Abstract:
An action of a connected Lie group G on a smooth manifold M is said to have cohomogeneity c if the maximal orbit dimension equals dim(M) - c. This lecture begins with a historical overview of research developments in Riemannian G-manifolds, tracing key contributions to the field. Subsequently, we discuss on isometric actions of low cohomogeneity on pseudo-Riemannian space-forms, focusing particularly on their structural properties and geometric implications. The presentation archives a complete classification of Lie groups admitting such actions on De Sitter space (for isometric actions) and on the Einstein Universe (for conformal actions), revealing constraints on admissible Lie groups.

Abstract:
Hsiang--Pati's 1985 paper \textit{\)L^2\)-Cohomology of Normal Algebraic Surfaces}, which proved there is a natural isomorphism between intersection cohomology of a projective normal algebraic surface \(X\subset\mathbb P^{n}_{\mathbb C}\) and the \(L^2\)-cohomology of the incomplete manifold \(M=X\setminus\mathrm{Sing}\, X\), includes monomialization of the induced Fubini--Study metric on \(M\subset X\subset\mathbb P^{n}_{\mathbb C}\) (the smooth part of \(X\)). Besides the application of this method to study \(L^2\)-cohomology, it initiated the algebro--geometric problem of desingularization of the cotangent sheaf of an algebraic variety. More precisely, given an algebraic variety \(X\), one can ask whether there exists a resolution of singularities \(\pi:\widetilde X\to X\) such that the pulled-back cotangent sheaf of \(X\) is generated by differential monomials in suitable coordinates, the so--called Hsiang--Pati coordinates, at every point of \(\widetilde X\). This problem has been proved only when \(\dim X\le slant 3\), and is closely related to the problem of monomialization or toroidalization of a morphism. In this exposition, the Hsiang--Pati problem and its applications will be discussed after introducing some basic resolution techniques.

Abstract:
Many applications of Heegaard-Floer theory in knot theory stem from its structure as a Topological Quantum Field Theory (TQFT). Among these applications, bounds on the unknotting number are particularly intriguing, as such results can often be described without recourse to advanced mathematics. In addition to fascinating results that address some long-standing open questions, there are also numerous conjectures with strikingly simple statements that remain unresolved. These topics will be explored in the talk. Earlier versions of Heegaard-Floer TQFT are constructed as functors from the category of (pointed) knots and links, along with (decorated) cobordisms between them, to the category of modules along with homomorphisms between them. However, the dependence of these modules and homomorphisms on the decorations introduces certain complexities in computations and applications. In this talk, we will review the applications of Heegaard-Floer TQFTs and introduce a modified Heegaard-Floer functor from the category of oriented links in closed three-manifolds and oriented surface cobordisms in four-manifolds connecting them to the category of F[v]-modules and F[v]-homomorphisms between them, where F is the field with two elements. Notably, this modified functor is independent of the decoration. We will discuss some of its basic properties, provide examples, and explore potential applications.

Abstract:
In this talk, we examine a two-parameter family of skew product systems featuring two invariant subspaces, each supporting a distinct chaotic attractor. Within an open region of the parameter plane, we observe that the corresponding basins of attraction are intermingled-meaning every neighborhood of a point in one basin contains points from the other with positive measure. This phenomenon reflects strong sensitivity to initial conditions and poses significant challenges for predictability. The basins are separated by a fractal boundary curve, whose intricate geometry we analyze both visually and quantitatively. We compute the stability index, which quantifies the likelihood of nearby points converging to different attractors. Using thermodynamic formalism, we study the statistical behavior of this index and perform a multifractal analysis of its level sets. This analysis reveals a spectrum of dimensions characterizing the fine-scale structure of the basin boundaries. The multifractal properties of the stabilty index offer a powerful framework for classifying basin complexity and for understanding how small perturbations can lead to dramatically different long-term behaviors.

Abstract:
We will survey a number of outstanding problems and recent results in the theory of Cartan-Hadamard manifolds, i.e., complete, simply connected spaces with nonpositive curvature. These are examples of CAT(0) spaces which generalize the Euclidean and hyperbolic spaces and share many of their basic properties. For instance any pair of points may be connected by a unique geodesic, and thus notions of convexity are natural to study. Yet many basic questions concerning isoperimetric inequalities, total curvature of hypersurfaces, or even the structure of the convex hull of 3 points remains open.

