We will sketch an approach to QFT using compact manifolds with allowable complex metrics. We will exhibit how nuclear spaces (and their duals) naturally arise and demonstrate how certain properties such as unitarity can be expressed in terms of these nuclear spaces.

In 1989 Mumford and Shah proposed a simple minimization problem in the plane: among functions which, roughly speaking, are piecewise smooth and undergo jump discontinuities on a set of finite length, one seeks those which minimize the sum of the length of the set of discontinuities and of the Dirichlet energy. A rich and successful theory was developed in the late eighties and in the nineties to deal with this and similar variational problems. However the main conjecture of Mumford and Shah, which details how regular a minimizer is expected to be, is still widely open. I will give an overview of the ideas in the existence theory and I will survey the regularity theorems proved in the literature in the past three decades, arriving to a recent joint work of mine with Matteo Focardi and Silvia Ghinassi.

The Figure-Eight Knot Complement, the Whitehead link complement, the complement of the Borromean rings, and the Magic manifold--what do they have in common? Well, they are nice 3-manifolds that we've given names to because they have had a lot of utility in the story of 3-manifold theory and the geometry of these objects. It turns out, part of what makes them so nice is that they are all arithmetic manifolds--but what does that mean? People have assured me that this property makes these manifolds very cool (some, like Thurston, write that they have a "special beauty"). Maybe this has happened to you too. In this talk, I hope to explain what they are, go over some examples, and talk about some cool results.

It is a fundamental principle in geometric group theory that large scale geometric properties of metric spaces on which a group acts should be considered properties of the group itself. If a given group acts on a hyperbolic metric space, then a particularly natural example of such a large-scale property is the Gromov boundary of that space: this is a topological space that compactifies the hyperbolic space at infinity. Connectivity properties of this boundary tell us how we can cut the group up into simpler pieces: a famous theorem of Stallings says that the group splits (as an amalgamated product or HNN extension) over a finite group if and only if the boundary is disconnected. In fact, one can say more: work of Brian Bowditch tells us that the structure of the collection of local cut points in the boundary determines a canonical JSJ decomposition for the group. In this talk I'll describe how Bowditch's JSJ decomposition is built from the structure of the boundary.

Despite seeing fluid vortices in our day to day lives, the mathematics of their behavior is poorly understood. I will introduce and motivate a model, explain some of the mathematical challenges in studying it, and discuss the stability of Oseen vortex, which is one of the only known exact solutions to the model.

In my talk I will give an introduction to Class S theories. I will first discuss the construction of Class S theories from the 6D (2,0) field theory. An important ingredient that goes into constructing these Class S theories are half-BPS defects called punctures, which I will introduce. Lastly I will introduce Seiberg-Witten theory and its relation to Class S theories.

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. In particular a knot can be untangled if and only if it has genus zero. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass–Lagarias–Pippenger, Agol–Hass–Thurston, Agol, and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. Marc Lackenby proved that the knot genus problem for the 3-sphere lies in NP. In joint work with Lackenby, we prove that this can be generalized to any fixed, compact, orientable 3-manifold, answering a question of Agol–Hass–Thurston from 2002.

A perfect cuboid is a box such that the distance between any two corners is a positive integer. A magic square is a grid filled with distinct positive integers, whose rows, columns, and diagonals add up to the same number. To date, we don't know if there exists a perfect cuboid, or a 3 x 3 magic square whose entries are distinct squares. What do these problems have in common? Secretly, they are both problems about rational points on algebraic surfaces of general type. I believe there is no such thing as a perfect cuboid or a 3 x 3 magic square of squares, and I will try to convince you that geometry suggests this is so. Zoom link: https://utexas.zoom.us/j/93570208579

Any smooth projective variety X over the complex numbers may alternatively be viewed as a compact Kähler complex manifold by passing to its analytification. As such, several cohomology theories are available to us when setting out to study this class of varieties, depending on whether we choose to work in the settings of algebraic topology (singular/de Rham cohomology), algebraic geometry (Chow cohomology), or complex geometry (Dolbeault cohomology). With that in mind, we will explore some of the various ways in which we can go about defining a theory of Chern classes on these varieties (a certain type of cohomological invariants associated to vector bundles, or more generally to coherent sheaves on X), and give some hints as to how these constructions relate to one another.

