GEOMETRY/TOPOLOGY SEMINAR (PAST TALKS)

Ever since I learned that a coffee cup accepts graph embeddings that a sphere cannot, I can’t stopped telling my students. Linking in the Gauss-Bonnet theorem, Morse theory, a little crochet, and some careful considerations of non-euclidean pizza, and you have a great conversation with a class. After several iterations, I have tested a few topics and can share what works for me and what draws yawns and crickets. I hope to describe ideas for other things I haven’t tried yet. Audience members are encouraged to share their experiences.

Dimension 4 is unlike the other dimensions. There are many simply-connected closed 4-manifolds that admit infinitely many distinct smooth structures, and surprisingly, there are no smooth 4-manifolds known to have only finitely many smooth structures. Also, classification problems for smooth, simply-connected 4-manifolds are far from fully understood. For example, the generalized Poincaré conjecture is true topologically in all dimensions, but unknown smoothly in dimension 4. Attempts to resolve this conjecture have led to constructions of 4-manifolds that are homeomorphic, but not diffeomorphic (such manifolds are called "exotic"), with the particular goal of constructing exotic 4-spheres. In this first talk, I will mostly focus on the simply-connected symplectic 4-manifolds, describe the topological invariants, and give examples.

Continuing from last time, we will get into more details about symplectic manifolds. Then, we will provide background about Lefschetz fibrations and mapping class groups, which have a very nice and useful interaction with symplectic manifolds. Finally, we will introduce the tools we use to construct exotic 4-manifolds, like the symplectic connected sum and Luttinger surgery.

In this part, we will talk about some results we obtained and what we have covered in the first two parts. First, we will present a new construction of symplectic 4-manifolds that are homeomorphic but not diffeomorphic to \((2h+2k-1)\mathbb{CP}^{2}\#(6h+2k+3)\overline{\mathbb{CP}}^{2}\) with \((h,k) \neq (0,1)\), via Lefschetz fibrations and Luttinger surgery on the product manifolds \(\Sigma_g \times T^2\). In the second half, we will construct families of Lefschetz fibrations over \(S^2\) using finite order cyclic group actions on \(\Sigma_g \times \Sigma_g\). These are joint works with Anar Akhmedov.

Manifolds appear in classical physics as space of states of a physical system in classical mechanics. When we move from classical world to quantum world, points as states are replaced by operators. In our talk, we will discuss manifold structure on such operator spaces and its geometry.

Manifolds appear in classical physics as space of states of a physical system in classical mechanics. When we move from classical world to quantum world, points as states are replaced by operators. In our talk, we will discuss manifold structure on such operator spaces and its geometry.

Complex hyperbolic space is a fairly natural complex analog of the more familiar real hyperbolic space. However, despite the similarities in construction, the geometric features of these two spaces can be quite different, and questions of the form "If [statement] is true in real hyperbolic space, is it also true in complex hyperbolic space?" might require vastly different techniques to answer. In this talk, I plan to give a gentle introduction to (complex) hyperbolic geometry.

As we saw last time, the boundary of complex hyperbolic 2-space is topologically \(S^3\), but due to the natural action of \(\operatorname{PU}(2,1)\), it geometrically inherits the structure of the (1-point compactification of the) Heisenberg group. In the early 2000's, Schwartz discovered that one could actually find a real hyperbolic 3-manifold in this strange \(S^3\), and so it seems natural to ask which other 3-manifolds can arise in the boundary of complex hyperbolic 2-space. In this talk, I will introduce Falbel's program for finding \(\operatorname{PU}(2,1)\) representations of 3-manifold groups and summarize some known results about 3-manifolds.

I'll use an elementary discussion of the nineteenth-century Riemann-Roch theorem to motivate later developments - vector bundles, characteristic classes, K-theory, and index theory - in algebraic topology and in the connections of algebraic topology with analysis.

I'll use an elementary discussion of the nineteenth-century Riemann-Roch theorem to motivate later developments - vector bundles, characteristic classes, K-theory, and index theory - in algebraic topology and in the connections of algebraic topology with analysis.

The mapping class group is an algebraic invariant of a topological space that detects the symmetries of that space. In the case of surfaces, this group has a deep connection with the fundamental group and the associated Teichmuller space. In this talk I'll give a introduction to the mapping class group along with some examples and intuition for some interesting results surrounding it. Time permitted, I'll also discuss some associated results for so-called "big" mapping class groups.