Abstract:
Let \(X\) be a reduced and closed subscheme of \(\mathbb{P}^n\) whose saturated defining ideal is denoted by \(I_X\subset \mathbb{K}[\mathbb{P}^n]\). Let \(d\ge m\) be two positive integers and \(P\) a general point in \(\mathbb{P}^n\). Let \(I_{X}\cap I_P^m\) be the ideal of those homogeneous polynomials \(F\) which vanish on \(X\) as well as all of its partial derivatives of order up to \(m-1\) vanish at \(P\). By a naive parameter count the coefficients of a homogeneous polynomial of degree \(d\), this vanishing conditions give rise to \({n+m-1 \choose n}\) linear conditions on the coefficient of \(F\) and it is expected \(\dim \big[I_X\cap I^m_P\big]_d =\dim \big[I_X\big]_d -{n+m-1 \choose n}\) and there is not any hypersurfaces that meet these conditions. However, this is not the case for any closed subscheme \(X\) and there are subschemes \(X\) such that the imposed linear conditions may not be independent and consequently\[\dim \big[I_X\cap I_P^m\big ]_d > \dim \big [I_X\big]_d -{n+m-1\choose n}.\]In this case, \(Z(F)\), the zero set of \(F\) is called an unexpected hypersurface and this inequality means that there are more degree \(d\) hypersurfaces than the expected ones that vanish on \(X\) and has multiplicity at least \(m\) at general point \(P\).
In this talk, we show how the geometry of \(X\), as well as the strong Lefschetz properties of the ideal associated to the ideal \(I_X\), can be used to obtain more information on the existence of unexpected hypersurfaces for \(X\) and on the reducibility or irreducibility of the unexpected hypersurfaces. Moreover, we use the ideal structure of the ideal \(I_X+I^m_P\) to show that by changing the multiplicity \(m\), how much the property will be remained stable.

Abstract:
In Euclidean plane geometry, it is known that the composition of three reflections with concurrent axes is again a reflection. Similar theorems exist in higher dimensions, and also for hyperbolic and elliptic geometries. Recalling Cayley-Klein geometries and Cayley's claim that ''projective geometry is all geometry'', we seek for corresponding prototypes in projective geometry. The prototype theorem is The Three Harmonic Homologies Theorem, where harmonic homologies are self-inverse projective transformations with a hyperplane of fixed points. It will be shown that this theorem has close relation with Pascal's Theorem. Then, in Mobius geometry, we generalize Steiner's Porism via Poncelet's Theorem and show that it is a consequence of the three reflections theorem in hyperbolic space. Also we prove Miquel's theorem in Laguerre geometry by using the three reflections theorem in pseudo-Euclidean space.This talk presents joint works with Fahimeh Heidari.

Abstract:
It is a celebrated result of Alexandroff that a locally compact Hausdorff space has a Hausdorff one-point compactification (obtained by adding a ''point at infinity'') if and only if it is non-compact. An analogous characterization for spaces that admit one-point connectifications is not known.
In this talk, we study one-point connectifications and, in analogy with Alexandroff's theorem, we show that in the realm of \(T_i\)​-spaces (for \(i=3\frac{1}{2},4,5\), and for \(i=6\) under the set-theoretic assumption \(\mathbf{MA}+\neg\mathbf{CH}\)), a locally connected space admits a one-point connectification if and only if it has no compact component. Unlike the case of one-point compactification, a one-point connectification (if it exists) need not be unique. We consider the collection of all one-point connectifications of a locally connected, locally compact space within the class of \(T_i\)-spaces (\)i=3\frac{1}{2},4,5\)). This collection, naturally partially ordered, forms a compact, conditionally complete lattice whose order structure is rich enough to determine the topology of all Stone--\v{C}ech remainders of the space's components.

Abstract:
In this talk we consider differential equations with only algebraic solutions, in particular, their solutions do not accumulate in the phase space and they are diffeomorphic to punctured Riemann surfaces, and so their topology is completely understood. Detecting such differential equations is a hard job. In this talk I will first remind the Grothendieck-Katz p-curvature conjecture which says that if the p-curvature of a differential equation is zero for all except a finite number of primes then such a differential equation must have only algebraic solutions. I will rise some doubts on this conjecture showing that differential equations with only algebraic solutions satisfy certain congruence properties modulo powers of primes. This seems to be neglected in the literature of p-curvature of differential equations.

Abstract:
There are several methods for coding geodesics on hyperbolic surfaces. The first method involves recording the sides of a given fundamental domain that are cut by geodesics. This method has a history dating back to the 1921 work of Morse and even earlier, to the 1898 work of Hadamard, who proposed representing a geodesic on a surface using a sequence of codes. In this lecture, we aim to explain some of these methods, including Morse code, geometric code, and arithmetic code. Interestingly, Gauss reduction theory for indefinite quadratic forms is equivalent to an arithmetic coding of closed geodesics on the modular surface. Furthermore, one can also generalize the concept of continued fractions to a finitely generated Fuchsian group by considering the coding of geodesics with respect to a fundamental region. We intend to illustrate this concept for the modular group, and if time permits, we will also discuss it for general Fuchsian groups. A significant application of geodesic coding on a surface of constant negative curvature is to provide a method for proving the ergodicity of geodesic flow on these Riemann surfaces. In this talk, we will survey these results and some related applications.