Following work of Danciger-Guéritaud-Kassel, we construct a family of affine actions with Fuchsian linear part acting on R^{4n-1}. We use the Margulis invariant to show that some of them are proper.

A hyperbolic group admits a compactification at infinity, called its Gromov boundary. These spaces can be quite complicated, and I will begin giving some examples to illustrate this. I will then describe a result that restricts the topological pathologies of a Gromov boundary by characterising the circumstances under which the boundary is locally simply connected. This can be thought of as a higher dimensional analogue of a theorem of Bestvina and Mess.

K3 surfaces are surfaces of "intermediate type"; they have a rich geometry, and their number theory is slowly starting to come into focus. In this talk I will survey results and conjectures that point towards the idea that there is an effective algorithm to determine if a K3 surface over Q has a rational point or not.

In this talk, I will introduce a recently constructed extension operator, called the "Shapley extension". This operator gives a Lipschitz extension for a function defined on a discrete subset of metric space to the whole space. The extension is based on the Shapley theorem, which comes from the Nash equilibrium in the game theory. We will give an application of the Shapley extension in 1-Wasserstein robust optimization problem. This is a joint work with Gao and Zhang.

I will describe work concerning the Thurston norm on the second homology of a 3-manifold and its interaction with foliations and flows. This norm is a 3-manifold invariant with connections to many areas: geometric group theory, foliation theory, Floer theory, and more. There are some beautiful clues due to Thurston, Fried, Mosher, Gabai, McMullen, and others that indicate there should be a dictionary between the combinatorics of the norm's polyhedral unit ball and the geometric/topological structures existing in the underlying manifold. The picture is incomplete, and mostly limited to the case when the manifold fibers as a surface bundle over the circle. I will explain some new results which hold not just in the fibered case but also the more general setting of manifolds admitting veering triangulations (introduced by Agol). The main focus will be on joint work with Yair Minsky and Samuel Taylor, in which we use a veering triangulation to define a polynomial invariant which generalizes McMullen's Teichmuller polynomial.

A construction of Donaldson assigns to a closed, oriented surface C a hyperkähler manifold M, that can be interpreted in several different ways: 1) as an open neighborhood of the zero section of Teichmüller space for C, 2) as the space of almost-Fuchsian 3-manifolds, 3) as an open subspace of the SL(2,C) character variety of C. I will explain what most of these words mean, and how they are connected. This will be accessible to anyone who likes hyperbolic or Riemann surfaces.

Continued discussion of the last section of the Kontsevich-Segal paper.

Continued discussion of the last section of the Kontsevich-Segal paper.

The structure of the cobordism group of surface automorphisms has been determined (Bonahon; Edmonds, Ewing, 1982) as an abstract group . However, the isomorphism is nonexplicit that one is not able to determine if a given automorphism represents 0. The obstruction resides in deciding if an automorphism 'compresses', and I will be showing you that there is an algorithm to decide this with some results from hyperbolic geometry (Casson, Long, 1985).

Relatively hyperbolic groups, first introduced by Gromov, generalize hyperbolic groups and have natural applications. For example, the fundamental group of a finite volume hyperbolic 3-manifold is hyperbolic relative to the cusp subgroups. In this introductory talk, I will discuss some of the many equivalent formulations of relative hyperbolicity and the relationship between a relatively hyperbolic group and its Bowditch boundary. I will also introduce CAT(0) cube complexes and give a brief survey of efforts to extend the machinery developed by Agol, Wise and others for cubical hyperbolic groups to relatively hyperbolic groups that act nicely on a CAT(0) cube complex.

Condensed-matter theorists are interested in classifying invertible phases of matter in which the symmetry group G can act on space. One approach uses "crystalline equivalence principles," describing isomorphisms between the abelian groups of invertible phases with spatial symmetries vs. with purely internal symmetries. Freed-Hopkins model these invertible phases using equivariant homotopy theory. I'll begin this talk by introducing (a slight generalization of) Freed-Hopkins' models of these classifications, then apply them to a theorem in homotopy theory which we interpret in physics as a crystalline equivalence principle for fermionic phases. If time remains, I'll discuss some techniques for computing these abelian groups of invertible phases. This is a thesis defense.

