Last Day of Class, Spring 2025

Reading Days and Exams, Spring 2025

Commencement 2025

Szego Seminar: 4/17/25

Abstract: TBA

Roever Lecture: The optimal paper Moebius band

Abstract: If you have a rectangular strip of paper with a large aspect ratio, meaning that it is long and thin, you can twist it around in space and tape the ends together to make a paper Moebius band. If the aspect ratio is small, you can't do this. What is the cutoff? In this talk, I will prove that you can make the Moebius band if and only if the aspect ratio is greater that sqrt(3). I'll also explain why the Moebius band looks very much like an equilateral triangle if the aspect ratio is near sqrt(3). These results answer the conjecture of B. Halpern and C.

Geometry & Topology Seminar: Pappus's Theorem, Patterns of Geodesics, and the Barbot Component

Abstract: Pappus's Theorem is a classic theorem in projective geometry. First, I will explain how the iteration of Pappus's Theorem leads to representations of the modular group into Isom(X), where X is the 5-dimensional symmetric space associated to SL₃(R). Second, I will explain how the famous Farey triangulation in the hyperbolic plane can be placed inside X and then bent like a pleated plane. The Pappus groups are symmetry groups of these high rank pleated planes.

Sonia Kovalevsky Math Day at WashU

The annual Sonia Kovalevsky Math Day at Washington University in St. Louis for Middle School Girls honors the first woman to receive a doctorate in mathematics and encourages young women to continue their study of mathematics. Join students and faculty from WUSTL and SLU for a fascinating look into the world of mathematics; including:

- Codes and Ciphers

- The Mathematics of Embroidery 

- Math-themed Puzzles and Games

- Coloring Maps

- Jobs Panel

Harmonic Analysis Working Seminar: 4/9/25

Host: Brett Wick

Harmonic Analysis Working Seminar: 4/16/25

Host: Brett Wick

Colloquium: 4/24/25

Abstract: TBD

Host: John McCarthy

Reception to follow at Cupples I, Room 200 (Lounge) from 3:00 - 4:00 pm.

Colloquium: The Topology of Algebraic Singularities

Singularities are a foundational object in algebra but it they also have a long and important history in topology. I will tell you a bit about that story along with a topological look at some of the examples (and interesting pictures) that arise in low dimensions.

Host: Minh Nguyen

Geometry & Topology Seminar: Singular Fibers in Families of Algebraic Curves

Kodaira gave a classification of singular fibers that can arise in algebraic families of genus 1 curves. This plays an important role in the symplectic topology of elliptic fibrations, where, via Harer-Kas-Kirby, one connects the symplectic topology of a configuration of curves to a monodromy factorization within the genus 1 mapping class group. I'll discuss a project to generalize this to families of genus 2 curves utilizing a classification by Namikawa and Ueno. These families were also studied by Matsumoto-Montesinos in the context of 3-manifolds and surface automorphisms.

Combinatorics Seminar: An Erdos-Szekeres Permutation Game

Abstract: Consider a two-player game where players take turns building a permutation. The first player to complete an increasing subsequence of length a or a decreasing subsequence of length b loses. While the Erdos-Szekeres Theorem gives an upper bound on how long this game lasts, we analyze strategies for optimal game play, especially in the cases where b is at most 5.

Host: Martha Precup

Geometry & Topology Seminar 4/18/25

Abstract: TBD

Host: Rachel Roberts

AAG: A new approach to the extendability of projective varieties: No Gaussian maps

Abstract: The topic of extendability of a projective variety has attracted a lot of attention among algebraic geometers. The approaches to tackle the extendability questions have almost always involved the Gaussian maps of the curve sections. In this talk we introduce a new approach that totally avoids the Gaussian maps of the curve section and a host of issues related to them.

First Summer Class Session Begins - Summer 2025

Memorial Day Holiday 2025 - no classes

Independence Day Holiday 2025 - no classes

Last Summer Class Session Ends - Summer 2025

First Day of Classes, Fall 2025

Labor Day 2025 - no classes

Fall Break 2025 - no classes

Thanksgiving Break 2025 - no classes

Reading Days and Exams - Fall 2025

Third Year Major Oral: What do we know about nonregular Hessenberg varieties?

Abstract: A Hessenberg variety is an important subvariety of the flag variety characterized by a linear operator and a Hessenberg function. It has rich properties related to representation theory, combinatorics, and geometry, and has been studied in multiple ways. 

In 2005, Tymoczko posed several open questions about the geometric structure of Hessenberg varieties. Those questions remain open in general, since most work on Hessenberg varieties focuses on the regular case. 

Awards Ceremony, 2025

Awards Ceremony for undergraduates and graduates in the Department of Mathematics.

There will be a tea reception preceding the ceremony will start at 3:00 in Cupples I, Room 200 (Lounge). The ceremony will take place in Cupples I, Room 199 from 4pm-5pm.

Colloquium: Gross and Zagier’s work revisited

Abstract: In 80s, Gross and Zagier discovered and proved a deep and well-known formula between derivative of an L-function and the height of some `CM’ point on an elliptic curve---Gross-Zagier formula, which gives partial answer the Birch and Swinnerton-Dyer conjecture. In the process, they also proved a beautiful factorization formula for the difference of CM values of the $j$-function. In addition, they also gave a conjecture about the algebraicity of the CM values of higher Green functions.

Third Year Major Oral: Squiggly Circles

Abstract: Given a self-homemorphism $\varphi$ of a surface $\Sigma$, we can form the mapping torus, $M_\varphi$, a 3-manifold fibering over the circle. In this talk, I will begin by covering the Nielsen–Thurston classification of surface automorphisms---discussing the topological properties of periodic, reducible, and pseudo-Anosov maps with an additional focus on the structure of the mapping torus. I will then discuss the work of Cannon and Thurston studying the geometric structure of the universal cover of hyperbolic surface bundles.

Combinatorics Seminar: Ehrhart polynomials of generalized permutohedra from A to B

Abstract: I’ll show how to derive a formula for the Ehrhart polynomials for the type-$B$ generalized permutohedra, which offer a more concise alternative to the recent formula obtained by Eur, Fink, Larson, and Spink from their study of delta-matroids. My approach utilized some of the techniques and tools introduced around 20 years ago by Postnikov from his study of generalized type-$A$ permutohedra, a family of polytopes that interconnects with many mathematical concepts such as matroids, graphs, and Weyl groups.