In the 1940's, Jakob Nielsen set about analyzing and classifying the elements in the mapping class group for closed orientable surfaces. In the 1970's, while working on his famous Geometrization Conjecture, Thurston managed to successfully complete what we now call the Nielsen-Thurston Classification of mapping classes. In this talk I'll motivate the classification on the torus, highlight some parallels with hyperbolic isometries, and present an overview of Thurston's proof for surfaces with genus g>1.

Although 4-manifolds are outside our imagination, there is a very nice way to encode the topological information in the so-called handlebody diagram. In this talk, we'll first start with basic descriptions and we'll see examples of handlebody diagrams for low-dimensional manifolds. Time-permitted, we will see how to obtain more suitable handlebody diagrams. Namely, we will modify it using two fundamental topological operations: handle cancelation/creation and handle sliding.

Last time we worked on handlebody diagrams of 1- and 2-dimensional manifolds. In the second part, we will describe how to obtain more suitable handlebody diagrams via two fundamental topological operations: handle cancelation/creation and handle sliding. In this talk we will see handlebody diagrams of 3- and 4-manifolds as well.

Braid group theory is an interesting and versatile subject with applications in many different fields of mathematics including algebra, topology, and quantum computation. In this talk, I will give an introduction to the braid groups and share my intuition for why and how these groups are used. In particular, I will discuss the representations of the braid groups and some of the motivating open questions that fuel my research. Many of the famous representations of the braid groups are parametrized by a variable \(q\) (these representations secretly come from quantum groups). I will share some of my results about choosing careful specializations of \(q\) with the aim of structural results about the image of the representation.

In this talk, I will discuss some of the building blocks of complex hyperbolic geometry. In doing so, I will point out some differences between real hyperbolic space and complex hyperbolic space including: the various analogous models in both settings, the difference in setup using symmetric bilinear forms and Hermitian forms, and the nature of distance, isometries, and geodesic subspaces. I will also discuss the connection of the boundary at infinity of the complex hyperbolic plane to the Heisenberg group. This introduction is aimed to provide the necessary background materials to discuss how to derive presentations for a special class of lattices in \(\mathbf{H}_{\mathbb{C}}^2\) known as the Picard modular groups.

In this talk, I will discuss a method for obtaining group presentations for a particular class of lattices in the complex hyperbolic plane known as the Picard modular groups. The method deals with finding an appropriate covering of \(\mathbf{H}_{\mathbb{C}}^2\) by translates of a collection of horoballs under the action of a discrete group, \(\Gamma\), and appealing to a theorem of Macbeath. We will focus on the application of this method to the case where our lattice has a single cusp, and as examples, we will derive presentations for the Picard modular groups when d=2 and d =11. The presentations for the Picard modular groups in the cases d=2,11 completes the list of presentations for Picard modular groups with entries from Euclidean domains. We will also discuss how the method changes when our lattice has more than one cusp.

First, we will give a brief introduction to symplectic manifolds and Lefschetz fibrations. Then we will talk about Mapping Class Groups and their connection with the Lefschetz Fibration. Finally, we will provide examples. This talk will be a preparation for the next talk, which I will talk about my recent work.

We construct families of Lefschetz fibrations over \(S^2\) using finite order cyclic group actions on the product manifolds \(\Sigma_g \times \Sigma_g\) for \(g>0\). We also obtain more families of Lefschetz fibrations by applying the rational blow-down operation to these Lefschetz fibrations. This is a joint work with Anar Akhmedov and Mohan Bhupal.

In this talk I will go through a combinatorial approach to understanding/measuring curvature. It is accessible to undergraduates and particularly interesting for those that want to understand a discrete version of the Gauss-Bonnet Theorem. There are a few related topics that I would like to cover in this short talk. As a theme, my focus will be how I engage undergraduates with topics that sit in the intersection of geometry, topology, and combinatorics.

At one time or another, most have probably found themselves bored in class with a pencil and ruler, doodling tilings of a plane with triangles. These triangular tilings can be realized in terms of group actions, and for a particularly nice family of triangles, we have the so-called "triangle groups." In this talk, I'll motivate triangle groups and we'll explore some of the interplay between the group theory and geometry (possibly with a sprinkling of arithmeticity, if time allows).

