Paolo Aluffi: “CSM classes for matroids”
Intersection theory has been extended to the combinatorial setting of matroids, overlapping the geometric setting in the case of hyperplane arrangements. I will review definitions of basic invariants in their combinatorial avatar, and focus on the CSM class of a matroid, defined by Lopez-Rincon-Shaw. A new formula for this class was obtained by applying tools from geometry, leading to a proof of a conjecture of Fife and Rincon on the coefficients of the class for arbitrary matroid. The talk will be based in part on the work of two of my students: Franquiz Caraballo-Alba and Jeffery Liu.

Jean-Paul Brasselet: “Manifolds and singular varieties, yesterday and tomorrow”.
The aim of the presentation is to assist in the discussions on the future of research based on the founding articles of the theory

José Luis Cisneros-Molina: Secondary characteristics classes for normal surfaces singularities I.
Cheeger-Chern-Simons classes are secondary characteristic classes of a vector bundle E over a smooth manifold M with a flat connection. They were defined by Cheeger and Simons using differential characters. In this talk, we show how to use these invariants to study new invariants of normal surface singularities.
i) In the case of a compact oriented 3-manifold L. For topologically trivial representations we will sketch the idea of how to compute the secondary characteristics classes using the Index Theorem for flat bundles by Atiyah, Patodi and Singer. Moreover, if L is a rational homology sphere we will provide a formula for these calculations.
ii) Our construction provides natural elements in algebraic K-theory. We will show this connection. We believe this may give a new perspective of the properties of normal surface singularities.
iii) We have some applications to open problems and to a new interpretation of some invariants. More precisely, we will show the relationship between the secondary characteristics classes and the spectrum to Klenian singularities, and how to use these invariants to complete the classification of maximal Cohen-Macaulay modules over quotient surface singularities.

Laurentiu Maxim: “Linear optimization via Chern-Mather classes”
The linear optimization degree gives an algebraic measure of the complexity of optimizing a linear objective function over an algebraic model. Geometrically, it can be interpreted as the degree of a projection map on the affine conormal variety. I will first explain how the geometry of this conormal variety, expressed in terms of its bidegrees, completely determines the Chern-Mather classes of the given affine variety. Moreover, these bidegrees are shown to coincide with the linear optimization degrees of generic affine sections. Relations to polar geometry and the nearest point problem will also be discussed. (Based on joint work with J. Rodriguez, B. Wang and L. Wu.)

Leonardo Mihalcea: Mather classes for cominuscule Grassmannians.
I will explain a recent calculation of Mather classes for cominuscule Grassmannians, and some positivity conjectures. The calculation, done in joint work with R. Singh, utilizes a desingularization of the conormal space of Schubert varieties in these Grassmannians, obtained earlier by Singh.

Irma Pallares: Thom polynomials: An exploratory discussion.
Thom polynomials and higher Thom polynomials are polynomials and power series in quotient Chern classes, respectively, with the latter also being closely related to the Chern classes of singular spaces. The aim of this talk is to discuss the possible relationships between the theory of Thom polynomials, other theories of characteristic classes, and their applications.

Guillermo Peñafort. “Cohomological connectivity of perturbations of map-germs”
Let f be a holomorphic finite map-germ, defined on a complete intersection. We show that the reduced cohomology of the image of a perturbation of f  is concentrated  in a range of degrees determined by the dimension of the instability locus of 𝑓. In the non-finite case, we obtain an analogous result, replacing finiteness by K-finiteness and the image of the mapping by the discriminant. This generalizes the previously known estimates for the concentration of the Milnor fibre of complete intersections in terms of the dimension of the singular locus.

Richard Rimanyi: “On a geometric problem in machine learning (with a singularity theory flavor).