K-multimagic squares and magic squares of k-th powers via the circle method

Abstract: Here we investigate K-multimagic squares of order N. These are N x N magic squares which remain magic after raising each element to the k^th power for all 2 ≤ k ≤ K. Given K  ≤  2, we consider the problem of establishing the smallest integer N₂(K) for which there exist nontrivial K-multimagic squares of order N₂(K). 

Thesis Defense: The Witten Deformation and Proper Cocompact Lie Group Actions

Thesis Defense: Paraproducts in Two Settings: Hankel Operators and Dyadic Paraproducts

Thesis Defense: Functional Equivariance and Backward Error Analysis

Abstract: Geometric, or structure-preserving, numerical integration has long been used as a framework for studying integrators that preserve a systems invariants. In backward error analysis, the approximate numerical flow of a system is viewed as the exact flow of a modified problem, allowing us to gain qualitative insights into the behavior of the numerical solution. The preservation of conservation laws by a numerical integrator can be generalized to F-functionally equivariant integrators, where F represents an observable of the  system in consideration.

Thesis Defense: Studies in Algebraic Cycles: Hodge Theory of Degenerations, Regulators, and Cluster Varieties

Abstract: TBD

Advisor: Matt Kerr

Seminar: Quasi-Representations of Discrete Groups

In this talk, I will explain how to classify quasi-representations of discrete groups using their characters. I will aim to make the lecture accessible to graduate students. This is joint work with Shmuel Weinberger and Jianchao Wu.

Host: Xiang Tang

Thesis Defense: Wavelet Representation of Singular Integral Operators

Abstract: The idea of representing singular integral operators as averages dyadic shifts has proven fruitful since Petermichl's representation of the Hilbert transform, and its generalization by Hytönen to prove the A_2 conjecture. An alternate approach to wavelet representation was provided by Di Plinio, Wick, and Williams (2022) in which the random dyadic grids are replaced by zero-complexity wavelet projections, providing finer control of smooth operators.

Colloquium: Spectral Learning for Dynamical Systems

Abstract: Koopman operators recast nonlinear dynamics as infinite-dimensional linear systems, enabling spectral analysis of time-series data. Over the past decade, they've found widespread use in fields ranging from robot control and climate variability to neural networks, epidemiology, and neuroscience. But which spectral features can actually be reliably computed from finite data-and which cannot? We show that the answer depends critically on the choice of observable space (e.g., L2 vs RKHS) and the nature of the underlying dynamics.

Seminar: Two Tales on M_{0,5} and M_{0,6}

Abstract: I will tell two stories where (the compactification of) moduli spaces M_{0,5} and M_{0,6} appear in a significant way. One is related to the volume preserving birational maps of P^2 and P^3 . The other one is related to the irrationality proofs for \zeta(2) and \zeta(3).  Both stories are related to each other via mirror symmetry for the del Pezzo surface of degree 5 and the Fano 3-fold V_{12}. However, I will speculate that the relations between these two stories might be subtler using the group structure for \zeta(2) and \zeta(3) studied by Rhin and Viola.

Algebraic Geometry and Combinatorics Seminar: d-elliptic loci and quasi-modular forms

Abstract:  Let N_{g,d} be the locus of curves of genus g admitting a degree d cover of an elliptic curve. For fixed g, it is conjectured that the classes of N_{g,d} on M_g are the Fourier coefficients of a cycle-valued quasi-modular form in d. A key difficulty is that these classes are often non-tautological, so lie outside the reach of many known techniques. Via the Torelli map, the conjecture can be moved to one on certain Noether-Lefschetz loci on A_g, where there is accesss to different tools.

Taibleson Colloquium: Quantum and classical low degree PAC learning, Harmonic analysis, and dimension-free Remez inequality

Abstract: The talk is based on recent works with Lars Becker, Ohad Klein, Joseph Slote, and Haonan Zhang. Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including noncommutative spaces (matrix algebras).

Roever Colloquium: Kähler-Einsten metric, K-stability and moduli of Fano varieties

Abstract: A complex variety with a positive first Chern class is called a Fano variety. The question of whether a Fano variety has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. The Yau-Tian-Donaldson Conjecture predicts the existence of such a metric is equivalent to an algebraic condition called K-stability.

First Day of Classes, Spring 2026

Colloquium: Tropical Algebra

Abstract: Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue.

Mathematics Department: Welcome BBQ

We’re excited to announce that registration is now open for the WashU Mathematics Department Welcome BBQ!

Algebraic Geometry and Combinatorics Seminar: Free curves on Fano varieties

Abstract: Rational curves play a pivotal role in understanding the birational geometry of varieties. Free curves are the easiest to work with, but on Fano varieties that are even mildly singular, it remains an open question whether these free rational curves exist. In this talk, we discuss an improvement in Mori's Bend-and-Break that achieves the optimal degree bound and allows us to improve our understanding of sweeping families of rational curves in singular varieties.

Analysis Seminar: Critical sets of harmonic functions and Almgren's frequency function

Abstract: Given a harmonic function on a nice domain in n-dimensional Euclidean space, its critical set (the set of points where its gradient vanishes) is known to have dimension at most n-2, with locally finite (n-2)-dimensional Hausdorff measure. For harmonic polynomials, the Hausdorff measure can be bounded in terms of the degree of the polynomial. For more general harmonic functions, Almgren's frequency function serves as an analog of the degree of a polynomial, measuring local growth properties of the harmonic function.

Analysis Seminar: Hyperbolic Analytic and Algebraic Curves

Abstract: Hyperbolic analytic curves are analytic generalizations of hyperbolic algebraic curves; ‘cut outs’ of algebraic sets intersected with bounded open sets in complex n space. To facilitate the analysis of these objects one can treat them as image sets of Riemann surfaces under holomaps; formally, holomorphic proper functions which are non-singular and injective away from an at most finite set.

