In this talk, I will discuss a conjectural characteristic class version of the Hodge index theorem for singular complex algebraic varieties, formulated by Brasselet, Schuermann, and Yokura, which expresses the Goresky-MacPherson homology L-classes in terms of suitable Hodge-theoretic L-classes. I will survey recent results on spaces for which the conjecture has been proved.

We introduce trim resolutions of complex algebraic varieties, a strengthening of the notion of small resolution. We prove that the characteristic cycle of the intersection cohomology sheaf of a variety admitting a trim resolution is irreducible and that for such varieties the stringy and Chern-Mather classes coincide.

According to Berstein and Edmonds, R. H. Fox showed in unpublished work that every closed connected non-orientable \( 2n \)-dimensional PL manifold admits a branched covering over the real projective \( 2n \)-space, for every \( n \in \mathbb{N}. \) However, no control is given on the degree of the covering and on the regularity of the branch set. In a joint work with Riccardo Piergallini and Daniele Zuddas, we improve such result for \( n=2. \) More precisely, we prove that every closed connected non-orientable PL 4-manifold X is a simple branched covering of \( RP^4 \), where the degree can be chosen to be any number \( d \geq 4\) with the same parity of the Stiefel-Whitney number \( w_1^4 [X].\) Moreover, we show that the branch set can be assumed to be non-singular if \( d \geq 5\) and to have just nodal singularities if \( d = 4.\)

A pseudomanifold is called a Witt space, if its middle intersection chain sheaf is self-dual. Examples include all pure-dimensional complex algebraic varieties. Suppose a finite group G acts on a Witt space by stratified diffeomorphisms. Then we will discuss how the equivariant signature operator and analytic bivariant KK-theory can be used to extend the Atiyah-Singer G-signature formula from G-manifolds to G-Witt spaces. This is joint work with Eric Leichtnam and Paolo Piazza. Based on the G-Witt bordism invariance of the G-signature, one obtains Zagier-type equivariant L-classes, whose average computes the Goresky-MacPherson L-class of the orbit space.

A contact structure on a 3-manifold M is a 2-plane distribution on M that is totally non-integrable. Knots and links in M that are everywhere tangent to the distribution are called Legendrian. Legendrian knots have two classical invariants, the Thurston-Bennequin and rotation numbers. A fundamental problem in contact topology is to classify Legendrian representatives of a given smooth knot type with given classical invariants; when there is more than one such representative, the knot type is called non-simple. However, the Thurston-Bennequin number is only defined when the knot is null-homologous, and the rotation number is only defined when either the knot is null-homologous or the contact structure is parallelizable. We will discuss appropriate generalizations of the classical invariants to arbitrary Legendrian knots in contact 3-manifolds. We also introduce several notions of non-simplicity for links. We then construct infinite families of examples of non-simple classes of Legendrian knots and links in overtwisted contact structures. This is joint work with Rima Chatterjee and Vladimir Chernov.

We study some general problems about which finite groups can act on which (localized) homotopy types, with given fixed point sets and with specified isotropy subgroup types. R. Oliver's groundbreaking work showed that for simply connected spaces, the Euler characteristic plays the key role in answering such questions. This surprisingly simple statement is extended to non-simply connected spaces using still just the classical Euler characteristic. We also consider semi-free actions, using classical Smith theory's homological results on fixed points and extending the groundbreaking results of L. Jones on converses to Smith theory; but answers to such questions in non-simply connected spaces will be seen to involve algebraic K-theories. However, for rational homotopy types, Smith theory doesn't reign. These results are part of a series on group actions jointly with Shmuel Weinberger and Min Yan.

We discuss recent progress on light bulb smoothing for surfaces and on "topological = smooth" for embedded disks up to isotopy in 4-manifolds. As an application, we present an invariant of homeomorphisms of 4-manifolds that detects rich structure in the mapping class group of both smooth and non-smoothable 4-manifolds.

The algebraic structure of the group of volume-preserving homeomorphisms of a manifold of dimension at least three has been well-understood since the work of Fathi from the 70s. However, the surface case has long remained a mystery. I will discuss joint work clearing up parts of this mystery. Some Weyl type laws that have recently been established in low-dimensional symplectic geometry play a key role in the arguments.

In the last century it was conjectured that the topological types of Calabi-Yau are finite, and in particular that the second Betti numbers are finite. The conjecture is still unsolved, but it is now known that it holds for Calabi-Yau elliptically fibered, and under some additional hypothesis for elliptic threefolds with trivial first Chern class.
Yet, there is no known explicit bound. An elusive summand of the second Betti number is the rank of the Mordell-Weil group or the fibration. I will discuss recent progress and prove explicit bounds for the rank of the Mordell-Weil group, beyond threefolds. We will see how the topology of a carefully chosen surface bounds the invariants of its ambient space, the higher dimensional variety of interest . If there is time I will discuss the torsion part. (Based on work in collaboration with Miranda, Paranjape, Srinivas and Weigand.)