Agustín Romano: “Secondary characteristics classes for normal surfaces singularities II”
Cheeger-Chern-Simons classes are secondary characteristic classes of a vector bundle E over a smooth manifold M with a flat connection. They were defined by Cheeger and Simons using differential characters. In this talk, we show how to use these invariants to study new invariants of normal surface singularities.
i) In the case of a compact oriented 3-manifold L. For topologically trivial representations we will sketch the idea of how to compute the secondary characteristics classes using the Index Theorem for flat bundles by Atiyah, Patodi and Singer. Moreover, if L is a rational homology sphere we will provide a formula for these calculations.
ii) Our construction provides natural elements in algebraic K-theory. We will show this connection. We believe this may give a new perspective of the properties of normal surface singularities.
iii) Applications: As a consequence of our work, we have some applications to open problems and to a new interpretation of some invariants. More precisely, we will show the relationship between the secondary characteristic classes and the spectrum to Klenian singularities, and how to use these invariants to complete the classification of maximal Cohen-Macaulay modules over quotient surface singularities.

Jakub Koncki: "How to compute higher Thom polynomials of multisingularities?"
Consider a map of varieties and a singularity germ. We study the locus where the map has the given singularity. Thom polynomial describes the fundamental (or ssm) class of such locus. In the talk, I will focus on methods of computing such polynomials. I will present an improvement in the interpolation method based on stable envelopes ideas. I will show a structure of higher Thom polynomials of multisingularities generalising results of Kazarian.
The talk is based on a joint project with R. Rimanyi.

Richard Rimany: "On a geometric problem of machine learning"
In this talk we will enumerate the main reasons for a collection of matrices multiplying to 0.  Our motivation is Bayesian Learning Theory, where one of the goals is to progressively approximate an unknown distribution using data generated from that distribution. A key component in this framework is a function K (relative entropy), which is often highly singular. The invariants of the singularities of K (in the style of `log canonical threshold') are related to how well the Singular Learning Theory "generalizes"---or, in Machine Learning terms, how efficiently the model can be trained (asymptotically). Computing the singularity invariants in real-life scenarios of Machine Learning is notoriously difficult. In this talk, we focus on an elementary example and compute the learning coefficients of Linear Neural Networks. Joint work with S. P. Lehalleur.

Jörg Schürmann: "Characteristic classes of singular spaces. Where we are and where can we go?"
We first summarize the development of the theory of characteristic classes of singular spaces up to now. Then we try to give an outlook for possible future applications and research directions

José Seade: “Three problems on the subject”.
I will explain, and give the motivations for, three problems (or lines of research) related to characteristic classes of singular varieties. One is about smoothings of singularities, another on indices of vector fields, and the latter about Milnor classes.

Andrzej Weber: "Characteristic classes with boundary conditions"
The process of resolving singularities produces smooth manifolds with smooth normal crossing divisors. The multiplicities naturally associated with the divisors play a key role in the analysis of geometry of algebraic varieties. I will report on the twisted motivic Chern classes which consider boundary multiplicities. I will discuss a connection with elliptic characteristic classes and show application to Schubert varieties and matrix Schubert varieties. It is a joint work with Jakub Koncki.

Matthias Zach: Some Le-Greuel type formulas on stratified spaces
The Le-Greuel formula for the Milnor number $\mu(f)$ of an isolated complete intersection singularity (ICIS) $f: (C^n, 0) \to (C^k, 0)$ is widely known and used in singularity theory. In this talk we discuss an extension beyond the classical setup to function germs $f: (X, 0) \to (C^k, 0)$ on an arbitrary reduced complex analytic space $(X, 0)$ with a Whitney stratification such that $f$ has an isolated singularity in the stratified sense. Since we may not assume anything about the rectified homological depth of $(X, 0)$, this requires a different approach based on Tibar's Handlebody Theorem. We give several topological definitions of counterparts of $\mu(f)$ in this setting which are based on the Nash modification, and we relate them to various ``homological indices'' via local Riemann-Roch-type formulas. In particular, this allows one to compute the numbers in concrete examples using computer algebra. This is work in progress.

Takahashi: Introductory lectures on singularities and mirror symmetry I
1. General Idea of Mirror Symmetry for Sigularities,
2. Invertible Polynomials and Topological Mirror Symmetry,
3. Matrix Factorizations and Homological Mirror Symmetry,
4. Possible Applications: Exponents vs. Vanishing Cycles. 