Analysis Seminar: Polynomials with no zeros on the unit ball

Abstract: There is a large and well-developed theory with numerous applications for polynomials with no zeros on the polydisk (and the equivalent family of polynomials with no zeros on the product of upper half planes).  The other standard model domain, the unit ball in C^n, has virtually no theory for the class of polynomials with no zeros there.  We will review a few things that are known and then discuss recent work about a local description of the zero set of a polynomial with no zeros on the unit ball but with a boundary zero.

Analysis Seminar: Invariant metrics

Abstract: An invariant metric is a metric that is invariant under biholomorphic mappings, and for which all holomorphic mappings are contractive. On bounded convex sets, a theorem of Lempert says that all such metrics coincide (and therefore equal the Caratheodory metric, which we will define). We will discuss subsets V of the polydisk with the property that their intrinsic Caratheodory metric agrees with the one inherited from the polydisk. This is the complex geometry analogue of looking at subsets of a Riemannian manifold that are totally geodesic.

Host: Alan Chang

Martin Luther King Jr. Day - no classes

Colloquium: The Arduous Path of AI Models to the Bedside

Abstract: In this talk, I will describe the fundamental challenges for researcher-developed AI models (both deep learning and generative AI) to be brought to the point-of-care. Specifically, using examples from my own research in clinical medicine, I will discuss the pragmatic challenges model innovation, causal reasoning, and clinician-AI collaboration. I will also highlight key areas for computational innovation in clinical medicine, and collaborative opportunities for federal and philanthropic funding.

Host: John McCarthy

Spring Break - no classes

Reading Days and Exams, Spring 2026

Algebraic Geometry and Combinatorics Seminar: Tight pairs, Kumar's criterion, inversion arrangements, and Bruhat intervals in type A and beyond

Abstract: Motivated by questions in representation theory, Lapid defined the notion of a tight pair of permutations and showed that tight pairs indexed smooth points in Schubert varieties.  His proof is entirely combinatorial and he asked for a geometric context for his statement.  I'll describe a conjecture based on Kumar's criterion for smoothness using restriction of cohomology classes.  The permutations that form a tight pair with the identity have also come up in other contexts.  Finally I'll also describe a potential generalization to other simple Lie groups.

Algebraic Geometry and Combinatorics Seminar: TBD

Abstract: TBD

Host: John Shareshian

Colloquium: Transverse Spheres in Flag Manifolds

Abstract: Flag manifolds are natural objects in algebraic and differential geometry, generalizing familiar examples like projective space and Grassmannians. A subset of a flag manifold is called transverse when every pair of points is in general position. Transverse circles abound. Conversely, recent restriction results show that in certain cases transverse circles are maximally transverse. We shall detail a complementary result. Using spinors, we build arbitrarily large transverse spheres in real full flag manifolds.

Algebraic Geometry and Combinatorics Seminar: Two techniques for understanding the intersection theory of Hurwitz spaces of degree-3 covers

Abstract: The Hurwitz space is a moduli space parametrizing branched covers of curves. Harris and Mumford introduced the ``admissible covers'' compactification of the Hurwitz space, in which the target and source curves of a cover degenerate into nodal curves when branch points come together. The boundary of the Hurwitz space is then stratified by lower-dimensional Hurwitz spaces. This structure is strikingly similar to the stratification of the Deligne-Mumford compactification of the moduli space of curves.

Algebraic Geometry and Combinatorics Seminar: Geometry and topology of universal compactified Jacobians

Abstract: For a family of smooth curves, the relative Jacobian parametrizing degree 0 line bundles is a fundamental example of a family of principally polarized abelian varieties. Over the moduli space of stable curves, the relative Jacobian admits several different smooth toroidal compactifications. The study of such abelian fibrations appears in the study of Higgs bundles, hyperkahler manifolds, enumerative geometry and logarithmic geometry.

Algebraic Geometry and Combinatorics Seminar: Singularities of Secant varieties

Abstract: Secant varieties are classical objects in algebraic geometry. Given a smooth projective variety inside a projective space, its secant variety is by definition the closure of the union of secant lines. It is almost always singular and sits inside the same projective space by its construction. In this talk, we will discuss the singularities of secant varieties when the embedding is sufficiently positive. In particular, we will study the Du Bois complex of secant varieties and will also discuss about its local cohomology modules.

Geometry & Topology Seminar: Torsion Invariants for Hecke Correspondences on Compact Hyperbolic Spaces

Abstract: We begin with the Reidemeister (R-) torsion, a combinatorial invariant of manifolds with flat vector bundles, and its analytic counterpart defined via Laplacian determinants. The  Cheeger–Müller theorem identifies these two notions of torsion in the acyclic case. We then turn to the Ruelle zeta function, a dynamical invariant built from the closed geodesics of the hyperbolic manifold, and discuss Fried’s conjecture — which relates its value at zero to analytic torsion.

Houston Kirk Colloquium: Incidence geometry and tiled surfaces

Abstract: We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a tiling of a closed oriented surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones. This is joint work with Pavlo Pylyavskyy.

Host: John Shareshian

Reception to follow at Cupples I, Room 200 (Lounge) from 2:00pm to 3:00pm

Analysis Seminar: From Subrepresentation to Sobolev: New Insights

Abstract: The classical subrepresentation formula shows that the size of a smooth function can be controlled pointwise by the Riesz potential of its gradient. The proof is a simple consequence of the fundamental theorem of calculus and the use of polar coordinates. When combined with the boundedness of the Riesz potential, this pointwise estimate gives rise to Sobolev inequalities in very general settings.

Colloquium: Tropical geometry and curve counting

Abstract: The goal of this colloquium is to introduce a circle of ideas and techniques used to deal with enumerative geometric problems of curves, concerned with finding the number of curves of a certain type that satisfy a certain number of geometric conditions.

Geometry & Topology Seminar: Expanding Ricci solitons asymptotic to cones with nonnegative scalar curvature

Abstract: In dimensions four and higher, the Ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature. It may be possible to resolve such singularities and continue the flow using expanding Ricci solitons asymptotic to these cones, if they exist. I will discuss joint work with Richard Bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding Ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.