Last time we looked at groups generated by reflections in the sides of triangles and ultimately were able to reconcile the geometric intuition with the algebra and classify these triangles by arithmeticity. It's then natural to ask if we can play this same game with other (convex) polygons or other higher-dimensional polytopes. As one might expect, keeping track of the geometric features becomes considerably harder as the combinatorial complexity of the polygon increases. Instead, the slightly more natural approach (which generalizes to higher dimensions) is to study which polygons (or polytopes) can arise from a given algebraic construction.

In 1975, E. Vinberg produced an algorithm for finding the convex polytopes, and in her 2015 PhD thesis, A. Mark gives an algorithm that improves upon Vinberg's original ideas in the setting of totally real quadratic number fields and obtains some effective bounds. In this talk, I'd like to motivate the algebraic setting and outline Mark's algorithm. Provided time allows, I'd like to also discuss some recent joint work with Mark for extending her algorithm to the more general case of totally real (Galois) number fields.

Continued fractions arise when trying to find rational numbers that approximate some fixed real number \(x\). Using a process which emulates the Euclidean algorithm, one produces a (possibly-finite) sequence of integers which encodes a sequence of rational numbers converging to \(x\), and the terms in this integral sequence are called a the continued fraction digits of \(x\). Given that there are other mathematical objects that somehow generalize the integers and the real numbers (a lattice in a real Lie group, for example), it is natural to ask if there is also a reasonable notion of continued fractions within these objects. In 2013, Lukyanenko and Vandehey affirmatively answer this question in the context of the Heisenberg group, which has a particularly interesting geometry associated with it. In this talk I'll go over the authors' construction, survey some of their results as compared with classical results, and (time-permitted) discuss some open questions for further generalizations in this direction.

• Anton Lukyanenko and Joseph Vandehey. “Continued fractions on the Heisenberg group”. In: Acta Arith. 167.1 (2015), pp. 19–42. issn: 0065-1036.
  (arXiv preprint)

We present a basic theory toward a hypothetical undergraduate class on complex geometry and symplectic geometry. Conventional wisdom says that the visualization of spaces of more than three dimensions is not feasible. However, when one focuses their studies on complex vector spaces, (finite-dimensional vector spaces over the field of complex numbers, or \( \mathbb{C}^n \) ), there is a visualization. We introduce the notion of complex lines, complex angles, and the utility of the symplectic product via this presentation and demonstrate some applications

In pursuit of a proof that topologically stable maps form an open dense subset of \(C^{\infty}(M, N)\), Thom and Mather laid the foundation for the theory of stratified spaces. In this talk, I will go over some of the key ideas they developed.

Complex geometry is now inseparable from modern mathematical physics. In this talk I'll discuss why that is and also show some basic constructions and calculations you can perform for common string vacua.

The Hirzebruch signature theorem is a special case of the Atiyah-Singer index theorem, and shows that the signature of a compact, oriented, even dimensional manifold (the signature of the intersection form) is equal to the index of a first order operator of Dirac type, the signature operator (the square root of the Hodge - de Rham laplacian). Hirzebruch showed that the signature vanishes for every \(n\)-manifold arising as the boundary of some \( (n+1) \)-dimensional manifold. Even so, interesting phenomena can arise if one varies the underlying Riemannian structure, by utilizing the family version of the index theorem in the form of the Riemann-Roch-Grothendieck-Quillen (RRGQ) formula and Quillen’s determinant line bundle. Such is the case of a two-torus and varying the conformal class of flat Riemannian metrics. By reinterpreting as the variation of complex structure on an elliptic curve, we construct an elliptic fibration over a rational base, the so-called j-line. Then the RRGQ formula for the family of (complexified) fibrewise signature operators results in a generalized cohomology class that is known in physics as a measure of the local and global anomaly. The anomalies will be explicitly computed, and if time permits, it will be shown how the anomalies can be resolved by combining several anomalous operators.

Based off https://arxiv.org/abs/1907.07313.

This talk will be a very basic introduction to connections (also called gauge fields in physics). We start with the definition of a connection, curvature, and gauge equivalence of connections. We then briefly discuss how Maxwell's equations can be written in terms of connections, together with their direct generalization as Yang-Mills equations. We also introduce the Chern-Simons functional as another example of a gauge equation.

