The seminar normally meets on **Tuesdays 14:30-15:40** in room **SCI 103**. An asterisk means special day/time/location.

Please contact uvarolgunes (at) ku.edu.tr for seminar-related matters.

Organizers: Türkü Özlüm Çelik, Umut Varolgunes.

Abstracts are listed at the bottom of the page.

## Date |
## Speaker |
## Title |
## Location |
## Slides |

Feb 13 | Volker Mehrmann (TU Berlin) | Energy based modeling, simulation and control of energy systems | SCI 103 | Here |

*Feb 15 at 14:30 | Volker Mehrmann (TU Berlin) | Hypocoercivity, hypocontractivity and short-time decay of solutions to linear evolution equations | SCI 103 | Here |

Feb 20 | Dimitrios Papathanasiou (Sabancı University) | Backward shift operators in Linear Dynamics | SCI 103 | |

*Feb 28 at 14:30 | Berkay Kebeci (Koç University) | A construction of Mixed Tate Nori Motives | SCI 129 | |

Mar 5 | Ali Süleyman Üstünel | Convex analysis on the Wiener space | SCI 103 | |

Mar 12 | Deniz Yılmaz (Bilkent University) | TBA | SCI 103 | |

Mart 19 | Mohammed Sadek (Sabancı University) | TBA | SCI 103 | |

Mar 26 | Barış Yıldız (Koç University) | TBA | SCI 103 | |

Apr 2 | Ekin Özman (Boğaziçi University) | TBA | SCI 103 | |

*Apr 25 at 14:30 | Asha Rao (Royal Melbourne Institute of Technology) | TBA | SNA 103 | |

Apr 30 | Çağrı Karakurt (Boğaziçi University) | TBA | SCI 103 | |

*May 6 | Ferit Öztürk (Boğaziçi University) | TBA | SCI 129 | |

May 21 | Gül İnan (Istanbul Technical University) | TBA | SCI 103 |

## Date |
## Speaker |
## Title |
## Location |
## Slides |

*Nov 9 at 16:00 | Daniele Agostini (University of Tübingen) | Algebraic Geometry at Sea | SNA 104 | Here |

Nov 21 | Tülay Ayyıldız (Gebze Technical University) | Hybrid: Symbolic-Numeric Computation | SCI 103 | Here |

*Nov 30 at 16:00 | Ezgi Kantarcı Oğuz (Galatasaray University) | Oriented Posets and Rank Matrices | SNA B173 | Here |

Dec 5 | Barış Coşkunüzer (University of Texas at Dallas) | Topological Machine Learning and Applications in Drug Discovery and Histopathology | SCI 103 | Here |

Dec 12 | Varga Kalantarov (Koç University) | Some Open Problems in Nonlinear Partial Differential Equations | SCI 103 | Here |

*Dec 14 at 16:00 | Claudia Fevola (INRIA Saclay) | Nonlinear algebra in particle physics | SNA B173 | Here |

Dec 19 | Enis Kaya (KU Leuven) | A journey into the world of p-adic heights | SCI 103 | Here |

*Dec 26 at 13:30 | Mohammad Sadek (Sabancı University) | Density Questions on Elliptic Curves | SNA 156 | CANCELLED |

*Jan 4 at 16:00 | Yusuf Barış Kartal (University of Edinburgh) | A Morse-Bott approach to the equivariant homotopy type | SCI Z07 | Here |

Jan 9 | Georgios Dmitroglou-Rizell (Uppsala University) | Floer homology and potentials for Lagrangians with conical singularities | SCI 103 | Notes |

Jan 16 | Tınaz Ekim (Boğaziçi University) | Extremal triangle-free graphs | SCI 103 | Here |

Jan 23 | Berkay Anahtarcı (Özyeğin University) | Learning Mean-Field Games | SCI 103 | Here |

Speaker: Volker Mehrmann (TU Berlin)

Title: Energy based modeling, simulation and control of energy systems

Abstract: Most real world dynamical systems consist of subsystems from different physical domains, modelled by partial-differential equations, ordinary differential equations, and algebraic equations, combined with input and output connections. To deal with such complex system, in recent years energy based modeling via the class of dissipative port-Hamiltonian (pH) descriptor systems has emerged as a very successful mathematical modeling methodology. The main reasons are that the network based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserve the pH structure and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Furthermore, dissipative pH systems form a very robust representation under structured perturbations and directly indicate Lyapunov functions for stability analysis. Another advantage of energy based modeling via pH systems is that each separate model of a physical system can be a whole model catalog from which models can be chosen in an adaptive way within simulation and optimization methods. We discuss the model class of constrained pH systems and show how many classical real world mathematical models in energy systems can be formulated in this class. We illustrate the results with some real world examples from gas transport, district heating and power systems and point out emerging mathematical challenges.