Host: Charles Ouyang

Algebraic Geometry and Combinatorics Seminar: Algebraic vector bundles of rank 2 over smooth affine fourfolds

Abstract: To what extent do Chow-valued Chern classes determine the isomorphism class of an algebraic vector bundle? In this talk, I'll discuss some progress on this question for algebraic vector bundles of rank 2 over smooth affine fourfolds. These results imply some concrete cohomological classification results (e.g., over the complex numbers, there are exactly 9 isomorphism classes of rank 2 vector bundles over the complement of a smooth degree 3 hypersurface in P^4). I'll also highlight some possible computations that, if completed, would shed further light on this problem.

Analysis Seminar: Tb Theorem for Singular Integral Operators in the Dunkl Setting

Abstract: In this talk, we will discuss a theorem for a family of singular integral operators in the Dunkl setting. These operators, which extend the behavior of the Dunkl–Riesz transforms, have kernels satisfying estimates that lie outside the Coifman–Weiss framework for singular integral operators on general spaces of homogeneous type. In analogy with the theorem of David, Journé, and Semmes, our ongoing work aims to establish boundedness criteria via testing conditions against para-accretive functions adapted to the Dunkl structure.

Host: Alan Chang

Math Chat: Prof. John McCarthy

This is an event for graduate students in the Department of Mathematics.

Get to know your professors! This week: John McCarthy.

Host: Ali Daemi

Math Club: Organizational Meeting

Algebraic Geometry and Combinatorics Seminar: Local inequalities for $cA_k$ singularities

Abstract: Proofs in birational geometry have traditionally involved a composition of blowups at smooth centres. At the same time, the most natural description of divisorial contractions in dimension 3 uses weighted blowups. In this talk, I will discuss a generalization of an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blowups. As an application, we prove nonrationality of many families of terminal Fano 3-folds. This is a joint work with Igor Krylov and Takuzo Okada.

Host: Matt Kerr

Geometry & Topology Seminar: The Diederich–Fornæss index and the ∂ ̄-Neumann problem

Abstract: In the study of complex geometry, a deep connection exists between the geometry of a domain and the analysis of functions within it. This talk explores domains in C^n that are pseudoconvex—the natural generalization of convex sets from Euclidean space.

Colloquium: TBD

Abstract: TBD

Host: Xiang Tang

Reception to follow at Cupples I, Room 200 (Lounge) from 2:00pm to 3:00pm

Geometry & Topology Seminar: Reflections on Bubbles and Films

Abstract: Since the work of Alexandrov in the 1950s, the presence of reflection symmetries has been a key to understanding the geometry of constant mean curvature and minimal surfaces embedded in 3-dimensional space forms.  Three decades later, this opened the door to a moduli space theory for such surfaces, particularly sub-moduli spaces of coplanar surfaces, with surprising connections to Teichmüller theory and CP^1-structures.  We'll discuss our long-standing work on these moduli spaces, along with recent work (with McGrath, and later with him, Karpukhin and Stern) employing an intri

Analysis Seminar: Semi-algebraic discrepancy estimates for skew-shift sequences

Abstract: The discrepancy of a sequence essentially measures how often a sequence visits a particular set. We show that the semi-algebraic discrepancy of a skew-shift sequence on the b-dimensional torus \mathbb{T}^b behaves differently for large b and small b, with a transition at b = 6. The key to our analysis will be the Weyl summation method for exponential sums and the Vinogradov mean value theorem recently proved by Bourgain, Demeter, and Guth. If time permits, we will discuss applications to problems in mathematical physics. This is based on joint work with W. Liu and X.

Geometry & Topology Seminar: Domains of discontinuity for Anosov representations

Abstract: We consider geometric structures on closed manifolds that are modeled over generalized flag manifolds, for example real or complex projective structures. We know that some manifolds admit a large deformation space of geometric structures, this can be seen from the theory of Anosov representations. A particularly interesting case comes from the Higher Teichm\"uller Spaces, spaces that generalize the classical Teichm\"uller spaces for higher rank Lie groups.

Algebraic Geometry and Combinatorics Seminar: Geometric and combinatorial connections between the Delta Theorem and the Rational Shuffle Theorem

Math Club: Magic Squares

Colloquium: From knots to 4-manifolds

Abstract: I will survey the connections between knot theory and four-dimensional topology. Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to progress in computing invariants of smooth four-manifolds that can detect exotic structures. I will explain how this was done in two contexts: Heegaard Floer theory and skein lasagna modules.

Host: Aliakbar Daemi

Algebraic Geometry and Combinatorics Seminar: Homotopical combinatorics and equivariant derived algebraic geometry

Abstract: Equivariant derived algebraic geometry (EDAG) seeks a “spectral algebraic geometry with norms,” where the basic functions are genuine G-E_∞ rings (from equivariant stable homotopy theory) and descent must remember restriction, transfer, and norm data. A central obstacle is conceptual and technical: how do we control which norms are present so that geometry is flexible enough to glue, but rigid enough to compute? This talk proposes transfer systems as the correct combinatorial lens.

Analysis Seminar: Attainability of the singular Wiener bound and leaf venation patterns

Abstract: I will introduce my recent work on the attainability of the singular Wiener bound and its applications in modeling the geometry of leaf venation patterns. This new model explains the reticulation phenomenon in the higher-order vein patterns by applying techniques that come from homogenization theory, geometric measure theory and homotopy groups. I will also introduce a novel application of stationary varifolds in modeling the leaf venation patterns based on a new principle that conductance maximality is equivalent to area criticality. 

Host: Ben Foster

Analysis Seminar: scalar and vector valued dyadic operators in the non-homogeneous setting

Abstract: In the classical setting, dyadic analysis has proven to be a powerful framework for studying weighted inequalities for both scalar and vector valued Calderón–Zygmund operators. While the scalar theory is now well understood, sharp matrix-weighted inequalities are not fully resolved. The best-known matrix-weighted L^p estimates (when Lebesgue is the underlying measure) were obtained in 2018 by Cruz-Uribe, Isralowitz, and Moen, and were shown to be sharp— though only for p=2 and for specific Calderón–Zygmund operators—by Domelevo, Petermichl, Treil, and Volberg in 2024.

Analysis Seminar: Nonlinear Projections and Quantitative Rectifiability

Abstract: From the delicate geometry found in a snowflake to the intricate patterns of a coastal shoreline, nature holds infinite patterns and scales. The world is not easily described using mere lines and cones, and classic Euclidean geometry falls short. The notion of fractals gives us a language and a set of tools to understand more complex phenomena. The modern application of fractals spans both pure and applied mathematics - from the study of lung vasculature to surprising constructions of counter examples.