It is a classical result of Grossman that mapping class groups of finite type surfaces are residually finite. In recent years, residual finiteness growth functions of groups have attracted much interest; these are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity. Residual finiteness growth functions detect many subtle properties of groups, including linearity. In this talk, I will discuss some recent joint work with Thomas Koberda on residual finiteness growth for mapping class groups, adapted to nilpotent and solvable quotients of the underlying surface group.
(Joint with Thomas Koberda)

Persistence theory is a very recent phenomena in mathematics originally developed independently by Ferri, Robins, and Edelsbrunner at the turn of the 21st century. This talk will focus more so on the work of the last author and the applications of persistence to Topological Data Analysis (TDA). The main tools here are persistent homology, simplicial complexes, and spectral sequences. There are also many applications outside of TDA such as in Riemannian Geometry and geometric group theory which we will explore if time permits.

Certain minimal blocking sets in various finite projective planes will be discussed. A blocking set is a set of points \( S \) for which every line of the plane meets \( S \) in at least one point, but \( S \) contains no line of the plane. A blocking set \( S \) is minimal if no proper subset of \( S \) is a blocking set. Along with the classical (Desarguesian) projective plane of order \(q\), coordinatized by a field of order \(q\), these sets will be sought in certain non-classical projective planes arising from a nearfield of order \(q\) . Such planes contain both \(q^2 + q + 1\) points and lines, where \(q\) is an integer power of a prime \(p\). For each plane, aspects of the \(p\)-code will also be discussed, such as the dimension, spanning sets from a geometric viewpoint, and the presence of the characteristic vectors of the aforementioned sets as codewords. The \(p\)-code of a projective plane \( \pi \) is the subspace of the vector space of dimension \(q^2 + q + 1\) over \( \mathbb{Z}_p \) spanned by the characteristic vectors of the lines of \( \pi \). In other words, the \(p\)-code of \( \pi \) is the rowspace over \( \mathbb{Z}_p \) of an incidence matrix of \( \pi \), where rows and columns are indexed by lines and points of \( \pi \), respectively.

We will consider operators in the classical sense as a linear map between two Hilbert spaces over a field of complex numbers and state (without proof) the classical Atkinson's theorem and Atiyah-Janich theorem. We will discuss a way to generalize these theorems when the field of complex numbers is replaced by an algebra of operators.

In the 1970's, Thurston found hyperbolic 3-manifolds which fibered over the circle (\(S^1\)) and his results prompted a natural follow-up question: are there higher-dimensional hyperbolic manifolds which also fiber over \(S^1\)? In 2023, Italiano-Martelli-Migliorini used a combinatorial result about right-angled Coxeter groups (RACGs) to construct a hyperbolic 5-manifold which also fibered over the circle. Might we finally have a way forward in tackling the big question??

In the immortal words of Vizzini (The Princess Bride) "You'd like to think that, wouldn't you?" Alas, the crucial ingredient in the aformentioned construction was a nice right-angled hyperbolic Coxeter polytope, and few things are even known about the existence of such objects in higher dimensions, so answering this question with geometric methods seems impractical at this juncture. In this talk, we'll look at recent results to the analogous group theoretic question about fibering and virtual cohomological dimension (VCD) of these RACGs.

This work is joint with Lafont, Minemyer, Sorcar, and Stover.

Stratified spaces are a class of spaces which generalize smooth manifolds and real/complex algebraic varieties. In studying their geometry, the need for spaces which generalize vector bundles to allow the rank of the fibers to drop over lower dimensional strata naturally arises. In this talk, we will discuss a possible framework for a theory of stratified vector bundles and construct several classes of examples.

The curious question of "Which Blacksburg am I in?" invites spherical point-inclusion tests (PITs) on \(S^2\). We discuss BAE-gons, i.e., the class of spherical polygons with antipode-excluding boundaries. A subset of \(S^2\) is said to be antipode-excluding if it does not intersect its antipode. We discuss the main proposition that spherical PITs for BAE-gons are decidable using planar PITs for polygons, if both notions of interiorness are defined using the winding number. We present two transformation algorithms from spherical PITs to planar PITs, illustrate their practicality in the broader context of geographic information science (GIS) and numerical simulation, and highlight some open questions.