Speaker: Volker Mehrmann (TU Berlin)

Title: Hypocoercivity, hypocontractivity and short-time decay of solutions to linear evolution equations

Abstract: For linear evolution equations (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time decay behavior of the norm of the fundamental solution matrix for linear continuous-time and discrete-time systems is characterized by an integer called hypocoercivity index or hypocontractivity index, respectively. The results extend to linear operators in Hilbert spaces and can be applied to the analysis of anisotropic flows.

Speaker: Dimitrios Papathanasiou (Sabancı University)

Title: Backward shift operators in Linear Dynamics

Abstract: Weighted backward shifts are among the most important classes of operators. Their dynamic behaviour has been well understood since the work of Salas. The need for creating more sophisticated examples of operators yet preserving some of the nice properties of weighted backward shifts has motivated the researchers to generalise the notion of a backward shift in several different directions. In the present talk, we will introduce weighted shifts on trees, weighted shifts on graphs, as well as an abstraction of those which we call generalized shifts. We will try to indicate the common properties enjoyed by all those shifts and mention some of their differences. The talk will be based on joint works with Karl Grosse-Erdmann, Quentin Menet, Anton Baranov and Andrei Lishanskii.

Speaker: Berkay Kebeci (Koç University)

Title: A construction of Mixed Tate Nori Motives

Abstract: Grothendieck proposed the category of motives as a Tannakian category, offering a universal framework for Weil cohomology theories. In this talk, we will consider motives in the sense of Nori. Beilinson conjectured that the Hopf algebra R of mixed Tate motives is isomorphic to the bi-algebra A of Aomoto polylogarithms. Our aim is to reconstruct A using limits of Nori motives coming from some special configurations. This allows us to write a morphism from A to R and gives a new approach to Beilinson's conjecture.

Speaker: Ali Süleyman Üstünel

Title: Convex analysis on the Wiener space

Abstract: Transformations of Legendre type are defined by taking the Cameron-Martin space as base space and the characterization of the maximizing elements are given as the solutions of the functional stochastic differential equations. Some applications to Physics are considered.

Speaker: Daniele Agostini (University of Tübingen)

Title: Algebraic Geometry at Sea

Abstract: Smooth algebraic curves give rise to solutions to the KP equation, which models waves in shallow water, via Riemann’s theta function. Singular curves produce solutions as well, but the theta function in this case becomes degenerate. I will give an introduction to this circle of ideas and present some theoretical and computational results on this topic, based on both smooth and singular curves.

Speaker: Tülay Ayyıldız (Gebze Teknik Üniversitesi)

Title: Hybrid: Symbolic-Numeric Computation

Abstract: There are two types of computation: symbolic computation and numerical computation.
Symbolic (algebraic) computation manipulates symbols (which may or may not contain numbers) to eliminate the error in the computation. Numerical computation involves the use of approximations and is inevitably subject to error. The tools are usually derived from analysis.
In this talk we will start with defining and comparing these two types of approaches. Then we will demonstrate how they can be used together. In particular, we will focus on the problem of polynomial root certification. Some symbolic (algebraic) methods can provide certification of the roots of polynomial systems; those roots can be computed very efficiently using numerical methods.

Speaker: Ezgi Kantarcı Oğuz (Galatasaray University)

Title: Oriented Posets and Rank Matrices

Abstract: Fence posets are combinatorial objects that come up in a
variety of settings, including cluster algebras, quiver
representations, snake graphs and continued fractions. We will show
that the rank polynomials corresponding to these posets are unimodal,
proving a conjecture by Ovsienko and Morier-Genoud. We will introduce
the more generalized framework of oriented posets where rank
polynomials are replaced with rank matrices, and show applications in
cluster algebras and polytopes. Partially based on joint work with Cem
Yalım Özel, Mohan Ravichandran and Emine Yıldırım.