Math Club: Magic Squares

Math Club: TBD

Math Club: TBD

Math Club: TBD

Math Club: Panel Discussion on Graduate Programs in Math

Math Club: TBD

Algebraic Geometry and Combinatorics Seminar: Slicing correspondences with high degree hypersurfaces.

Abstract: In this talk I will briefly discuss the concept of correspondence degree as defined by Lazarsfeld and Martin. After that I will discuss a result that  (under some numerical assumptions) computes the correspondence degree between two hypersurfaces in a smooth variety- this answers a question of theirs under some extra hypotheses.

Host: Roya Beheshti

Analysis Seminar: Gradients of single layer potentials for elliptic operators with coefficients of DMO-type and applications to elliptic measure

Abstract: We study a uniformly elliptic operator L_A in divergence form, associated with an (n+1)×(n+1) matrix A with real, bounded, and possibly non-symmetric coefficients. Assuming that a suitable L^1-mean oscillation of the coefficients of A satisfies a Dini-type condition, we establish a rectifiability criterion for Radon measures in terms of the operator

T_μ f(x) = ∫ ∇_x Γ_A(x, y) f(y) dμ(y),

where Γ_A(x, y) denotes the fundamental solution associated with L_A.

Analysis Seminar: Harmonic extensions on ℤ × ℕ and a discrete Hilbert transform

Abstract: The classical Hilbert transform of a function on the real line arises as the boundary value of the function's conjugate-harmonic extension. In this talk, harmonic extensions to the upper half-integer lattice ℤ × ℕ are constructed for a given boundary sequence on the integers; these serve as discrete analogs of the Poisson and conjugate-Poisson integrals. The construction is characterized by: (i) discrete harmonicity with respect to a two-dimensional Laplacian, (ii) a Cauchy–Riemann system, and (iii) boundary values involving a discrete Hilbert transform.

Geometry & Topology Seminar: Marked boundary rigidity and Anosov extension

Abstract: In this talk we will show how a sufficiently small geodesic ball in any Riemannian manifold can be embedded into an Anosov manifold with the same dimension. Furthermore, such embedding exists for a larger family of domains even with hyperbolic trapped sets. We will also present some application to marked length spectrum rigidity. This is a joint work with Alena Erchenko and Andrey Gogolev.

Host: Yanli Song

Geometry & Topology Seminar: Kähler-Einsten metric, K-stability and moduli of Fano varieties

Abstract: Constructing moduli spaces for algebraic varieties has roots in many different fields, e.g. number theory, complex analysis, mathematical physics etc. It is often related to stability notions. The first framework of the construction is Mumford’s geometric invariant theory (GIT), which provides a successful moduli theory for algebraic curves. For higher dimensional case, one needs to seek for new theory beyond the GIT.

Honors Thesis Presentation: Symplectic Geometry and Quantum Measurement

Abstract: TBA

Advisor: Renato Feres

Algebraic Geometry and Combinatorics Seminar: Quasisymmetric and Coxeter flag varieties

Abstract: In algebraic combinatorics, \emph{quasisymmetric} and \emph{Coxeter--Catalan} describe two common philosophies for generalizing classical objects and deepening results.  One area where neither philosophy has made serious headway is the Schubert calculus of the complete flag variety.  In this talk I will introduce a toric complex $\mathrm{QF}\ell$ that is simultaneously a quasisymmetric and Coxeter-Catalan generalization of the complete flag variety.

Mathematics Department: Holiday Potluck

Details: Coming Soon!

Algebraic Geometry and Combinatorics Seminar: Equivariant gamma-positivity of Chow rings of matroids

Abstract: Chow and augmented Chow rings of matroids played important roles in the settlement of the Heron-Rota-Welsh conjecture and the Dowling-Wilson top-heavy conjecture. Their Hilbert series have been extensively studied ever since. It was shown by Ferroni, Mathern, Steven, and Vecchi, and independently by Wang, that the Hilbert series of Chow rings of matroids are gamma-positive. However, no interpretation for the gamma-coefficients was known.

Algebraic Geometry and Combinatorics Seminar: Algebraic cycles constructed from curves

Abstract: In this talk, I will discuss various problems and results on the theory of algebraic cycles. In particular, I will discuss the construction and properties of some concrete cycles lying inside the Jacobian varieties of algebraic curves and the self-product of algebraic curves. 

Host: Matt Kerr

Colloquium: Reductions of abelian varieties

Abstract: Given an abelian variety defined over a number field, a conjecture attributed to Serre predicts that its set of ordinary primes is of positive density. This conjecture had been proved for elliptic curves, abelian surfaces, and certain higher dimensional abelian varieties following the work of Serre, Katz, Ogus, Pink, Sawin, Suh, Fite, etc. On the opposite direction, Elkies proved that an elliptic curve over Q has infinitely many supersingular reductions.

Taibleson Colloquium: Finding ellipses: Blaschke products and the numerical range

Abstract: The numerical range of an n × n complex matrix A is defined by
W(A) = { Ax, x >: x ∈ Cⁿ, ||x|| = 1}.
In general, it’s not easy to compute the shape of the numerical range. In
this talk, we investigate the question of when numerical ranges of matrices
are elliptical by connecting this phenomenon to two seemingly different
settings: function theory and projective geometry. Starting with n = 2
and extending to general n leads to a class of operators known as compressions of the shift operator. This viewpoint provides new insight into the

Algebraic Geometry and Combinatorics Seminar: Plane partitions in quantum chemistry

Abstract: The space of quantum states of a system of electrons may be modeled mathematically as an exterior power of a complex vector space. This space carries a natural SU(2)-action, and chemists are often particularly interested in the space of invariants with respect to this group action, called the space of spin adapted quantum states. We describe an explicit basis for this invariant subspace using plane partitions and find its dimension as a complex vector space via an explicit bijection with certain Dyck paths. This is joint work with Ada Stelzer and Svala Sverrisdóttir.

