We develop algebraic geometry for the coupled cluster (CC) theory of quantum many-body systems. Coupled cluster theory provides a large class of high-accuracy methods that are widely employed in the computational many-body community. In this ansatz, the high-dimensional eigenvalue problem that encodes the electronic Schrödinger equation is approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. The talk is on Zoom - https://utexas.zoom.us/j/91039414253 - and will also be broadcast in PMA 12.166.

Neural operators are a class of deep learning data-driven architectures that learn maps between function spaces. Typically, an operator like that is capable of learning how to solve a family of a particular PDE problem by learning the operator that maps the known function(s) of an equation to the unknown solution. One of the challenges I would like to address in this talk is how to obtain data in order to train such models. Classical numerical method (such as finite element/differences) have been used to solve instances of PDEs for training and testing models. However, if we wish to outperform classical solvers, how can we really still depend on them to generate training data? This talk will be based on a recent preprint with Prof. Rachel Ward (see https://arxiv.org/pdf/2401.02398.pdf).

The Levine-Tristram signature of a link is a classical invariant with many definitions and applications – it is closely related to the Alexander polynomial, provides lower bounds on topological invariants such as the slice and doubly slice genus, and is almost-everywhere a concordance invariant. In 2018 Kashaev introduced a link invariant using a simple algorithm on link diagrams which he conjectured also computes the Levine-Tristram signature. In this talk we give a proof of Kashaev’s conjecture using the original Seifert surface definition of the Levine-Tristram signature. The proof also gives another way of computing the Alexander polynomial.

I will speak about the paper by June Huh and Botong Wang with the same title, where the Bellinson-Bernstein-Deligne decomposition theorem is used to prove Dowling and Wilson’s ‘top-heavy’ conjecture in combinatorial geometry.

In this series of two talks talk I will survey the construction of canonical (intrinsic) mirror families via the canonical wall structure, obtained in joint work with Mark Gross. The construction is based on punctured Gromov-Witten invariants previously defined in joint work with Abramovich and Qile Chen, that I will also comment on. The first talk is targeted at a general audience with an algebraic geometric background.

The talk originally scheduled at this time will have to be rescheduled.

The (Hilbert metric) geometry of properly convex domains generalize real hyperbolic geometry. This generalization is far from the Riemannian notion of non-positive curvature, but they have some intriguing similarities. I will explore this connection from a coarse geometry viewpoint. The focus will be on Morse geodesics ("negatively curved directions", in a coarse sense) in properly convex domains. I will show that Morse-ness can be characterized entirely using linear algebraic data (i.e. singular values of matrices that track the geodesic). Further, I will discuss how this coarse geometric notion of Morse is related to the symmetric space geometry as well as the smoothness of boundary points. This is joint work with Theodore Weisman.

The reconstruction of flexible proteins is one of the most critical challenges in structural biology, allowing us to gain insights into the functions and mechanisms of biomolecules by observing their motion. Cryogenic electron microscopy (cryo-EM) stands out as an ideal technique for studying the dynamic conformational landscape (i.e., range of motions) as it can capture a snapshot of the entire conformational ensemble. However, this reconstruction task comes with notable mathematical and computational challenges due to massive datasets, sometimes exceeding terabytes, high dimensionality, and substantial noise. After delving into the basics of the cryo-EM reconstruction problem, I will introduce a framework for reconstructing the protein distribution in a dataset by representing it in a linear subspace. The initial step can be viewed as a linear algebra problem: how can one compute a basis for proteins (3D volumes) from only incomplete and noisy measurements (2D images)? I will propose a method based on a Nyström extension of a regularized estimator of the covariance of the volumes. In subsequent steps, we will use this low-dimensional basis to reconstruct individual volumes using standard statistical methods and infer motions using elements of optimal control. I will conclude by discussing remaining challenges and open problems.

There are two perspectives we often get on a ribbon disk for a knot K in the 3-sphere: A ribbon disk is defined as either an immersed disk in S^3 with only ribbon self-intersections, or an embedded disk in B^4 with no maxima with respect to the radial Morse function. These definitions are often called "equivalent," but more technically, the difference between them is like the difference between a knot and a knot diagram; one is an embedding, and one is a lower-dimensional projection of an embedding. The "ribbon number" r(K) for a ribbon knot K is the smallest number of ribbon intersections among any projection of any ribbon disk bounded by K, and r(K) can then be seen as a sort of crossing number for ribbon disks. The Alexander polynomial of K can be used with rather surprising accuracy to give lower bounds for r(K), and we discuss our work in understanding and cataloguing ribbon numbers for low-crossing ribbon knots. Parts of this talk are joint with Stefan Friedl, Jeffrey Meier, Filip Misev, and lots of undergrads.