This talk will begin with an introduction to Witten-Reshetikhin-Turaev (WRT) invariants and a related q-series invariant which first appeared in the work of Lawrence and Zagier. This q-series was one of the first key examples of a quantum modular form, and unified the WRT invariants of the Poincaré homology sphere. I will then discuss results based on joint work with Eleanor McSpirit. We show that the series of Lawrence and Zagier is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3-manifolds whose radial limits toward roots of unity may be thought of as a deformation of the WRT invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS q-series of 3-manifolds. As part of this work, we provide the first calculation of the AJK series for an infinite family of 3-manifolds.

Transfer systems are mathematical objects that encode transfers within algebras over specific types of structures known as equivariant operads. These systems allow us to employ combinatorial tools to investigate equivariant homotopy theory. One significant aspect is the study of compatible pairs of transfer systems, which correspond to multiplicative structures that align with an underlying additive structure.

In this talk, I will introduce G-transfer systems, which are transfer systems defined for a given group G. I will discuss the fundamental concepts of saturation and pairs of compatible transfer systems. Additionally, I will present joint work with Kristen Mazur, Angelica Osorno, Constanze Roitzheim, Rekha Santhanam, and Danika Van Niel on the compatibility of \(C_{p^rq^s}\)-transfer systems, where \(C_{p^rq^s}\) denotes the cyclic group of order \(p^r q^s\). Specifically, we will outline a criterion for determining when transfer systems only form trivially compatible pairs.

An awkward fact about manifolds is that the intersection of two submanifolds isn't necessarily another submanifold. Various generalizations of manifolds remedy this problem. One of these generalizations, Smooth Sets, is defined from the key insight that the set of smooth functions from cartesian spaces into a manifold completely determines the manifold in an appropriate sense. The "appropriate sense" here requires understanding a sheaf condition, which I will discuss extensively.

Smooth sets fix the intersection problem, but many more issues are also fixed, such as Lie group quotients being much more well-behaved and the new existence of a natural way to do homotopy theory called a model structure. Along the way, we'll explore how some of these fixes give rise to interesting objects and ideas, such as the irrational torus and concordance.

This is a continuation of my work in making complex and symplectic geometry accessible to undergraduates. We review a method to visualize complex affine space as a means to visualize standard symplectic space, and discuss the standard construction of the Kodaira-Thurston manifold, the simplest non-Kahler symplectic manifold. We will then connect the two presentations.

This is a cultural talk. I’ll talk about a few areas I’ve worked in where it felt like one was exploring playgrounds. The first area will be growth series and growth functions for groups. The second will be constructing 3-manifolds by face pairings.

This will be a brief talk about a problem I worked on in graduate school. The task was to produce an example factory for what are called hyperbolic expansion complexes. The main difficulty of this task is a version of a classical problem called the type problem: The Uniformization theorem asserts conformal equivalence of any simply connected Riemann surface to one of the following surfaces: the open unit disk, the Riemann sphere, or the complex plane. The Type Problem is to, given such a surface, figure out which one.

K3 surfaces are an important class of 4-manifolds, appearing in many diverse places in both modern and classical mathematics. In this talk I will describe some fundamental features of their geometry, and demonstrate how these features lead to the appearance of rich geometry on their complex structure moduli spaces. Based on arXiv:2401.10950.

The Ptolemy relation (circa 100-170 AD) says that for a cyclic quadrilateral (a quadrilateral inscribed in a circle) the product of the lengths of the diagonals is equal to the sum of the products of the lengths of opposite sides. We will discuss the central importance of the Ptolemy relation in the (more contemporary) branch of mathematics called cluster geometry. Fundamental examples come from hyperbolic geometry.

In 1957 Bott showed that there is a surprising periodicity in the homotopy groups of the infinite unitary group. In this expository talk we outline Bott's original proof, give a reformulation of the result in terms of K theory, and discuss its role in the Atiyah-Singer index theorem.

This will be an image-based and non-technical talk about a topic nearing its 40th anniversary. Bill Thurston’s notion of circle packing first reached a broad audience (and me) in a 1985 talk in which he conjectured a connection with classical analytic function theory. I will give a brief visual tour of the rich “discrete conformal geometry” that has subsequently emerged, largely due to the software package CirclePack. I will also encourage you to steal this software! Its open-ended experimental capabilities are, IMHO, unprecedented in mathematics. Moreover, the discreteness, computability, and visual nature of circle packing allows conformal geometry to reach far beyond its pure mathematics base. I’ll mention results in graph embedding, particle physics, 3D printing, brain imaging, emergent behavior, and even artistic expression. And I’ll mention several open avenues for new discoveries. Serendipity plays an outsized role in these developments, so steal this topic, play with CirclePack, make and exploit your own mistakes!!