Analysis Seminar: Cancellative Sparse Domination

Abstract: We present a sparse domination result for linear operators that respects the cancellation present in the input function. In particular, the H^1 \to L^1 boundedness of Calderón-Zygmund operators follows easily. Our method of proof provides a path to sparse domination results in situations where the weak type (1,1) endpoint is lacking. Based on joint work with Emiel Lorist (TU Delft) and Guillermo Rey (Universidad Autónoma de Madrid).

Host: Brett Wick

Colloquium: Bad Student Matrix Products Using Singular Integrals

Abstract: A Schur multiplier S_M is a linear operator associated to a matrix M=[m_{i,j}]_{i,j}, called its symbol. It acts on other matrices by entrywise multiplication, that is,

S_M(A) = [m_{i,j}a_{i,j}]_{i,j}, \quad A=[a_{i,j}]_{i,j}.

Analysis Seminar: From Well-Mixed to Networked Populations: Stability Gaps in Replicator Dynamics

Abstract: Evolutionary game theory provides a mathematical framework to describe how the distribution of competing strategies evolves over time in a population under selection. In its classical formulation, the population is assumed to be well mixed, meaning that all individuals are intrinsically identical and interact uniformly with the population as a whole, without spatial or structural constraints. Under this assumption, the dynamics are governed by the replicator equation. For a game with $N$ strategies, this equation defines a nonlinear dynamical system on the $(N-1)$-simplex.

Analysis Seminar: Weighted and Unweighted estimates for the Bergman projection on planar domains

Abstract: Boundedness of the Bergman projection on simply connected planar domains has a direct connection to weighted estimates on the disk through the domain’s Riemann map. We generalize the weighted theory from the disk to general planar domains, and precisely characterize the range of weighted estimates in terms of the conformal map. We also obtain some partial results for a family of novel weak-type estimates.

Host: Brett Wick

Colloquium: Projections of random Cantor sets

Abstract: The four-corner Cantor set (a.k.a. Garnett set) is a planar analogue of the classical Cantor set and arises in several areas of analysis, including the study of Kakeya sets and removable singularities for analytic functions. A central problem is to understand how this set behaves when projected onto lines. This turns out to be a very difficult question, so we study a random variant of the Cantor set, where we are able to obtain sharp estimates. This is joint work with Pablo Shmerkin and Ville Suomala.

Analysis Seminar: Weighted Inequalities and the Testing Philosophy in Harmonic Analysis

Abstract: To show that an operator T from X to Y is bounded, one typically must compute the norm of Tx for all unit vectors x in X. Testing theorems tell you that in certain cases it is enough to compute the norm of Tx for all vectors in a "nice" subclass of X. This approach proved very useful in studying the two-weight problem, i.e., when operators are bounded from L^p(w) to L^p(v) for two different weights w and v. 

Note: This talk is given in partial fulfillment of the Third Year Candidacy Requirement. 

Geometry & Topology Seminar: Adiabatic Limit and Analytic Torsion of Vector Bundles

Abstract: For a closed manifold, the analytic torsion is a secondary topological invariant that can be defined in terms of the determinant of Hodge Laplacian. In this talk, I will explain how Witten Laplacian can be used to generalize this construction to vector bundles over closed manifolds. I will also discuss how to relate the index and the analytic torsion of the total space to those of the base manifold. This is a joint work with Xianzhe Dai.

Host: Xiang Tang

Geometry & Topology Seminar: TBA

Abstract: TBA

Host: Yanli Song

Analysis Seminar: Müntz-Szasz Approximation Theorems

Abstract: Polynomial approximation theory was initiated by Weierstrass in the late 1800s. Bernstein conjectured what would eventually become the C[0,1] Müntz-Szasz theorem (Müntz gave a proof of the continuous version and Szasz gave an L^2 formulation). Although purely real in statement, the theorem proves particularly amenable to complex analytic techniques. An asymptotic form of the theorem was later derived by Agler and McCarthy; who also introduced a convenient formalism for discussing limits of monomial spaces.

Professor Steven Frankel Presents at the A&S Research Innovation Showcase

From quantum technologies to augmented reality -- see how bold ideas are making a real-world impact. The Research Innovation Showcase is an opportunity to hear from innovative A&S faculty who are driving innovation on campus—and far beyond. Our own Steven Frankel will present on Crystal Facets and Fractals at Infinity.

Expect quick & creative presentations from faculty along with refreshments and great conversation. After hearing from the finalists, the audience will vote to award three $1000 prizes.

Colloquium: Irrational ways of manufacturing numbers

Abstract: This is meant to be light mathematical entertainment, exploring simple yet unusual phenomena involving rational, irrational, transcendental numbers.  We will also glean a few open problems.

Host: Brett Wick

Reception to follow at Cupples I, Room 200 (Lounge) from 2:00pm to 3:00pm

A Magic Show

Abstract: A magic show should not steal its own thunder with an abstract, but this one is easy for curious children and difficult for seasoned scientists, and you can try it yourself afterwards on friends and family.

Loeb Lecture: A world from a sheet of paper

Abstract: Starting from just a sheet of paper, by folding, stacking, crumpling, tearing, we will explore a rich variety of phenomena, from magic tricks and geometry through elasticity and the traditional Japanese art of origami to medical devices and ‘h-principle’.  Much of the lecture consists of table-top demos.

Host: Brett Wick

There will be a pre-talk tea in Cupples I, Room 200 (Lounge) from 3:00pm to 3:50pm

Algebraic Geometry and Combinatorics Seminar: Shioda’s Conjecture on Unirationality

Abstract: In characteristic zero, Castelnuovo proved that every unirational surface is rational. In positive characteristic, this fails dramatically: there exist many non-rational, often even general-type, surfaces that are nevertheless unirational. In 1977, Shioda conjectured that such phenomena are completely explained by the Galois representation—i.e., a simply-connected surface is unirational if and only if it is supersingular. In this talk, I will present a counterexample to this conjecture.

Analysis Seminar: The Dirichlet problem as the boundary of the Poisson problem.

Abstract: We will describe a novel approximation result about how solutions to the Dirichlet problem for second-order real elliptic PDEs with boundary data in  Lp on rough domains may be globally approximated, up to the boundary in a precise sense, by a family of solutions to certain corresponding inhomogeneous Poisson problems with null boundary data. This result is new even for the Laplacian on the unit ball, but is shown in high geometric generality, as well as with minimal assumptions on the coefficients.