Let F be a p-adic field and G the F-points of a reductive group over F. If U is a compact open subgroup of G, and R is a ring, then the Hecke algebra attached to U, G, and R is the space of compactly supported R-valued functions on G that are left and right U-invariant; convolution gives this space the structure of an R-algebra. When R is a field of characteristic zero, a celebrated result of Bernstein shows that such Hecke algebras have good finiteness properties; in particular they are Noetherian. For R a more complicated ring such as Z_{\ell}, the question has been open for over thirty years. I will explain how recent results of Fargues and Scholze can be used to resolve this question.

Chern-Weil theory is, in my opinion, one of the all-time great mathematical theories. It unites the hard differential geometry concepts of connections and curvature with the soft algebraic topological idea of a characteristic class. The prerequisites are a familiarity with smooth manifolds, vector bundles, and de Rham cohomology.

Phase transitions of interacting particles in statistical mechanics can be seen in computer simulations, but cannot be mathematically proven. A few ‘spin’ models, chiefly the 2-dimensional Ising model, can be proven to show a transition. We show the underlying difficulty, and discuss a new class of models in which some examples can be proven to show transitions. Relevance to scissors congruence and the Banach Tarski paradox will be discussed if there is time.

The Boltzmann hierarchy is an infinite system of coupled equations that are instrumental for the rigorous derivation of the Boltzmann equation from many particles. In this talk we will first show uniqueness of the mild solutions to the Boltzmann hierarchy by combining, for the first time, a combinatorial technique known as the Klainerman-Machedon board game argument together with an -based estimate. Then we will show existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data. The proof of existence is constructive, and employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.

In this talk we will focus on the derivation and the analysis of the inhomogeneous kinetic wave equation (IKWE). Although there is an extended amount of literature for the space homogeneous KWE, the inhomogeneous problem has barely been addressed. In the first part we examine the validity of the IKWE for quadratic nonlinearities and dispersion relations close to Laplacian, based on joint work with Collot, Germain. In the second part of the talk, we investigate the global well-posedness and stability of the 4-wave inhomogeneous kinetic wave equation near vacuum, employing techniques from classical kinetic theory, such as the Kaniel-Shinbrot iteration.

Two knots are called n-friends if they share the same n-surgery. Manolescu and Piccirillo defined RBG links, a kind of three-component framed links in S^3, which provide a systematic way to look for 0-friends. In this talk, we will explore how to generalize their RBG-link technique to find n-friends and how certain n-friends have the potential to create exotic simply-connected definite 4-manifolds. Finally, we will focus on a special subclass of RBG links, called k-special RBG links, which can be used to obstruct knots from being (-k)-slice (i.e. bounding a disk with self-intersection number k) in mCP^2.

We consider an abelian variety A defined over various base fields F, and discuss its arithmetic over the cyclotomic Z_p-extension and more general p-adic Lie extensions. After reviewing some known results over number fields, we shift our focus to the case of global function fields. In this context, we compare the arithmetic of A over different p-adic Lie extensions without assuming the finiteness of the Selmer group of A over the base field F.

We will show that a conic in P^2 has a solution in Q if and only if it has solutions in Q_p for all primes p. If you don’t know what some of these words mean it’s no problem, everything will be introduced. The proof will rely on the volume of an ellipsoid and the fact that pi 3. Along the way we will be introduced to the p-adic numbers and the Minkowski bound.

This is the first of a two-talk series about Atiyah Bott localization in enumerative geometry. We will first (quickly) review equivariant cohomology and then discuss Atiyah Bott localization for smooth complex varieties with a torus action and finitely many fixed points. Anyone with a basic background in algebraic geometry or topology should be able to follow this talk.

We study an optimal transport problem with a backward martingale constraint in a pseudo-Euclidean space S. We show that the dual problem consists in the minimization of the expected values of the Fitzpatrick functions associated with maximal S-monotone sets. An optimal plan γ and an optimal maximal S-monotone set G are characterized by the condition that the support of γ is contained in the graph of the S-projection on G. For a Gaussian random variable Y, we get a unique decomposition: Y = X +Z, where X and Z are independent Gaussian random variables taking values, respectively, in complementary positive and negative linear subspaces of the S-space. Based on joint work with Dmitry Kramkov: https://arxiv.org/abs/2209.04664.

Far-from-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system out-of-equilibrium. In this talk, we will focus particularly on the fluctuations of certain interacting particle processes about their hydrodynamic limits. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

As many a depressed undergraduate learning ring theory has been unlucky to learn, not all rings are UFDs. Primary decomposition allows for a different type of factorization in a more general class of rings. In my talk, I will introduce the primary decomposition of an ideal and discuss some of its properties. I will then describe the classification of rings where primary decomposition holds (called Lasker rings). Finally, I will end with a discussion of the geometric interpretation of primary decomposition.

The Hofstadter Hamiltonian describes an electron moving on a two-dimensional lattice under the influence of a perpendicular magnetic field. It has been important for the theory of integer quantum Hall effect. This model, and the related almost-Mathieu maps, are a popular playground in the area of Schrödinger operators, dynamical systems, quantum groups, etc. A famous plot, known as the Hofstadter butterfly, displays striking self-similarity properties of the spectrum. The appropriate approach to such phenomena, developed first in the theory of critical phenomena, is renormalization. In this talk, besides introducing the models, describing phenomena, and showing interesting plots, I will describe a "universality" conjecture and proofs in special cases. Some of these proofs are computer-assisted.