One problem of interest in modular representation theory of finite groups is in computing the group of endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by Carlson-Thévenaz using the theory of support varieties. However, jointly with Jeroen van der Meer, we provide new, homotopical proofs of their results for the quaternion group of order 8, generalized quaternion groups, and cyclic p-groups using descent methods. This is a thesis defense.

I will discuss generalized global symmetries and higher p-form symmetries from the perspective of spontaneous symmetry breaking. I will mostly focus on continuous (non-discrete) symmetries, and discuss the generalizations of Goldstone's theorem, the Goldstone modes, and the Coleman-Mermin-Wagner theorem to generalized global and higher form symmetries.

Manolescu and Piccirillo recently initiated a program of constructing potential counterexamples to the smooth 4-dimensional Poincare conjecture via a homeomorphism of the zero surgeries of two knots. They mainly focus on examples of pairs where one knot is not slice in and the other is unknown if slice. We instead consider pairs where one knot is slice and study the resulting homotopy spheres.

It has been known since their definition due to Labourie in 2004 that Anosov representations are stable- any sufficiently small perturbation of an Anosov representation is still Anosov. In 2014, Kapovich, Leeb and Porti characterized Anosov representations as those sending geodesics in the word hyperbolic group \Gamma to “Morse quasigeodesics” in the associated symmetric space G/K. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. Supplementing their proof with explicit estimates in the symmetric space allows us to describe explicit neighborhoods of Anosov representations. This is a thesis defense. https://utexas.zoom.us/j/93336434967?pwd=eDZEUi9EdEhLcHJXQjk4eHBuNThMQT09 Meeting ID: 933 3643 4967 Passcode: morse

The study of hyperbolic and relatively hyperbolic groups acting on CAT(0) cube complexes has produced exciting recent results in geometric group theory. I will talk about a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex called a relatively geometric action. In joint work with Daniel Groves, we develop analogues of tools used to construct and study geometric actions of hyperbolic and relatively hyperbolic groups on CAT(0) cube complexes, including a relatively geometric version of Agol's Theorem. I will also discuss some of the structural theorems we hope to prove and a potential application to the Relative Cannon Conjecture.

I will describe joint work with Abouzaid which constructs a stable homotopy theory refinement of Floer homology that has coefficients in the Morava K-theory spectra. The classifying spaces of finite groups satisfy Poincare duality for the Morava K-theories, which allows us to use this version of Floer homology to produce virtual fundamental chains for moduli spaces of Floer trajectories. As an application, we prove the Arnold conjecture for ordinary cohomology with coefficients in finite fields.

Leray in 1934 and Hopf in 1950 have proved the existence of weak solutions of Navier stokes. To have the result, we can actually make use of eigenfunctions of Stokes operator and also the Bogovskii operator. This is now a classical process to find existence on a bounded domain and can inspire a lot of other existence results.

I will describe some results of 1501.01310 and 1608.01761 concerning quantization of moduli space of Higgs bundles for simply connected groups. Then I will describe how this story can be extended to the case of non-simply connected groups; this is based on an ongoing project with Charlie, Du Pei and Sergei Gukov.

In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk) in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Algebraic cycles on a variety tell us a lot about the structure and behavior of that variety. They give rise to many important invariants such as Chow groups, Picard groups, and cohomology. As usual, there is a tropical analogue to algebraic cycles in tropical geometry, which are called tropical cycles. Unlike in the classical world, tropical cycles are less understood. This talk will discuss work by Allermann and Rau who laid the groundwork for tropical cycles as well as outline their proof that the bounded tropical Chow group of is the group of fan cycles.

I'll attempt to discuss some aspects of the speculative idea that quantum gravity can be thought of as an emergent topological phase of matter.

Abstract: Mostow Rigidity says that a closed hyperbolic 3 manifold has only one hyperbolic structure. What happens if our hyperbolic 3 manifold has flaring ends with negative euler characteristic? I found in Chapter 11 of Thurston's notes that hyperbolic structures on such a 3 manifold are parametrized by the product of Teichmuller spaces of its boundary components. I will attempt to explain how this works

The talk will focus on the following results: a finitely generated group quasi-isometric to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group contains a finite index subgroup isomorphic to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group. This talk is based on joint work with Cyril Lecuire.