Algebraic Geometry and Combinatorics Seminar: TBA

Abstract: TBA

Host: Carl Lian

Roever Colloquium: A tale on geometric structures

Abstract: A Riemannian metric on a closed manifold M can be thought of as providing M with a notion of length and size. An old but vague question is the following: is there a best ge metric structure which allows to recover topological features of a manifold? I'll discuss what an Einstein metric is and why sometimes such a metric serves this purpose. I then discuss some new families of examples and how they fit into a largely speculative broader picture. Mostly based on joint work with Frieder Jaeckel and Henri Guenancia.

Host: Charles Ouyang

Geometry & Topology Seminar: Geometric structures on surfaces

Abstract: Discrete and faithful homomorphisms of the fundamental group of a closed surface into a simple Lie group of non-compact type often provide surfaces with geometric structures, for example using minimal surfaces. On the other hand, an a priori understanding of such geometries helps to understand the space of all such homomorphisms in a given connected component, blending geometry and dynamics. We'll explain how this idea can be used to show that the diameter of quasi-Fuchsian space for the so-called pressure metric is finite.

Houston Kirk Colloquium: Nonabelian Hodge theory for Riemann surfaces and the P=W conjecture

Abstract: Often in algebraic geometry, there are nontrivial identifications between algebraic structures and purely topological structures.   Given a compact Riemann surface $C$, a very interesting example of this phenomenon comes from nonabelian Hodge theory (developed in the 1980's) which relates topological and algebro-geometric structures on $C$. Namely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field.

Third Year Major Oral: The Scalar Curvature Problem via Dirac Operators: From Closed Manifolds to Manifolds with Boundary

Abstract: The interplay between positive scalar curvature and topology is a central theme in global analysis. One powerful tool for addressing these problems is the use of the Dirac operator and index theory, which are rooted in the classical Lichnerowicz Vanishing Theorem and the Atiyah-Singer Index Theorem. In this talk, I will first review these foundational results and introduce the method of using twisted Dirac operators to study the curvature properties of a Riemannian metric.

Third Year Major Oral: How Geometric PDEs and Symplectic Geometry Meet?

Abstract: Morse homology is a way to compute homology of finite-dimensional manifolds M by looking at the downward gradient flow lines of the function on M. In 1988, Andreas Floer adapted the idea of Morse homology to infinite-dimensional setting, and introduced two different approaches to construct the invariants.

Algebraic Geometry and Combinatorics Seminar: The Multipermutohedral Chow ring

Abstract: The multipermutohedral Chow ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of permutohedral varieties and type-B Coxeter arrangements. In this talk, we establish the combinatorial structure of the multipermutohedral Chow ring through an explicit Gr"obner basis, yielding a Feichtner-Yuzvinsky-type monomial basis and a formula for the Hilbert series. Using this formula, we refine the palindromicity of the Hilbert series.

Senior Honors Thesis Presentation: Introduction to Fourier Restriction Theory

Abstract: In this talk, we will introduce the basics of Fourier restriction theory. We will describe the general question and motivate the theory with an example from physics. We will then pay particular attention to the setting of the sphere with its surface measure. We will examine endpoint cases and discuss the Tomas-Stein theorem, one of the first major results in this area. We will then demonstrate necessary conditions for restriction inequalities on the sphere, which will motivate the restriction conjecture.

Algebraic Geometry and Combinatorics Seminar: Szego kernels and the Scorza map on moduli spaces of spin curves

Abstract: The Scorza correspondence was first studied by Scorza. Starting with a spin curve of genus 3 (i.e., a curve of genus 3 with an even theta-chracteristic with no global sections), Scorza used his correspondence to construct a second plane quartic which gave a birational map from the moduli space of curves of genus 3 to the moduli space of spin curves of genus 3. Scorza’s results were further used by Mukai to construct the family of Fano threefolds of genus 12 and degree 22. Scroza’s correspondence is in fact well-defined in all genera.

Commencement 2026

Friday, May 15, 2026
Francis Olympic Field, Danforth Campus
9:00 a.m.

Gates open at 7:00 a.m.

The May university-wide Commencement ceremony will include a graduate procession, remarks from Chancellor Andrew Martin, Keynote Speaker remarks, Conferral of Honorary Degrees, Student Speaker remarks, Conferral of Academic Degrees, and a graduate recession.

Family and guests are welcome to attend this ceremony. Seats are available first come, first served. This ceremony is a rain or shine event.

Analysis Seminar: Mandelbrot Cascade on a planar curve

Abstract: Mandelbrot cascades have been widely studied as models for multifractal sets. In this talk, we consider Mandelbrot cascades supported on planar curves with nonvanishing curvature and show that the Fourier dimension of each cascade equals the minimum of the local dimensions. This is joint work with Ville Suomala.

Host: Ben Foster

Analysis Seminar: Hypersurfaces on which few harmonic functions vanish

Abstract: It is easy to see that if one harmonic function vanishes on a given set in the Euclidean plane, then infinitely many linearly independent harmonic functions do. Perhaps surprisingly, this is no longer true in higher dimensions. We will show that in any dimension greater than two, there exist cones on which exactly two linearly independent harmonic functions vanish. This result holds on the level of germs at the origin.

Colloquium: p-adic strings and arithmetic

Abstract: The p-adic string worldsheet action in genus zero has a family of deformations, which turns out to be closely related to Tate's thesis. Very recently, we found that the Green's function of the p-adic string action in genus one, defined on a Tate curve, coincides with the Neron-Tate local height function. Therefore, this basic height function in arithmetic geometry acquires a physics meaning, as well as an interpretation as a Green's function of a pseudo-differential operator defined directly on the Tate curve.

Host: Yanli Song

Algebraic Geometry and Combinatorics Seminar: Springer fibers, Richardson varieties, and Schubert calculus

Abstract: In this talk, we will introduce two important subvarieties of the flag variety: Springer fibers and Richardson varieties. Both play central roles in the interplay between combinatorics, geometry, and representation theory. We will discuss recent results on their geometry, as well as open questions. This talk is based on joint work with Cristina Sabando Álvarez and with Steven Karp.