Cheol Hyun Cho 1: Introductory lectures on singularities and mirror symmetry II
1. Disk potential as a singularity.
2.  Jacobian ring and Quantum cohomology.
3.  Matrix factorization and Lagrangian Floer complex.
4.  Symmetries in mirror symmetry. 

Nemethi/Romano 1: Cohen Macaulay Modules and matrix factorizations on surface singularities 1.
In these talks, we will discuss results regarding the classification of maximal Cohen-Macaulay modules over normal surface singularities. We will focus on three cases: known results and techniques for general surface singularities, the classification of rank-one modules over rational and minimally elliptic singularities, and the construction of matrix factorizations for weighted homogeneous singularities.

Burban: Maximal Cohen-Macaulay modules over degenerate cusp singularities
Degenerate cusp singularities is a special class of non-isolated Gorenstein surface singularities obtained by an appropriate gluing of cyclic quotient singularities. In my talk I am going to show that the category of maximal Cohen-Macaulay modules over a degenerate cusp  is representation tame. The proof is based on a reduction to a certain matrix problem. I am going to illustrate the corresponding classification of indecomposable objects  on the case of the ring k[[x,y,z]]/(xyz). This is a joint work with Yuriy Drozd.

Filip: Smoothing Gorenstein toric singularities and mirror symmetry
We establish a correspondence between one-parameter deformations of an affine Gorenstein toric variety, defined by a polytope P, and mutations of a Laurent polynomial f, whose Newton polytope is equal to P. If the Newton polytope P of f is two dimensional and there exists a set of mutations of f that mutate P to a smooth polygon, then, under certain assumptions, we show that the Gorenstein toric variety, defined by P, admits a smoothing. This smoothing is obtained by proving that the corresponding one-parameter deformation families are unobstructed and that the general fiber of this deformation family is smooth. Our assumptions hold for all polygons that are affine equivalent to a facet of a reflexive three-dimensional polytope Q, and thus we are able to provide applications to mirror symmetry and deformation theory of the Fano toric variety corresponding to Q.

Felten 1: Global logarithmic deformation theory 1
WIn this series of two lectures, we explain how a curved dg Lie algebra controls infinitesimal deformations of logarithmic varieties, why these deformations are unobstructed for logarithmic Calabi-Yau varieties, and how this unobstructedness allows us to construct smoothings of not necessarily d-semistable normal crossing spaces.

Drozd: Kahn construction for Cohen-Macaulay modules over noncommutative surface singularities
We generalize the results of Kahn about a correspondence between Cohen–Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen–Macaulay finite, but are Cohen–Macaulay tame. It is a joint work with V.Gavran.

Kyungmin Rho: Matrix factorizations from Fukaya categories of surfaces
Burban-Drozd (2017) classified all indecomposable objects in the category of maximal Cohen-Macaulay (MCM) modules over the non-isolated surface singularity xyz=0, and hence, in the category of matrix factorizations of xyz. Under homological mirror symmetry (HMS), these categories are also equivalent to the Fukaya category of the pair-of-pants surface, as proven by Abouzaid-Auroux-Efimov-Katzarkov-Orlov (2013), and the equivalence is explicitly realized by Cho-Hong-Lau's localized mirror functor (2017). In this talk, we find the objects in the Fukaya category that correspond to the indecomposable MCM modules in Burban-Drozd's classification. They are given by immersed curves in the surface equipped with local systems. Using this geometric description, we derive an explicit canonical form of matrix factorizations of xyz, and present applications to algebraic operations through geometric approaches. This is based on joint works Cho-Jeong-Kim-Rho (2022) and Cho-Rho (2024).

Aguilar 1: An introduction to variations of Hodge structures 
We will define the basic notions of variations of Hodge structures. This is intended as a preparation for the next talks in this lecture series.

Felten 2: Global logarithmic deformation theory 2
In this series of two lectures, we explain how a curved dg Lie algebra controls infinitesimal deformations of logarithmic varieties, why these deformations are unobstructed for logarithmic Calabi-Yau varieties, and how this unobstructedness allows us to construct smoothings of not necessarily d-semistable normal crossing spaces.

Nemethi/Romano 2: Cohen MacaulayModules and matrix factorizations on surface singularities 2. 