HOMFLY homology is a triply-graded vector space-valued invariant of links which recovers the HOMFLY polynomial upon taking Euler characteristic. Various authors have promoted this to an invariant of colored links, in which each component carries the additional data of a Young diagram. I will introduce colored HOMFLY homology and discuss a symmetry conjecture relating the behavior of this invariant under different colorings of the same link. I'll end by discussing the status of this conjecture and outlining a recent proof for a large family of colored links. See this webpage for more information about the speaker https://tarheels.live/lukegconners/.

What does a typical quotient of a group look like? Gromov looked at the density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for d < 1/2, the typical quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov’s result to show that for d < 1/2, the typical quotient of many hyperbolic groups is also non-elementary hyperbolic. Zuk and Kotowski–Kotowski proved that for d > 1/3, a typical quotient of a free group has Property (T). We show that (in a closely related density model) for 1/3 < d < 1/2, the typical quotient of a large class of hyperbolic groups is non-elementary hyperbolic and has Property (T). This provides an answer to a question of Gromov (and Ollivier).

A metric space has small Urysohn 1-width if it admits a continuous map to a 1-dimensional complex where the preimage of each point has small diameter. An open problem is, if a space's universal cover has small Urysohn 1-width, must the original space also have small Urysohn 1-width? While one might intuitively expect this to be true, there are strange examples that suggest otherwise. In this talk, I will explore the motivations behind this question and discuss some partial progress we have made in understanding it. This is a joint work with H. Alpert and P. Papasoglu.

It is well-known that the moduli space of smooth n-pointed algebraic curves of genus g is not compact. The celebrated Deligne-Mumford-Knudsen compactification fixes this by adding some mildly singular "stable" curves to the moduli space. Yet this is not the only such compactification. In this talk, I will present a classification of the modular compactifications in genus one by curves with Gorenstein singularities. Time permitting, we will discuss how logarithmic tropical geometry allows us to find this classification.

This talk provides an introduction to the various objects and operations found in projective rigid and conformal geometric algebras. It first discusses the exterior (Grassmann) algebra as an extension to an ordinary vector space, identifies the homogeneous geometries that arise in the projective case, and demonstrates the kinds of geometric manipulation that can be performed with different forms of multiplication. That is followed by an introduction to the projective geometric (Clifford) algebra and a discussion of the motion operators it contains, highlighting the roles of quaternions and dual quaternions. Emphasis is placed on practical utility and efficient implementation.

Classically, a partial differential equation is said to be Darboux integrable (DI) if its general solution can be found by only solving a system of ordinary differential equations. In this talk, we describe a new transformation group-theoretic approach to the study of DI equations and highlight how this approach can be used to solve equivalence problems for DI f-Gordon equations. The main result of this approach is that a complete list of all f-Gordon equations which are DI at order 3 can be determined from a complete list of rank 2 distributions in 5 dimensions which admit intransitive 5-dimensional symmetry groups. Through this correspondence, we have uncovered a new class of DI equations which leads to a complete classification of all DI f-Gordon equations at order 3. This talk is based on joint work with Ian M. Anderson, Utah State University.

The Calculus of Variations involves problems in which the quantity to be minimized (or maximized) appears as an integral. I will begin with a brief refresher course on the Euler - Lagrange equation, and illustrate its use in some basic problems of physics such as the brachistochrone, the path of a projectile, the shape of a liquid in a rotating cylindrical container, the quantum mechanical harmonic oscillator and, if time allows, a brief discussion of some of the difficulties such applications can encounter as these are exemplified in the shape of soap-films. The problem of the shape enclosing a fixed volume with minimum surface area is then discussed and the results of some calculations I have made opened to welcome discussion.

Kontsevich showed (based on the work of Deligne and Mumford) that if C_t is a family of *smooth* algebraic curves *embedded* in projective space, depending on a nonzero parameter t, there is a limiting *map* (not necessarily an emedding) from a *nodal* curve C_0 to projective space. However, not every map from a nodal curve to projective space is the limit of a family of smooth embedded curves. Can the maps from nodal curves that are limits of smooth embedded curves be characterized directly? The answer to this question is known only in low genus, and is due to Vakil--Zinger and Hu--Li. I will discuss new perspectives on this question coming from tropical geometry, due to Santos-Parker, Bozlee, Neff, Battistella--Carocci, and Ranganathan.