Abstract:
I will present a (cohomological) Fourier-Mukai transform for tropical Abelian varieties and give some applications, including a (generalized) Poincaré formula (for non-degenerate line bundles on tropical Abelian varieties).
This is a joint work with Soham Ghosh.

Abstract:
In this talk we first give an overview to certain results on statistical properties of dynamical systems, focusing on well-known families such as Logistic family, Henon family, and Lorenz flows. Then I will discuss our ongoing project with Muhammad Ali Khan, on the existence and prevalence of a strange type of statistical behavior, called Non-statistical behaviour (!) within maps in the Logistic family, and contracting Lorenz family.

Abstract:
An abstract simplicial complex \(\Delta\) on a finite vertex set \(V = \{x_1, \dots, x_n\}\) is a collection of subsets of \(V\) (faces) closed under inclusion. Its geometric realization \(|\Delta| \subset \mathbb{R}^n\) is the union of convex hulls \(\text{conv}\{e_i \mid x_i \in F\}\) for all \(F \in \Delta\), where \(e_i\) are the standard basis vectors in \(\mathbb{R}^n\). When a topological space \(X\) is homeomorphic to \(|\Delta|\), we call \(\Delta\) a triangulation of \(X\). Crucially, two simplicial complexes with homeomorphic geometric realizations may differ drastically in their combinatorial structure. A property \(P\) of \(\Delta\) is topologically invariant if it depends only on the homeomorphism type of \(|\Delta|\); that is, if \(\Delta_1\) and \(\Delta_2\) satisfy \(|\Delta_1| \cong |\Delta_2|\), then \(\Delta_1\) has \(P\) if and only if \(\Delta_2\) does. This talk explores topological invariance for both combinatorial and algebraic properties of simplicial complexes: Combinatorial properties like shellability or vertex decomposability are inherently sensitive to the complex's combinatorial structure and often fail to be topologically invariant. Algebraic properties defined via the Stanley-Reisner ring \(K[\Delta] = K[x_1, \dots, x_n]/I_\Delta\), where \(I_\Delta\) is generated by monomials corresponding to non-faces of \(\Delta\), may or may not reflect topological invariance. For instance, we examine when Cohen-Macaulayness, Buchsbaumness, or Gorensteinness of \(K[\Delta]\) are preserved under homeomorphism of \(|\Delta|\). We survey known results, highlight open questions, and contrast invariant versus non-invariant properties, emphasizing connections between combinatorics, commutative algebra, and topology.

Abstract:
The famous and celebrated work of Rene Thom introduces an equivalence relation between closed manifolds of a fixed dimension, known as bordism. It then shows that the resulting set is equipped with a ring structure and computes this ring by determining its structure and determining its generator. This formulation, however, is in the stable range, meaning that it does not speak of where and how these manifold live (are embedded or immersed). The bordism of immersions takes these information into the account. By the famous Pontrjagin-Thom construction, this problem is translated into a computational problem in stable homotopy theory. In this talk, after a brief review of the aforementioned results of Thom, we show that how various results from algebra, namely from Hopf algebras to linear algebra, help to do computations related to this problem.

Abstract:
We revise Hamiltonian mechanics, Morse Homology and Riemannian geometry by replacing either anti-symmetric or symmetric nondegenerate 2-tensors with a non-symmetric nondegenerate 2-tensor.

Abstract:
The main objective of this article is to extend the concept of transversality to supergeometry. Transversality has two important properties in classical case, namely '' stability'' and '' genericity'', which we show in the following that in the category of smooth supermanifolds, supertransversality has stable property. In the last section, we demonstrate that supertransversality is generic in the subcategory of \(\sqcap\)-symmetric supermanifolds. These result is a step towards an extension of the concept of Euler-Poincaré characteristic to \(\sqcap\)-symmetric supermanifolds.
This is a joint work with Mehdi Ghorbani and Saad Varsaie.

Abstract:
Generalizing the Martens theorem for line bundles over a curve \(C\), we obtain upper bounds on the dimension of the Brill--Noether locus \(B^k_{n, d}\) parametrizing stable bundles of rank \(n \ge 2\) and degree \(d\) over \(C\) with at least \(k\) independent sections. This proves a conjecture of the second author and generalizes bounds obtained by him in the rank two case. The statements are obtained chiefly by analysis of the tangent spaces of \(B^k_{n, d}\). As an application, we show that for \(n\geq 5\) the locus \(B^2_{n, n(g-1)}\) is irreducible and reduced for any \(C\).
This is a joint work with Ali Bajravani and George H. Hitching.