We consider axisymmetric incompressible inviscid flows without swirl. When the axial vorticity is non-positive in the upper half space and odd in the last coordinate, we call the flow anti-parallel and we may expect a head-on collision of anti-parallel vortex rings. By establishing monotonicity and infinite growth of the vorticity impulse on the upper half-space, we obtain infinite growth of vorticity maximum at infinite time for certain classical vorticity. On the other hand, a finite but faster growth for some smooth vorticity is obtained thanks to the stability of Hill's vortex. This talk is based on joint work with In-Jee Jeong(SNU).

Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.

This is joint work with Abert and Fraczyk. Given a percolation E of the Cayley graph of a group G (thought of as a random subgraph), Burton-Keane and Lyons-Schramm showed the existence of the trajectory density, which is the almost sure limit (1/n)|\&ob;1\leq j\leq n:g_&ob;1&cb;\cdots g_&ob;j&cb; is connected to the identity in E&cb;|. for a random walk g_&ob;1&cb;\cdots g_&ob;j&cb; in the Cayley graph of G. Moreover, this limit is a.s. independent of the random walk. We give a separate proof of this result in the more general context of inclusion of equivalence relations S inside of R (where R is given by a free action of G). Moreover, we give a complete computation of this quantity in terms of the index of R restricted to the ergodic components of the subrelation. When G has Property (T) we show that zero trajectory density implies a uniform upper bound (independent of S) on the co-spectral radius (in the sense of our previous work) of S inside of R.

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin—Selberg, with applications to the Bloch—Kato conjecture and to Iwasawa theory. For this talk, I'll consider Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I'll discuss a new "first explicit reciprocity law" in this setting and its application to the corresponding Bloch—Kato conjecture, focusing on new obstacles which arise from the lack of local multiplicity one.

We know them from algebra class, but we never see them around and we just can’t multiply them very well. To bring them out of hiding, I will explain some four-dimensional geometry and algebra using the quaternions.

In the third and final talk in a series on Atiyah-Bott localization, we will count some lines. We'll discover that there are 27 lines on a general cubic surface in P^3 and, time permitting, will see that there is a unique line through two general points of P^2. It seems Euclid must have known about torus actions.

This will be an informal talk about a class of equations that arise in various areas of applied probability, such as optimal stochastic control, stochastic differential games, and mathematical economics/finance, but are also related to stochastic analysis on manifolds and Karcher means. Formally, they resemble classical stochastic differential equations but come with a terminal condition instead of an initial one. This 'minor' difference leads to a class of significantly different problems, which require distinct techniques. Unlike SDEs, BSDEs are an active area of research. Indeed, while the one-dimensional case has been well understood for over 20 years, basic questions of existence and uniqueness about systems of BSDEs are still wide open.

The breakthrough of Mouhot and Villani (2011) on the so called Landau Damping for the collision-less Vlasov equation \begin{eqnarray} & \partial_t F +v\cdot \nabla_x F \pm \nabla_x V\cdot\nabla_v F=0 \; \hbox{with} \; (x,v) \in (\Omega \subset (\mathbf R_v)^d\times \mathbf R^d)\\ & -\Delta V =G+f = F- \langle F\rangle, \; \langle F \rangle =\int_{\Omega} F(x,v,t) dx \end{eqnarray} has generated in the mathematical community many activities around the qualitative behavior (in particular for large time) of their solutions and for the behavior of the space independent average: \begin{equation} G(v,t) = \int_{\Omega} G(t,v,x)dx\,. \end{equation} As I intend to show, starting from physical motivation, such behavior depends on the spectra of the linearized operator and on the size of the perturbation. Existing mathematical results are in line with a very rich and diverse behavior.

Schubert calculus involves studying intersection problems among linear subspaces of C^n. A classical example of a Schubert problem is to find all 2-dimensional subspaces of C^4 which intersect 4 given 2-dimensional subspaces nontrivially (it turns out there are 2 of them). In the 1990's, B. and M. Shapiro conjectured that a certain family of Schubert problems has the remarkable property that all of its complex solutions are real. This conjecture inspired a lot of work in the area, including its proof by Mukhin-Tarasov-Varchenko in 2009. I will present a strengthening of this result which resolves some conjectures of Sottile, Eremenko, Mukhin-Tarasov, and myself, based on surprising connections with total positivity, the representation theory of symmetric groups, symmetric functions, and the KP hierarchy. This is joint work with Kevin Purbhoo.

