for nontrivial solutions of two families of nonlocal equations. The equations investigated include those satisfied by the cusped highest steady waves for both the uni- and bidirectional Whitham equations. The problem is therefore analogous to that of capturing the 120∘ interior angle at the crests of classical Stokes’ waves of greatest height. In particular, our results partially settle conjectures for such extreme waves posed in the series of recent papers by Ehrnström, Johnson, and Claassen (2019), Ehrnström and Wahlén (2019), and Truong, Wahlén, and Wheeler (2022). Our methods may be generalised to solutions of other nonlocal equations, and can moreover be used to determine asymptotic behaviour of their derivatives to any order.

Speaker: Kristoffer Varholm (University of Pittsburgh)
Pre-print Algebra Seminar
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place
MSB Room 12
The Asymptotic Samuel Function of a Filtration (Part 2)

We continue the discussion on projective equivalence of filtrations. We further look at the discrete valued filtrations and show that they have particularly nice properties . We generalize some of the theory of Rees valuations of ideals to these filtrations.

Speaker: Mifron Fernandes
Data Seminar
event
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place
MSB 110
Degree Bounds for Rational Invariants

Degree bounds have a long history in invariant theory. The Noether bound on the degrees of algebra generators for a ring of invariants is over a century old, and there is a vast literature sharpening and generalizing it. In the last two decades, there has also been an active program on degree bounds for invariants which are able to distinguish orbits as well as algebra generators can (known as separating invariants).

In this talk I make the case that generators for the field of rational invariants represent an exciting avenue for research on degree bounds as well. I present new lower and upper bounds. It will transpire that even the case of G=Z/pZ, uninteresting from the point of view of generating and separating invariants, has a story to tell for rational invariants. The methods involve the classical Minkowski “geometry of numbers”. I argue that this domain of inquiry is both interesting in itself and well-motivated by applications.

This talk is based on joint work with Thays Garcia, Rawin Hidalgo, Consuelo Rodriguez, Alexander Kirillov Jr., Sylvan Crane, Karla Guzman, Alexis Menenses, Maxine Song-Hurewitz, Afonso Bandeira, Joe Kileel, Jonathan Niles-Weed, Amelia Perry, and Alexander Wein.

Speaker: Benjamin Blum-Smith (Johns Hopkins)
Analysis Seminar
event
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place
Math Sci 111
Multiplier Weak-Type Inequalities for the Maximal Operator

We discuss a kind of weak-type inequality for the Hardy-Littlewood maximal operator that was first studied by Muckenhoupt and Wheeden. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior; in particular, as shown by Muckenhoupt and Wheeden, the class of weights characterizing such inequalities is strictly larger than $A_p$. In this talk, I will present recent work on the full characterization of the weights for which these inequalities hold for the Hardy-Littlewood maximal operator. Connections to weak-type estimates for the Bergman projection on bounded simply connected planar domains will also be discussed.

Speaker: Brandon Sweeting (Washington University in St Louis)
Algebra Seminar
event
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place
MSB 110
Noncommutative geometry over dg-algebras

The general motto of noncommutative algebraic geometry is that any category sufficiently close to the derived category of a variety should be regarded as a noncommutative variety. Using this principle, pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative setting. I’ll talk about extensions of this theory to noncommutative spaces associated to dg-algebras with a focus on how it feeds back into the commutative setting. In particular, I’ll discuss a generalization of a celebrated theorem of Orlov concerning the derived category of a projective complete intersection. Joint work with Michael K. Brown.

Speaker: Prashanth Sridhar, Auburn University
Geometry and Topology Seminar
event
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place
MSB 312
Orthogonal Mobius Inversion and Grassmannian Persistence Diagrams

Signed persistence diagrams provide a concise representation of the rank invariant (persistent Betti numbers) or birth-death invariant for filtrations of topological spaces. When these filtrations are indexed by a linear poset, the signed persistence diagrams consist of nonnegative values, allowing for straightforward interpretation as the multiplicity of the bars in the barcode of the filtration. However, for filtrations indexed by non-linear posets, signed persistence diagrams can include negative values, which complicates their interpretability. In this talk, we introduce Grassmannian persistence diagrams, a notion that extends the capabilities of signed persistence diagrams while preserving interpretability. Moreover, when the indexing poset is linear, Grassmannian persistence diagrams also reveal canonical cycle spaces for each bar in the barcode associated with the filtration.

Speaker: Aziz Burak Gülen (Duke University)
Differential Equations Seminar
event
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place
MSB 111
group
A two-boson lattice Hamiltonian with interactions up to next-neighboring sites

The goal of this talk is to discuss spectral theory for quantum Hamiltonians describing  a system of two  identical spinless bosons on the two-dimensional lattice particles.

We assume that the particles are interact via on-site and first and second nearest-neighboring site interactions between the bosons are only nontrivial and that these interactions are of magnitudes \(\gamma\), \(\lambda\), and \(\mu\), respectively.

Introducing the quasi-momentum \(k\in (-\pi,\pi]^2\)  for the system of two identical bosons,  and decomposing the  Hamiltonians into the direct von Neumann integral reduces the problem to the study  two-particle fiber lattice Schrödinger operators that depend on the two-particle quasi-momentum \(k\).

In this model, we determine both the exact number and location of the eigenvalues for the lattice Schrödinger operator \(H_{\gamma\lambda\mu}(0)\), depending on the interaction parameter \(\gamma\), \(\lambda\) and \(\mu\).

We find a partition of the \((\gamma,\lambda,\mu)\)-space into connected components such that, in each connected component, the two-boson Schrödinger operator corresponding to the zero quasi-momentum of the center of mass has a definite (fixed) number of eigenvalues, which are situated below the bottom of the essential (continuous) spectrum and above its top. Moreover,  for each connected component, a sharp lower bound is established on the number of isolated eigenvalues for the two-boson Schrödinger operator corresponding to any admissible nonzero value of the center-of-mass quasimomentum.

We will reveal the mechanisms of emergence and absorption of the eigenvalues at the thresholds of the essential (continuous) spectrum of \(H_{\gamma\lambda\mu}(0)\), as the parameters \(\gamma\), \(\lambda\) and \(\mu\) vary.

In addition, we have identified the sufficient conditions for the exact number of eigenvalues of the lattice Schrödinger operator for all values of the interaction parameters. 

Speaker: Saidakhmat Lakaev (Samarkand State University)
Pre-print Algebra Seminar
event
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place
MSB Room 12
The Asymptotic Samuel Function Of A Filtration (Part 1)

We extend the asymptotic Samuel function of an ideal to a filtration and study some properties of the function. We look at the relation between integral closure of filtrations and the asymptotic Samuel function. We further study the notion of projective equivalence of filtrations. The talk follows the pre-print ‘The Asymptotic Samuel Function of a Filtration’ by Dale Cutkosky and Smita Praharaj.

Speaker: Mifron Fernandes