Abstract:
We present an axiomatic, algebraic framework -lifted geometry- which extends the differential-geometric structures familiar on finite dimensional manifolds to a broad range of infinite dimensional spaces associated to a smooth manifold X. This framework requires neither a topology nor local charts on the space in question, but instead builds its algebra of smooth functions and Lie algebra of vector fields by canonical lifting from X. As principal examples we treat spaces of Radon measures, measurable mappings, embedded submanifolds, tilings, and path spaces on X. In each case we exhibit definitions of smooth functions, derivations, differential forms, and cohomology; for Radon measures we construct a gradient operator and discuss the associated Dirichlet form. We also show that classical results such as Stokes' Theorem arise naturally from the differentiability of the boundary operator in this lifted setting.
This is a joint work with Maysam Maysami Sadr.

Abstract:
The Euler-Poincare characteristic, or Euler characteristic in short, is a fundamental topological invariant of compact manifolds that plays a crucial role in a variety of geometric and topological situations. From this point of view, we tried to expand this important concept in supergeometry. In this article, we introduce Euler-Poincare characteristic pair in the category of \(\sqcap\)-symmetric supermanifolds. This is done by finding the self-intersection pair of each supermanifold.
This is a joint work with Fatemeh Alikhani and Saad Varsaie.

Abstract:
The Gromov-Hausdorff (GH) metric is a metric on the space of all compact metric spaces (up to isometry). This metric has gained the focus of probabilists in the study of scaling limits of discrete objects. The latter is motivated by Aldous's novel work, which showed that a random spanning tree on \(n\)‌ vertices converges to the so called Brownian continuum random tree. Since then, various generalizations of the GH metric have been introduced, which define a metric on the space of compact metric spaces \(X\) equipped with an additional structure on \(X\). Examples include when the additional structure is a measure on \(X\) (i.e., the Gromov-Hausdorff-Prokhorov (GHP) metric), a closed subset of \(X\), a continuous curve on \(X\), a c\`adl\`ag curve on \(X\), an isometry of \(X\), etc.
In this talk, we provide a general functorial framework for defining GH-type metrics, which unifies all such metrics already defined in the literature. It is also useful for defining new GH-type metrics without redoing everything and helps to avoid the pitfalls where some previous works had fallen in. We then use such metrics to define and study the notion of unimodular random measured metric spaces, which is a continuum analogue of unimodular random graphs.

Abstract:
We introduce a moduli space of stable commuting Higgs bundles on a smooth, projective, complex curve. Inspired by the Hausel-Thaddeus conjecture for Higgs bundles, we conjecture the equality of the \(E-\)polynomials for the moduli space of stable \(SL_n\) and \(PGL_n\) commuting Higgs bundles. The conjecture uses an equivariant localization. We prove the conjecture for \(n=2\) and even degree line bundles by providing explicit formulas of the \(E-\)polynomial on both sides.

Abstract:
We give a definition of weak geodesics on prox-regular subsets of Riemannian manifolds as continuous curves with some weak regularities. Then we characterize weak geodesics on a prox-regular set with assigned end points as viscosity critical points of the energy functional. Moreover, we prove that every locally minimizing curve with constant speed in a prox-regular set is a weak geodesic and under certain assumptions, every weak geodesic is locally minimizing.

Abstract:
The investigating of Ricci bi-conformal vector fields and their associated outcomes is crucial for gaining insights into the geometric and topological characteristics of the underlying manifolds. The study of conformal vector fields and their extensions is highly valuable in the realms of geometry and physics. In this manuscript, we study the topological properties of the Ricci bi-conformal vector field. The goal of this paper is to find some results of the Ricci bi-conformal vector fields. We prove that a complete manifold admits the Ricci bi-conformal vector fields have a finite fundamental group. For this purpose, we first state the definition and lemma, and then use them to prove our theorems. This is a joint work with Shahroud Azami.

Abstract:
We study the geometry of dynamically defined Cantor sets in arbitrary dimensions, introducing a criterion for \(C^{1+α}\) stable intersections of such Cantor sets, under a mild bunching condition. This condition is naturally satisfied for perturbations of conformal Cantor sets and, in particular, always holds in dimension one. Our work extends the celebrated recurrent compact set criterion of Moreira and Yoccoz for stable intersection of Cantor sets in the real line to higher-dimensional spaces. Based on this criterion, we develop a method for constructing explicit examples of stably intersecting Cantor sets in any dimension. This construction operates in the most fragile and critical regimes, where the Hausdorff dimension of one of the Cantor sets is arbitrarily small and both Cantor sets are nearly homothetical. All results and examples are provided in both real and complex settings. This is a joint work with Meysam Nassiri.