Computing transport is a fundamental problem in engineering and a fertile area of applied mathematics methods and ideas. Although significant progress has been made, the complexity and importance of the problem call for novel methods to overcome limitations of current approaches. The models of interest in this presentation are Fokker-Planck type equations in which local (e.g., diffusive) transport is represented by differential operators and nonlocal transport by integrodifferential operators (e.g., non-diffusive turbulent transport closures and fractional diffusion). Computational approaches for these problems can be roughly classified as deterministic methods (e.g., finite-difference) and stochastic methods (e.g., Monte-Carlo). Although extensively used, continuum deterministic methods face stability and scalability challenges specially in the case of nonlocal operators that result in dense (non-sparse) matrices. On the other hand, particle-based stochastic methods face poor convergence due to statistical sampling. Here we present an alternative approach based on the Feynman-Kac theory that establishes a link between the Fokker-Planck equation and the stochastic differential equation of the underlying stochastic process. The proposed method reduces the computation to the deterministic evaluation of mathematical expectations (i.e., integrals) bypassing the need of discrete sampling of individual stochastic trajectories. The resulting algorithm is unconditionally stable and parallelizable. We present applications of the method to the study of the exit time problem and the initial value problem for local and nonlocal Fokker-Planck equations modeling transport in fluid dynamics and plasma physics.

A braided link is the link formed by a braid closure along with the braid’s axis. The bi-orderability of such links is equivalent to asking whether or not the free group automorphism induced by the braid preserves a bi-ordering of the free group. Kin-Rolfsen produced many examples of both order-preserving and non-order-preserving braids including a complete classification of periodic braids. In this talk, we'll survey some key results and techniques for finding order-preserving braids and obstructing order-preservingness. This includes work from an ongoing project with Nancy Scherich and Hannah Turner. This work is partially funded by the NSF grant DMS-2213213. Part of this research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University.

You’ve likely heard of the Erlangen program, Felix Klein’s 19th century effort to unify geometry with group theory. A thing of the past you say? Well, not quite. As a philosophy, the Erlangen program has certainly invaded many corners of math. More directly, the legacy of Klein’s program is alive and well today in the study of homogeneous and locally homogeneous spaces. Important examples of these spaces include the Euclidean, affine and projective geometries. In my talk, I will motivate locally homogeneous spaces and introduce (at least) three different perspectives from which they can be understood.

This talk will be divided in two parts. In the first, we will establish negative-order estimates for the accuracy of discontinuous Galerkin (DG) approximations to smooth solutions for the Vlasov-Maxwell (VM) system of equations. For the approximated solutions, we are able to extract this “hidden accuracy” through the use of a Smooth-Increasing Accuracy-Conserving (SIAC) filter which is a convolution kernel that is composed of a linear combination of B-splines. We provide rigorous error estimates for the DG solutions that show improvement to (2k + 1/2)-th order in the negative-order norm. In the second part we will analyze a hybrid method for radiation transport. The method that consists of (i) splitting the kinetic equation into collisional and non-collisional components; (ii) applying a high-order method to the non-collisional component and a low order method for the collisional component; and (iii) re-partitioning the kinetic distribution after each time step in the algorithm. In practice the hybrid method yields a more efficient method and provides a level of accuracy that is comparable to a uniform high-order treatment of the entire system. We provide for the first-time rigorous estimates for the hybrid method when we perform semi-discretization in angle in the collided component.

In this talk, we discuss how GIT quotients by reductive groups depend on a choice of linearisation of the action, known as variation of GIT, by following the works of Thaddeus and Dolgachev-Hu. To this end, we shall discuss that there is a finite wall-and-chamber decomposition of the cone of linearisations such that the GIT quotient does not change when varying the linearisation in a chamber, and undergoes a blow-up blow-down transformation, similar to a Mori flip, when crossing adjacent chambers. Furthermore, we shall illustrate the birational transformations between the GIT quotients produced by wall-crossings in explicit toric situations. In the following week, we shall discuss wall crossings for certain degenerations of logarithmic Hilbert scheme of points.

In my talk last fall, I went over the "thermodynamic formalism" for studying random graphs. In this talk I'll go over some specific results regarding the edge-triangle model. Specifically, the existence and structure of several phases and the boundaries between them.

In this presentation, I will establish the global existence of entropy weak solutions for scalar balance laws with nonlocal singular sources, along with a partial uniqueness result. A detailed description of the solution is provided for a general class of initial data in a neighborhood where two shocks interact.

Let f,g be newforms. For odd prime p, we let $\chi$ be a primitive Dirichlet character of p-power conductor and order. Due to the work of Shimura, it is known that there exist algebraic modular L-values $L_f(\chi)$ and $L_g(\chi)$. We prove that the Hecke field $Q_fQ_g(\chi)$ is generated by the products $L_f(\chi)L_g(\chi)$, for χ with large conductor. This result was derived by estimating a Galois average analogue of the second moment of central modular L-values, due to Blomer-Mili\'cevi\'c. This is a joint work with Ashay Burungale, Philippe Michel and Valentin Blomer.