We will introduce tensor networks and discuss their relation to vertex and edge colorings of planar graphs, in particular in the context of the Four Color Theorem.

Vaughan Jones showed how to associate links in the $3$-sphere to elements of Thompson’s group $F$ and proved that $F$ gives rise to all link types. This talk will discuss two recent extensions of Jones’ work– the first is a method of building annular links from Thompson’s group $T$, which contains $F$ as a subgroup, and the second is a method of building $(n,n)$-tangles from $F$ . Annular links from $T$ arise from Jones’s unitary representations of the Thompson groups, and tangles from $F$ give rise to an action of $F$ on Khovanov’s chain complexes. This talk includes joint work with Slava Krushkal and Yangxiao Luo

The log etale topology is a natural analogue of the etale topology for log schemes. Unfortunately, very few things satisfy log etale descent -- not even vector bundles or the structure sheaf. We introduce a new rhizomic topology that sits in between the usual and log etale topologies and show most things do satisfy rhizomic descent! As a case study, we look at tropical abelian varieties and give some exotic examples.

Error-correcting codes are crucial for ensuring reliable data transmission and storage in the presence of errors or data corruption. Designing efficient, high-performance codes requires understanding their structural and combinatorial properties. A key aspect of classical coding theory is determining bounds on coding parameters and identifying invariants that describe code structures. One powerful tool for analyzing these properties and their performance is the theory of anticodes. Within the 'quantum revolution,' quantum communication and computation are transforming information processing through the principles of quantum mechanics. However, quantum systems introduce complex error models, making it challenging to directly apply classical coding theory techniques.In this talk, we present a new framework for analyzing stabilizer codes from an anticode perspective. We extend the notion of generalized distance to quantum anticodes and derive the McWilliams identities. This approach leads to a powerful generalization of the quantum cleaning lemma, which can be interpreted as the quantum counterpart of a classical duality result. The new results in this talk are joint work with C. Cao and B. Lackey.

We consider the homogeneous mean-field Bose gas at temperatures proportional to the critical temperature of its Bose--Einstein condensation phase transition. Our main result is a trace-norm approximation of the grand canonical Gibbs state in terms of a reference state, which is given by a convex combination of products of coherent states and Gibbs states associated with certain temperature-dependent Bogoliubov Hamiltonians. The convex combination is expressed as an integral over a Gibbs distribution of a one-mode Φ4-theory describing the condensate. We interpret this result as a justification of Bogoliubov theory at positive temperature. Further results derived from the norm approximation include various limiting distributions for the number of particles in the condensate, as well as precise formulas for the one- and two-particle density matrices of the Gibbs state. Key ingredients of our proof, which are of independent interest, include two novel abstract correlation inequalities. The proof of one of them is based on an application of an infinite-dimensional version of Stahl's theorem. This is joint work with Phan Thành Nam, Marcin Napiórkowski.

The Sageev construction is a method for building group actions on CAT(0) cube complexes using codimension-one subgroups. It has been especially useful in the study of hyperbolic and relatively hyperbolic groups, thanks to the work of Agol and Wise. In this talk, we will describe how to construct a group action on a CAT(0) cube complex from a countable group acting on a sufficiently nice topological space M. When M is the Gromov boundary of a hyperbolic (relatively hyperbolic) group, our more general approach retrieves Sageev’s construction. This is joint work with Jason Manning.

The volume conjecture connects a large $n$ limit of the colored Jones polynomial $J_n(K)$ of a knot $K$ to the hyperbolic volume of its complement. In recent work, Agarwal, Lee, Gang, and Romo proposed the length conjecture, which connects the large $n$ limit of a colored Jones polynomial of the link $K\cup L$ to the ``length" of $L$ in the complement of $K$. A careful statement of this conjecture requires many ingredients, the most important of which are a state integral model for perturbative $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons theory and a $3$d quantum trace map. In this talk, we will begin with an overview of the necessary ingredients to define the volume conjecture. Then, we will discuss the modifications required to extend the volume conjecture to the length conjecture, along the way giving a working definition of the aforementioned state integral and 3d quantum trace. Time permitting, I will discuss ongoing work, joint with Mauricio Romo, on proving this conjecture for twist knots.