(Joint with Nick Sheridan.) Fukaya categories can be constructed for arbitrary closed symplectic manifolds, but at considerable technical cost. I’ll explain a comparatively simple construction that works for Calabi-Yau manifolds, leveraging the complex geometry of a system of divisors close to a specified ample normal crossing divisor. An unusual feature is that the coefficients lie in a multivariable power series ring with integer coefficients. The construction is used in the proof of homological mirror symmetry (HMS) for “generalized Greene-Plesser mirrors” by Sheridan-Smith - the current cutting-edge for HMS.

In this talk, I hope to go over the basics of gradient flows. Starting from gradient flows on Euclidean space, I will motivate the various notions of gradient flows on general metric spaces. If time permits, I will give a few examples of PDEs which appear (sometimes naturally, sometimes not) as gradient flows on metric spaces.

Abstract: Strip deformations along embedded arcs were used by Danciger-Guéritaud-Kassel for parametrising those infinitesimal deformations of a convex cocompact hyperbolic surface that uniformly lengthen all closed geodesics of the surface. During my thesis, I have generalised this result firstly for some small non-compact surfaces (polygons), and then to general hyperbolic surfaces with spikes (missing points from the boundary) that are decorated with horoballs. I shall talk about some simple examples and state the main theorems. One of the applications is to parametrise the space of all Margulis space-times with convex cocompact linear part and decorated with pairs of non-intersecting light-like lines called photons. https://utexas.zoom.us/j/99824268289

I will introduce Painleve equations which are first discovered by Painleve and Gambier in classifying degree 1 2nd order ODEs that doesn’t have movable branch points. I’ll show how Painleve equations and 4d rank one N=2 theories are related. In particular, the realization of Painleve equation as isomonodromic deformations of system of linear ODEs in isospectral deformations limit can be related to the oper limit of flat connection of Hitchin systems. The relation to 4d gauge theory provides a way to construct general solutions to Painleve equations in terms of 4d gauge theory quantities like instanton partition function.

I will introduce Painleve equations which are first discovered by Painleve and Gambier in classifying degree 1 2nd order ODEs that doesn’t have movable branch points. I’ll show how Painleve equations and 4d rank one N=2 theories are related. In particular, the realization of Painleve equation as isomonodromic deformations of system of linear ODEs in isospectral deformations limit can be related to the oper limit of flat connection of Hitchin systems. The relation to 4d gauge theory provides a way to construct general solutions to Painleve equations in terms of 4d gauge theory quantities like instanton partition function.

Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of stable rational curves, introduced by Castravet and Tevelev, appear as numerators of scattering amplitude forms for n massless particles in N=4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. We re-interpret and generalize leading singularities of MHV scattering amplitude forms as probabilistic Brill-Noether theory: the study of statistics of images of n marked points on a Riemann surface under a random meromorphic function. This leads to a beautiful physics-inspired geometry for various classes of algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.

In this talk I will discuss my attempts to interpret Serre duality using 2d topological field theory. To a smooth projective variety we will assign a framed 2d TFT and observe that the Serre functor arises as the image of a certain one dimensional bordism. We will interpret Serre duality using boundary theories for this 2d TFT.

The left-orderability of a 3-manifold group is closely connected to the topological properties of the manifold. However, link groups are always left-orderable. Interestingly, there are link groups which are known to be bi-orderable, as well as link groups known not to be bi-orderable. In this talk, I will discuss a couple of results on the bi-orderability of link groups and how these results are related to properties of the cyclic branched covers of a knot. If a two-bridge link has Alexander polynomial with coprime coefficients and all real positive roots, then its link group is bi-orderable. This result shows that a large family of knots whose cyclic branched covers are known to be L-spaces have bi-orderable knot groups. Additionally, many genus one pretzel knots have bi-orderable knot groups including the $P(-3, 3, 2r+1)$ pretzel knots. Issa-Turner showed that all the cyclic branched covers of these knots are L-spaces. There is also a family of genus one pretzel knots with bi-orderable knot groups and double branched covers which are not L-spaces. This talk is a thesis defense. ID: 952 3915 2304 Code: Bi-order