Joint work with Riccardo Pedrotti (UT Austin). A finite sequence of simple closed curves on a surface determines a symplectic 4-manifold with boundary, as the total space of a Lefschetz fibration over the disc with the given surface as reference fiber and the curves as vanishing cycles. The monodromy around the boundary of the disc is the composite of Dehn twists. A positive relation in the mapping class group – a composite of Dehn twists isotopic to the identity – gives an example with trivial monodromy, in which case one can complete the 4-manifold to a closed symplectic 4-manifold with a Lefschetz fibration over the 2-sphere. This construction is known to account for (a blow-up of) every closed symplectic 4-manifold. It is natural, then, to ask for a recipe for the Seiberg-Witten invariants of these 4-manifolds in terms of the given sequence of curves. Donaldson-Smith interpreted the SW invariants of a Lefschetz fibration as counts of pseudo-holomorphic multi-sections representing a fixed homology class. We use symplectic Floer theory to obtain a “formula” for the count of pseudo-holomorphic sections (not yet multi-sections), over the disc, in a given homology class (rel boundary). The answer is expressed in terms of domains in the fiber surface.

Dirichlet domains provide polyhedral fundamental domains for discrete subgroups of the isometries of hyperbolic space on the hyperbolic space. Selberg introduced a similar construction of a polyhedral fundamental domain for the action of discrete subgroups of the higher rank Lie group SL(n,R) on the projective model of the associated symmetric space. His motivation was to study uniform lattices, for which these domains are finite-sided. We will address the following question asked by Kapovich: for which Anosov subgroups are these domains finite-sided? Anosov subgroups are hyperbolic discrete subgroups satisfying strong dynamical properties that have infinite covolume in higher rank. We will first consider an example of an Anosov subgroup for which this fundamental domain can have infinitely many sides. We then provide a sufficient condition on a subgroup to ensure that the domain is finitely sided in a strong sense. This is joint work with Colin Davalo.

Many authors discovered special phenomena in dimension 4 that do not occur in other dimensions, but today it has become difficult to find such phenomena in classical problems about exotic structures on manifolds. We present a new special phenomenon in dimension 4 from the point of view of infiniteness of mapping class groups and characteristic classes of fiber bundles. The proof uses a new series of characteristic classes obtained from Seiberg-Witten theory for families.

Many authors discovered special phenomena in dimension 4 that do not occur in other dimensions, but today it has become difficult to find such phenomena in classical problems about exotic structures on manifolds. We present a new special phenomenon in dimension 4 from the point of view of infiniteness of mapping class groups and characteristic classes of fiber bundles. The proof uses a new series of characteristic classes obtained from Seiberg-Witten theory for families.

The twisted Gan--Gross--Prasad conjecture considers the restriction of representations of GL_n to a unitary group. In this talk, I will explain the relative trace formula approach to the global twisted GGP conjecture. In particular, I will discuss how the fundamental lemma that arises can be reduced to the Jacquet--Rallis fundamental lemma, which allows us to prove the global twisted GGP conjecture under some unramifiedness assumptions and local conditions.

Proving two objects are “same” or “equivalent” (with respect to any notion of equivalence) is generally a taller order than proving they are not. A ubiquitous strategy for proving two objects are different is to cook up an invariant that "captures the difference". In this talk we will introduce the Grothendieck Ring of Varieties and see how the equivalence of two varieties in the Grothendieck ring guarantees equality of many of the known invariants (for example topological Euler Characteristic, Hodge-Deligne polynomial and even point counting functions!). Time permitting we will show that this ring is not a domain.

In the second part of this talk, we shall continue from the end of last time; namely understanding wall-chamber structure for the Atiyah flop/ blowing up two general points in $\mathbb{P}^{3}$ and a sextic del Pezzo surface in the language of toric geometry. Next, we shall recall the Hilbert-Mumford criterion for detecting GIT (semi)stable points, focusing on the toric case of understanding the weight polytope. Using this, we sketch the main wall-crossing theorem for topic GIT quotients (in the sense of Thaddeus and Dolgachev-Hu) and extrapolate the analysis to the general setup. Next, if time permits or in the third part of the talk, we shall use the above background to discuss logarithmic Hilbert scheme (in the sense of Siebert-\'et al) of points and wall crossings for certain degenerations of logarithmic Hilbert scheme of points.

In this talk, we give an overview of several recent results where quantitative estimates play a key role. In the first part of the talk, we discuss convex integration constructions for fluid systems with external forcing. We will then discuss a novel application of these ideas to the surface quasi-geostrophic (SQG) equation. Moving forward with the theme of quantitative estimates, in the second part of the talk we will describe new bounds for the defocusing energy-supercritical Nonlinear Schrödinger equation (NLS) and use these to give a universal blow-up criteria which goes below the scaling invariant threshold. These results are in line with a recent breakthrough construction of finite-time blow-up solutions, and in particular give the first generic result distinguishing potential defocusing blow-up phenomena from many of the known examples of blow-up in the focusing setting. At the end of the talk, we will briefly describe applications to related models.