A real form of the exceptional simple Lie group of rank 2 can be obtained from the local symmetries of a pair of spheres rolling along each other, but only when the ratio of the radii of the spheres is 1:3 or 3:1. While this result might seem miraculous, we will attempt to provide a fairly simple visual explanation for why such a ratio of radii is needed, using a bit of basic Lie theory and a lot of pictures.

Topological and piecewise linear manifolds require extremely different methods, but the global structures are almost the same. A (compact, connected) topological manifold M has an invariant ks[M] in H^4(M;Z/2), and for dimensions greater than 4, M has a PL structure if and only if km[M]=0. In dimension 4 this reduces to an invariant in Z/2. A PL structure implies the invariant is trivial, but the converse is dramatically false. The question here is: when does the homotopy type of M determine the Kirby-Siebenmann invariant, and in these cases how can it be calculated? The answer is known for oriented manifolds; we speculate on the unorientable case.

We study the scaling limit of the dimer model on ‘critical’ graphs. We establish a connection between the dimer model and the free Dirac Fermion quantum field theory in various ways by studying them in a background ‘gauge field’. We speculate on how to leverage the fundamental nature of the Dirac Fermion in 2D CFT to study general CFTs from the dimer point-of-view.

In 2020, Naisse and Putyra gave the first extension of odd Khovanov homology to tangles. To answer this relatively longstanding open question, they introduced new algebraic objects called grading categories. Recently, we have shown that grading categories, while slightly unfriendly to their user, admit flexible generalizations which allow for other interesting constructions. These include categorified Jones-Wenzl projectors and, ultimately, an "odd" colored link homology theory. In this talk, I’ll introduce grading categories and aim to discuss ongoing work regarding the Hochschild (co)homology of algebras graded by grading categories, and its application to link homology.

A real algebraic set is a set of common zeros of polynomials with real coefficients. For example, a cuspy cone is given by z^3=x^2+y^2. It has a singular point at its tip. A geodesic in the cuspy cone might loop around this tip an arbitrarily large number of times. The first question is: “ Does there exist a geodesic that loops around a singular point infinitely many times?”. The answer is yes. I will show an evidence (2025) of the existence of such a geodesic.  Moreover, a geodesic that starts and ends on the smooth part of an algebraic surface might pass through the singular part to achieve the shortest distance. For example, a surface with a cuspy dent is given by (x-z)^2 =(z^3 -y^2)^2. We will look at a picture to see the existence of a geodesic that passes through the dent more than once. The second question is: “Does there exist a geodesic that passes through the singular part of a real algebraic set infinitely many times? The conjecture is no. I will again provide partial results. You can skip the next two lines. In my thesis (2021), I proved that any semi-algebraic set in R^2 has a cell decomposition such that every geodesic passes through the cell at most twice. In a paper (2023), I showed that in a closed region in R^3 with two analytic hypersurfaces as boundary, a geodesic consists of finitely many interior line segments alternating with boundary hypersurface geodesics. In this talk, I will include as many pictures as possible for better understanding.

Data consisting of a graph with a function mapping into R^d arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this talk, I will discuss how the interleaving distance can be used to measure distance between geometric graphs, where functor representations of the data can be compared by finding pairs of natural transformations between them. In many cases, computation of the interleaving distance is NP-hard, so we provide an upper bound on the interleaving distance between graphs that can be computed in polynomial time. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts. This is joint work with Erin Wolf Chambers, Ishika Ghosh, Elizabeth Munch, and Bei Wang.

In early 2010s, Andersen and Kashaev defined a TQFT based on quantum Teichmuller theory. In particular, they define a partition function for every ordered ideal triangulation of hyperbolic knot complement in $\mathbb{S}^3$ equipped with an angle structure. The Andersen-Kashaev volume conjecture suggests that the partition function can be expressed in terms of a Jones function of the knot which, in its semi-classical limit, decay exponentially with decay rate the hyperbolic volume of the knot complement. In this talk, we will introduce a purely combinatorial condition on triangulations which, together with the geometricity of the triangulations, imply the Andersen-Kashaev volume conjecture and its generalization. This talk is based on the joint work with Fathi Ben Aribi.