We explore the role of community in understanding neural networks and their capabilities. Notably, large language models are beginning to reason about physics. Working together, they can automate scientific workflows and attack longstanding theoretical problems. These agentic models open new possibilities for the shape of physics research but also sharpen the need for rigor and reliability through benchmarks, formalization, and verification. Community is also essential for understanding neural networks, which evolve in a swarm that defines a field theory. In fact, this new approach to field theory is very general and can recover numerous known results, including in string theory. Together, we will see that neural networks don't only model physics: they can also do physics and be physics.

We define a new integer-valued invariant of framed concordance between "evenly" framed knots in closed oriented 3-manifolds. As an application, we derive a formula relating our new invariant to the signatures of certain 4-manifolds realized as branched covers of surfaces in \( B^4.\) We also give an interpretation of the signature formula in which the terms are classical knot invariants. Joint work with Julius Shaneson.

We discuss classification of curves on simply-connected surfaces for which the fundamental groups of the complement admit free group of rank greater than 6 as a quotient assuming that classes of irreducible components are in linear systems with small degree of discriminant. As application we obtain a complete classification of curves with such fundamental groups on surfaces with classes of components in linear systems with degree of discriminant not exceeding 6.

I will explain how various facets of singularity theory can be used to understand the algebraic optimization degrees appearing in nearest point problems. (Based on joint work with J. Rodriguez, M. Tibăr and B. Wang.)

It is a major problem in singularity theory to understand the relationship between the combinatorics of a hyperplane arrangement \( \mathcal{A} \) in \( \mathbb{C}^{n+1} \) and the topology of the arrangement's Milnor fiber. In this talk, I will explain an interesting connection of this problem with vanishing cohomology and the topology of projective hypersurfaces, and show some new results which can be found from this perspective.

In the 1970s, Cappell and Shaneson conjectured equality between the meridional rank of a link and its bridge number. I will discuss two approaches to this Meridional Rank Conjecture (MRC). For the first approach, we will choose a Coxeter quotient of a link group which gives a lower bound for its meridional rank that is equal to the link's bridge number, proving MRC for a certain class of links. For the second, we will explore a diagrammatic method which approximates bridge number from above, and in fact realizes the bridge number in some diagram for any given link.

In this talk, we prove that every negative amphichiral link is slice in a rational homology ball, generalizing a result due to Kawauchi in 2009. The proof relies on a systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior. Using the same techniques, we show that every fibered negative amphichiral knot is strongly negative amphichiral, answering a question asked by Kim and Wu in 2016 on Miyazaki knots. This is joint work with Jaewon Lee (KAIST, Daejeon) and Oguz Savk (METU, Ankara).

In this talk, we discuss Milnor's link invariants and their extensions to invariants of concordance of knots, links, and surfaces in non-simply-connected 3- and 4-manifolds. We will focus on recent applications to concordance of knots modulo local knotting and concordance of 2-spheres. Part of this work is joint with Maggie Miller.

Abstract

Any connected, finite graph can be imbedded in \( S^3 \) so that its complement has free fundamental group. Some graphs can be imbedded so that the complement of every subgraph also has free fundamental group. Planar graphs have such imbeddings, and so do some non-planar graphs, such as the complete graph on 5 vertices. However some graphs, such as \( K_7,\) are intrinsically knotted; i.e., no matter how \( K_7 \) is imbedded in the 3-sphere, it contains a knotted cycle. I'll talk about an integer invariant that measures how intrinsically knotted a graph is, and relate this invariant to a long-standing conjecture in graph theory, the orientable cycle double cover conjecture.

The best model for a discrete group is a closed aspherical manifold with that as fundamental group. This exists, according to a conjecture of Johnson and Wall, iff the group is a Poincaré duality group over \( \mathbb{Z} \), and it is unique according to the Borel conjecture. When the group has torsion, this cannot be true as it has infinite homological dimension. However, with respect to fields with characteristic prime to the order of the torsion in the group, one could have had a chance of a parallel picture -- except that for lattices with elements of odd order, Fowler showed that "rationally aspherical manifolds" (or even ANR rational homology manifolds) don't exist. We discuss what we know about the other cases. (Joint work with Cappell and Yan; if time permits, also joint with Manin and Tshikshiku.)

One advantage of toric varieties is that they carry an associated fan/polytope, which acts as a combinatorial blueprint, allowing one to find combinatorial formulas for their invariants. This yields results not only in algebraic geometry, but also in the study of polytopes. However, not every polytope can be associated to a toric variety, so to remedy this, Barthel-Brasselet-Fiesler-Kaup threw away the geometry and introduced "combinatorial intersection cohomology" for an arbitrary fan that is not in general the blueprint for a toric variety. In doing so, BBFK open the door to studying more general convex polytopes such as those whose combinatorial type does not contain polytopes with all rational coordinates. Using their combinatorial intersection cohomology framework, I will give a purely combinatorial proof (one that does not refer back to the geometry after passing to the fan) of the formula for the intersection cohomology signature of a (possibly non-simplcial) complete toric variety, proved by Maxim-Schürmann geometrically with mixed Hodge modules.