I will discuss a very vague strategy to construct a non sofic group, and then prove the following result in the direction of the first step of this strategy: every non amenable initially subamenable group admits 2 sofic embeddings that are not conjugate by any automorphism of the universal sofic group. The proof surprisingly crucially uses elementary von Neumann algebra theory. This is join work with Ben Hayes.

Discrete subgroups of SLn(R) are called Anosov if they exhibit "good" eigenvalue growth properties, which persist under small deformation. This makes them suitable higher-rank generalizations of convex-cocompact Fuchsian groups, which have a rich ("Teichmüller" or "Fricke-Klein") deformation theory. In this talk, we will try to answer the question: what are the possible directions in which the tuple of eigenvalues can grow? What are the most extreme directions, and which group elements achieve them? Thurston's results on the Lipschitz distance in Teichmüller space give a surprisingly intricate answer in a special case. Guided by these, we will describe some generalizations and challenges in the general case. Joint work with J.Danciger and F.Kassel.

I will discuss some combinatorial methods used to study the geometry of moduli spaces: spaces which parametrize geometric objects. These spaces play a central role in the field of algebraic geometry, the study of spaces that are patched together from zero sets of systems of polynomial equations. This talk will be accessible to all, including undergraduate math students and graduate students working in other fields.

For a given eigenform, how many other eigenforms of the same weight and level are congruent to it modulo a prime l? We will explain how modular representation theory can be used to help answer this question. We focus especially on the case when the level is the square of a prime p such that p = -1 mod l. This is ongoing joint work with Robert Pollack and Preston Wake.

A colouring rule is a way to colour the points x of a probability space according to the colours of finitely many measure preserving transformations of x. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We show that there is a paradoxical colouring rule when the rule is continuous and the measure preserving transformations generate a group. This implies that the only fixed points (as functions of a function space) involve extreme non measurability.

In 2006, Labourie introduced Anosov representations to investigate a special connected component of the character variety of representations of Fuchsian groups into PSL(d,R) discovered by Hitchin, shown to consist entirely of discrete and faithful representations. Guichard-Wienhard generalized Labourie’s definition to word-hyperbolic groups, and Anosov representations have since grown to become a widely accepted generalization of convex cocompact Fuchsian groups to higher rank, exhibiting interesting and desirable dynamical and geometric properties at infinity along with a deformation theory that has many similarities and differences with classical Teichmuller theory. In this talk, we will introduce Anosov representations and survey recent results towards understanding the hyperbolic groups which admit Borel Anosov representations into SL(d,R), which are the "most Anosov” kinds of Anosov representations. This is a candidacy talk.

The L-space conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology. In this talk, I'll discuss the motivation for this conjecture, and describe some new results which produce taut foliations by leveraging special properties of positive braid knots. Along the way, we will produce some novel obstructions to braid positivity. Finally, I will describe some surprising apparent weaknesses in the construction, and how this potentially identifies some new avenues to study the L-space conjecture. I will not assume any background knowledge in Floer or foliation theories; all are welcome!

Gromov-Witten theory probes the geometry of projective manifolds by studying maps into them, from algebraic curves. The theory is a natural source of cohomology classes on the moduli space of stable curves, and includes several classical constructions in algebraic geometry. I will explain a speculation, going back to Levine and Pandharipande, that the cohomology classes produced by Gromov-Witten theory always lie in a small and predictable part of cohomology, known as the tautological ring. I will then outline some recent advances that give us some control over the classes produced by Gromov-Witten theory, and then speculate on what this might mean for the speculation of Levine-Pandharipande.

This talk will begin by explaining a few avatars of the notion of a family of TQFTs parameterized by a certain space. I will then explain the connection with symmetry, anomalous symmetries, and projective TQFTs. The second half will be primarily concerned with the three-dimensional setting. This will involve the Morita theory of fusion categories along with their higher groups of symmetries and the homotopy theory of these higher symmetries. (This is my dissertation defense.)

Among the groups arising as orthogonal groups of integer quadratic forms, which are maximal under inclusion? And what are the simplest quadratic forms from which each such group arises? We explain the answers for indefinite forms with rank at least 3. The proofs involve a novel non-positively curved simplicial complex, and a detailed analysis of certain 2-adic integral orthogonal groups.

We relate symplectic manifolds to Poisson manifolds. Specifically, we consider the splitting theorem.

We study families of Poncelet maps on a pencil of ellipses, which are conjugate to a corresponding family of billiard maps. This conjugation unveils a one-parameter family of continuous symmetries that Poncelet maps inherit. We show that each pencil of ellipses has a single parameter, the pencil eccentricity, which becomes the modulus of the Jacobi elliptic functions used to construct a covering space. We find that the rotation number of the Poncelet map for any element of a pencil can then be written in terms of elliptic integrals. The resulting expression for the rotation number gives an explicit condition for Poncelet porisms, the parameters for which the rotation number is rational. For such parameters, an orbit of the corresponding Poncelet map is periodic: it forms a polygon for any initial point. These universal parameters also solve the inverse problem: given a rotation number, which member of a pencil has a Poncelet map with that rotation number? We also analyze how rotation numbers are related to Cayley's conditions.