Algebraic Geometry and Combinatorics Seminar: Expansions of chromatic quasisymmetric functions via the A_{q,t} algebra

Abstract: The Stanley-Stembridge conjecture asserts that the chromatic symmetric functions of incomparability graphs of (3+1)-free posets are e-positive. In 2024, Hikita proved the conjecture by guessing a remarkable formula for the expansion coefficients and proving they satisfy the modular law recurrence. We give a second proof of Hikita's formula by direct algebraic methods involving the A_{q,t} algebra of Carlsson and Mellit. In fact, we prove a "master formula" for chromatic symmetric functions in terms of Macdonald polynomials that involve an extra parameter t.

Senior Honors Thesis Presentation: An Introduction to the Decomposition Theorem

Abstract: Among the most beautiful results concerning the topology of algebraic varieties is the Decomposition Theorem, but as it has substantial prerequisites in algebraic geometry, it can be difficult to approach. In this talk, we will introduce the theorem from an intuitive, geometric point of view, emphasizing the motivation behind it and working out concrete examples to see the theorem in action.

Faculty Advisor: Matt Kerr

Senior Honors Thesis Presentation: Measure concentration in complex projective space and quantum entanglement

Abstract: We explain, after Patrick Hayden, how concentration inequalities can be used in quantum theory to show that random quantum states associated to high-dimensional bipartite systems are highly entangled with high probability. A geometric interpretation of the concentration phenomenon can be given in terms of Michael Gromov's notion of the observable diameter of families of measure-metric (MM) spaces. We use Hayden's result to obtain estimates for the observable diameter of complex projective spaces.

Faculty Advisor: Renato Feres

Geometry & Topology Seminar: The flow product and integrating algebroids

Abstract: In this talk we will give background on the "integration problem" for Lie algebroids. We will then explain the traditional construction of the fundamental groupoid of an algebroid used by Cranic and Fernandes in their well-known article on integrating Lie algebroids. We will then present an alternative construction based on what we call the "flow product" of sections. We will then explain some of the advantages and applications of this approach. This talk is concerned with a joint work with Pooja Joshi.

Host: Xiang Tang

Analysis Seminar: Smoothing inequalities for corner-type averaging operators

Abstract: In this talk, we will introduce smoothing inequalities for corner-type averaging operators over a wide class of curves and discuss their applications to several objects in harmonic analysis.

Host: Ben Foster

Senior Honors Thesis Presentation: Introduction to the Riemann-Roch Theorem

Abstract: The \textit{Riemann-Roch Theorem} is a cornerstone result in complex analysis, algebraic geometry, and differential geometry, exploring the link between the topology and analysis of a surface. The theorem transforms the challenge of constructing meromorphic functions with certain poles into a topological and combinatorial problem. In this talk, I will introduce the Riemann-Roch theorem, the mathematical machinery to understand it, and explain some of its applications.

Faculty Advisor: Xiang Tang

Senior Honors Thesis Presentation: Multiresolution Analysis for Singular Integrals: One-Parameter, Product, and Entangled Settings

Abstract: This is an introduction to the study of singular integrals via dyadic multiresolution analysis. The idea is to produce “representation theorems” that exhibit singular integrals as sums of simpler dyadic operators. These methods are advantageous in part because they are adaptable to various settings. Here we look at three settings: the one-parameter case where we view $\R^d$ as a one-parameter space, the multi-parameter case where we view $\R^d$ as a product of lower-dimensional spaces, and the Zygmund case that lies somewhere between the other two.

Senior Honors Thesis Presentation: Methods in Topological Data Analysis

Abstract: We introduce core methods in topological data analysis and show how topology can be used to understand high dimensional data. The main tool is persistent homology, which tracks when topological features appear and disappear across a filtration and summarizes this information using persistence diagrams and barcodes. We develop the theory behind persistent homology and introduce distances such as the bottleneck and Wasserstein metrics to compare datasets through their topological signatures. These ideas are applied to image analysis by studying the topology of painting styles.

Colloquium: TBA

Abstract: TBA

Host: Yanli Song

Awards Ceremony, 2026

Light refreshments will be offered in Room 200 at 3pm.

The Awards Ceremony will begin promptly at 4pm in Room 199.

Algebraic Geometry and Combinatorics Seminar: TBA

Abstract: TBA

Host: Martha Precup

Analysis Seminar: Some frame theory questions in the manifold setting

Abstract: The talk will start at the elementary level and introduce some basics of frame theory.  We will make the distinction between a frame and a basis in a Hilbert space and familiarize ourselves with the basic ideas of reconstruction.  We will then expand the theory to the manifold and vector bundle setting.  As we delve deeper, we will present the question and solution of approximating a signal from finitely many samples in a compact manifold.

Geometry & Topology Seminar: The period map for smooth 4-manifolds

Abstract: The period map of a smooth 4-manifold assigns to a Riemannian metric the space of self-dual harmonic 2-forms.  This map is from the space of metrics to the Grassmannian of maximal positive subspaces in the second cohomology. It is conjectured that the period map is always surjective. I'll explain a proof of this conjecture for 4-manifolds with b^+ = 1, as well as some results that address the more general case.

Host: Ali Daemi

Algebraic Geometry and Combinatorics Seminar: The geometry of algebras associated to hyperplane arrangements 

Abstract: The space of sections of a line bundle on an equivariant compactification of a commutative algebraic group naturally has a coalgebra structure. We show that a well-studied family of algebras associated to hyperplane arrangements can be realized in this way. This allows us to apply powerful tools from sheaf cohomology to some combinatorial algebra problems. Joint work with Colin Crowle.

Host: Carl Lian

Thesis Defense: Complexity of the zero set of a matrix Schubert ideal

Abstract: T-varieties are normal varieties equipped with an action of an algebraic torus T. The complexity of a T-variety is the codimension of the largest T-orbit. Matrix Schubert varieties are T-varieties consisting of n-by-n matrices that satisfy certain constraints on the ranks of their submatrices. Every matrix Schubert variety can be written as the product of an affine subvariety and a k-dimensional complex space, where k is as large as possible. In this talk, we focus on the complexity of these affine subvarieties endowed with a naturally defined torus action.