Griffiths 1: Positivity in Hodge theory and algebraic geometry 1
The Hodge-Riemann bilinear relations define metrics in the vector bundles associated to a family of polarized Hodge structures. These metrics have associated curvatures that have very interesting geometric and analytic properties. These properties suggest, and in many cases lead to proofs, of many deep results concerning the variation of the cohomology of the fibers in families of algebraic varieties. In these two lectures we will discuss these results from the perspctive of the global differential geometry of period mappings.

Cheol hyun Cho 2: Transpose polynomial from Lagrangian Floer theory
Given an invertible singularity of two variables, we explain how to obtain its Berglund-Hubsch transpose polynomial via Lagrangian Floer theory.  We will focus on a relation between hypersurface restrictions in matrix factorizations or coherent sheaves and its symplectic geometric counterpart, which is one of the crucial steps in the construction.

Ruddat: Smoothing and resolving toroidal crossing Fano varieties using log structures obtained from zero-mutable Laurent polynomials
In a series of joint works with Alessio Corti, we construct generically log smooth Fano varieties from reflexive polytopes. The log structures are obtained locally from a zero mutable Laurent polynomial and we conjecture that all log structures of this type are smoothable. A related conjecture for toric singularities was recently stated by Corti-Filip-Petracci. We also conjecture the existence and uniqueness of log resolutions alongside some interesting new mutation structure of the resolutions. A proof of the conjecture for admissible log singularities, as well as for the singularities known as Tom and Jerry, is current joint work with Tim Gräfnitz. The prospective outcome of this program is a unified construction of compact Fano manifolds, possibly in all dimensions but certainly for Fano 3-folds. The method is also believed to work for Q-Gorenstein Fano varieties. 

Budur: The local structure of the generic theta divisor
A principle governing deformation theory with cohomology constraints in characteristic zero, generalizing Deligne's well known deformation theory principle, was developed together with B. Wang in terms of differential graded Lie modules, and with M. Rubio in terms of L-infinity modules. This principle has been illustrated many situations with complicated moduli spaces, but surprisingly, it has also recently led to a result about one of the oldest topics in algebraic geometry: for a generic compact Riemann surface the theta function is at every point on the Jacobian equal to its first Taylor term, up to a holomorphic change of local coordinates and multiplication by a local holomorphic unit.

Griffiths 2: Positivity in Hodge theory and algebraic geometry 2
The Hodge-Riemann bilinear relations define metrics in the vector bundles associated to a family of polarized Hodge structures. These metrics have associated curvatures that have very interesting geometric and analytic properties. These properties suggest, and in many cases lead to proofs, of many deep results concerning the variation of the cohomology of the fibers in families of algebraic varieties. In these two lectures we will discuss these results from the perspctive of the global differential geometry of period mappings.

Katzarkov 1+2: Theory of atoms, singularity theory and applications to birational geometry I, II
In these talks we will introduce new birational and G birational obstructions to rationality - the theory of atoms, Mendeleev tables, spectra.

Takahashi 2: Maximally-graded matrix factorizations and the Gamma integral structure for an invertible polynomial of chain type
An invertible polynomial of chain type is a weighted homogeneous polynomial having nice combinatorial properties. Motivated by the Orlik-Randell conjecture and the homological mirror symmetry conjecture,  we construct a full exceptional collection in the category of maximally-graded matrix factorizations. As an application, we show that the Gamma integral structure defined through the full exceptional collection is isomorphic to the natural one from the homology group of the Milnor fiber of its mirror dual.
This talk is based on my joint work with Daisuke Aramaki and the one with Takumi Otani.

Aguilar 2: Infinitesimal methods in mixed Hodge theory
After recalling, giving examples and applications of infinitesimal variations of Hodge structures, we will define its mixed counterpart. Then we associate to it two invariants and describe them more explicitly in the case of pairs (X,Y) where X is a Fano 3-fold and Y an anticanonical surface. We will conclude with a Torelli type theorem for general pairs (X,Y) where X is a cubic threefold and Y as before. Joint work with M. Green and P. Griffiths.