We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. If the velocity field is autonomous or time periodic, long-time behavior follows by studying the spectral properties of the transfer operator associated with the finite time flow map. When the flow is uniformly hyperbolic, it is well known that it is possible to construct anisotropic Sobolev spaces where the transfer operator becomes quasi-compact with a spectral gap, yielding exponential decay in these spaces. In the non-autonomous and non-uniformly hyperbolic case this approach breaks down. In this talk, I will discuss how in the stochastic ergodic velocity setting one can recover analogous results under the expectation using pseudo differential operators to obtain exponential decay of solutions to the transport equation from $H^{-\delta}$ to $H^{-\delta}$ -- a property we call annealed mixing. As a result, we show that the Markov process obtained by considering advection with a source term has a unique stationary measure. This is a joint work with Jacob Bedrossian and Patrick Flynn.

One classical incarnation of Stone duality is an anti-equivalence between the categories of profinite sets and Boolean algebras respectively. In this talk, we will discuss how this can be extended to a fully faithful embedding of condensed sets into fpqc sheaves over any base field. The resulting algebro-geometric spaces are not as pathological as one might expect, and many properties of a condensed set reflect in terms of algebro-geometric properties on the other side.

The dynamics for the interface in many free boundary problems is driven by the normal derivative of a corresponding pressure function, and often the free boundary is the boundary of the positivity set of this unknown pressure, which evolves in time. One well-known example of these types of models is called the Hele-Shaw problem. In this talk we will describe the set-up of some of these free boundaries and show how in many situations, their solution becomes equivalent to solving a nonlinear fractional heat equation (in one fewer space variables). These fractional heat equations fall into the general scope of what are called Hamilton-Jacobi-Bellman equations, which have enoyed extensive study in the past 20 years or so (at least for the fractional setting, and much longer for the first and second order settings). Furthermore, many of the well established properties about existence, uniqueness, and regularity for Hamilton-Jacobi-Bellman equations can then be transferred back to the original free boundary problem. We will discuss various recent results in this direction.

This is the first part of a series talk on symplectic duality. Symplectic duality is a duality between a pair of symplectic resolutions where a measurement on one resolution matches with another measurement on its dual resolution. We first introduce the notion of symplectic resolution and explain how it naturally arises from the study of Coulomb/Higgs branches of 3D N=4 gauge theory, leading to the understanding of symplectic duality as 3D mirror symmetry in physics. After covering those generalities, we restrict our concern to quiver gauge theory and give explicit descriptions of both of the branches. Throughout, the Springer resolution and resolution of Kleinian singularity will serve as our running examples.

This is the second part of a series talk on symplectic duality. Symplectic duality is a duality between a pair of symplectic resolutions where a measurement on one resolution matches with another measurement on its dual resolution. We introduce the notion of symplectic resolution and some interesting properties they enjoy. Next, we explain the idea of symplectic duality, i.e., what do we expect for symplectic dual pairs. Finally, we confirm some of the expectations with a concrete example, namely the duality between the Springer resolution of nilpotent orbit closure and the resolution of A type Kleinian singularity.

In this talk, we will discuss Euler classes of orthogonal bundles and then see how they can be used to define algebraic Donaldson-Thomas invariants counting sheaves on Calabi-Yau fourfolds.

After introducing the Landau-Fermi-Dirac equation, we discuss the theory of moments, higher integrability, and boundedness of solutions. The strategy produces estimates independent of the quantum parameter which are used to prove algebraic convergence to thermal equilibrium, via an entropy method, in a regime of mild saturation.

We will present the conjecture of vortices concentrated around a filament for the 3D Euler equations. While this question is open in general, we will give several geometric configurations where results have been obtained: 2D Euler, axisymmetric 3D Euler without swirl, lake equations and helical 3D Euler without swirl. I will show the link between the kernels of the Biot-Savart law for these equations and for the 2D Euler equations. I will present the main arguments to show the persistence of the concentration towards filaments that move according to the "binormal curvature flow". This work is in collaboration with Martin Donati, Lars Eric Hientzsch and Evelyne Miot.

For every $n \geq 3$, we construct links in $S^{n+1}$ which are split by an integer homology -sphere, but not by the standard . This is joint work with Marco Golla.

We will start with the formula of Siegel and Weil, which identifies a weighted average of theta series attached to positive definite lattices (over integers) with a certain more explicit generating series, called Eisenstein series. A special case gives Siegel’s mass formula expressing the weighted number of such lattices of a given dimension in terms of Bernoulli numbers. For non-definite lattices, there is an “arithmetic” analog of theta series, as a generating series of special topological cycles on certain locally symmetric spaces. When these locally symmetric spaces admit algebraic structure (in our case, they are moduli spaces of abelian varieties or K3 surfaces), the topological cycles are algebraic cycles (defined over rational numbers or integers). The arithmetic intersection numbers (an arithmetic analog of “linking numbers”) of these cycles have recently been determined and are shown to be related to the derivative (with respect to a suitable complex parameter) of suitable Eisenstein series. We will finish with mentioning some applications of these results to quetions about algebraic cycles on such moduli spaces and L-functions. No prior knowledge on any of these is required!

Berkovich spaces give a formalism for constructing spaces of valuations on varieties over nonarchimedean fields. As such they encode a great deal of information from birational geometry. The most notable invariant is the essential skeleton, a subset of the Berkovich space corresponding to the valuations monomial on strata of a dlt minimal model of . Inspired by Mori's conjecture in birational geometry, we conjecture that the essential skeleton is the complement of the images of all virtual disks, which are analytic objects analogous to families of rational curves. I will present some progress on this conjecture in joint work with Jiachang Xu and Muyuan Zhang.

In this talk we will discuss some aspect of the question: given a group, what is the range of the Furstenberg entropy of ergodic stationary actions of it? For the special linear group and its lattices, constraints on this spectrum come from Nevo-Zimmer structure theorems; and entropy values can be realized based on Poisson bundles over stationary random subgroups.

The kinetic theory of plasma features a surprising variety of equations such as the Landau equation, the Vlasov-Poisson equation and the Balescu-Lenard equation. This wealth of different dynamics is due to the onset of collective effects and the scale-invariance of the Coulomb potential. In this talk, we discuss the theory in the framework of scaling limits of interacting particle systems and present recent advances.

If Δ is a contractible compact d-manifold, then its boundary Σ will be a homology (d-1)-sphere, but the boundary need not be simply connected and Δ need not be homeomorphic to the d-disk. In joint work with Randal-Williams, we show that the topological group consisting of homeomorphisms of Δ fixing the boundary pointwise, is nevertheless contractible assuming d≥6. In the special case of the d-disk this was proved in 1923 by Alexander, by writing down an explicit homotopy now known as the Alexander trick.

I plan to spend time talking about the boundary at infinity for surfaces, geodesic currents (a measure on the space of geodesics), and why they are both interesting tools to help to understand geometric notions. Afterwards, I will go more in depth about how I use these tools to study the marked length spectrum. If time allows, I will also give examples of some interesting length spectrum results.

We sketch and reprove the main equidistribution instance in the Ferrero-Washington proof of the vanishing of Iwasawa mu-invariant, based on the ergodicity of a certain p-adic skew-product extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan).

In this talk, I firstly explain the definition and properties of variation of Hodge structures as a topic in transcendental algebraic geometry. Then, I will use this theory to formulate the Hodge-Theoretic mirror symmetry, which is the earliest version of mirror symmetry. If time permits, I will use the mirror symmetry of quintic threefolds as an example.

Having discussed square root Euler classes for special orthogonal bundles last week, we will see how they can be used to define sheaf-counting invariants for Calabi-Yau fourfolds.

Consider labelings of the vertices a large regular graph by some finite set of symbols. I'll explain how fixing some average local quantity (like what fraction of edges have the same symbol on either side) can sometimes "essentially" determine all other average local quantities, and what this has to do with maximal-(sofic)-entropy measures. For some physics-inspired examples I'll say what the maximal-entropy measures are and sketch how they can be determined. For the Potts model, it turns out that the entropy is not a concave function of the entropy. This means that standard exponential families miss a range of energy values.

When von Neumann introduced the concept of amenable groups in 1929, he showed that no group containing a nonabelian free subgroup was amenable. The von Neumann-Day conjecture was a converse to this, that any nonamenable group must contain F2 as a subgroup. This was originally disproved by Ol'shanskii in 1980, and since then a wide range of counterexamples have emerged. One may then ask, are there any large classes of group where the von Neumann-Day conjecture holds? Or, does some similar statement hold if we weaken what is meant by having a nonamenable free subgroup? In this talk, I will discuss some positive answers to the first question, in particular groups which satisfy a Tits alternative. Then, I will discuss progress towards the second in the field of measured group theory, and the Gaboriau-Lyons problem.

In this talk we show that for any smooth 4-manifold X homeomorphic to a Gompf nucleus N(2n) and any odd prime p, the standard Z/p-action given by rotation in the fibers of the Seifert-fibered boundary cannot extend smoothly to a Z/p-action over X (with one exceptional case), whereas in some cases these actions do extend topologically. In particular, we show that for each prime p=5 and each n=1 there exists a non-smoothable Z/p-action on N(2pn) extending the standard Z/p-action on its boundary. Furthermore these actions remain non-smoothable after arbitrarily many equivariant stabilizations with S^2 x S^2 of a certain type, showing that non-smoothable Z/p-actions do not satisfy a Wall stabilization principle with respect to this class of stabilizations. The proof makes use of invariants coming from Seiberg-Witten Floer K-Theory as well as some equivariant index theory.