Speaker: Barış Coşkunüzer (University of Texas at Dallas)

Title: Topological Machine Learning and Applications in Drug Discovery and Histopathology

Abstract: In this talk, we'll introduce fundamental techniques in topological machine learning and showcase their application in two specific contexts. The first application is on computer-aided drug discovery, utilizing Multiparameter Persistence for graph representation learning. Our second application revolves around cancer detection from histopathological images via cubical persistence. We apply our methodologies across five distinct cancer types, demonstrating superior performance compared to state-of-the-art deep learning methods. The talk is accessible to graduate students in science and engineering, assuming no prior background in either topology or machine learning.

Speaker: Varga Kalantarov (Koç University)

Title: Some Open Problems in Nonlinear Partial Differential Equations

Abstract: The talk will be devoted to open problems concerning glonal behavior of solutions to the initial boundary value problems for Navier-Stokes equatons and related systems, the Kirchhoff equation, Kortweg-de Vries equation, nonlinear Klein-Gordon and nonlinear Schrödinger equation.
Some open problems in the theory of infinite-dimensional dynamical systems generated by nonlinear parabolic and damped nonlinear wave equations will be also discussed.

Speaker: Claudia Fevola (INRIA Saclay)

Title: Nonlinear algebra in particle physics

Abstract: Feynman integrals play a central role in particle physics in the theory of scattering amplitudes.
In this talk, I will show some examples of how the interplay between algebro-geometric methods and fundamental physics problems leads to advances in both disciplines. In particular, I will discuss vector spaces associated with a family of generalized Euler integrals and the study of their singular locus.

Speaker: Enis Kaya (KU Leuven)

Title: A journey into the world of p-adic heights

Abstract: In Diophantine geometry, height functions measure the “size” of rational points on algebraic varieties. Such functions play a central role in defining and computing several interesting invariants in arithmetic geometry. A p-adic height function can be regarded as a “local” analogue of a classical height function, where p is a prime number, and there are several p-adic height functions in the literature. In this talk, we will journey into the world of p-adic heights. In particular, we will discuss the relationships among different p-adic heights and algorithms to compute them numerically.

Speaker: Mohammad Sadek (Sabancı University)

Title: Density Questions on Elliptic Curves

Abstract: In number theory, there are many important questions that have been withstanding our attempts to answer. While a complete answer to these questions may seem out of reach in the near future, certain conjectural answers are widely believed due to significant theoretical and numerical evidence. Arithmetic statistics provides a plausible approach to test these conjectures. In this talk, we plan to overview some of the results offered by arithmetic statistics in the direction of elliptic curves.

Speaker: Yusuf Barış Kartal (University of Edinburgh)

Title: A Morse-Bott approach to the equivariant homotopy type

Abstract: Morse theory provides an effective way to calculate the homology of smooth manifolds, in terms of critical points of a function and its gradient flow. Floer applied this idea in the infinite dimensional setting to produce new invariants in symplectic and low dimensional topology, and motivated by this, Cohen, Jones and Segal has shown how to obtain finer information about the topology of a smooth manifold from the Morse theory, thus providing a framework for refining Floer's invariant too. However, neither Morse theory nor the framework of Cohen-Jones-Segal are compatible with the compact group actions on the underlying manifold. In this talk, I will explain how to define a new framework for Morse-Bott functions in order to extract information about the equivariant stable homotopy type. In the remaining time, I will discuss applications to equivariant Floer theory. Joint work with Laurent Cote.

Speaker: Georgios Dmitroglou-Rizell (Uppsala University)

Title: Floer homology and potentials for Lagrangians with conical singularities

Abstract: We explain how to extend the definition of the superpotential and wrapped Fukaya category from closed embedded Lagrangians to Lagrangians with conical singularities, by relying on techniques from Symplectic Field Theory. In particular we introduce the refined potential as defined in joint work with T. Ekholm and D. Tonkonog. We apply this technique to the singular Lagrangian skeleton given as the complement of z_1z_2...z_n=1 in C^n, which can be considered as a monotone singular Lagrangian inside CPn. This is joint work with P. Ghiggini.

Speaker: Tınaz Ekim (Boğaziçi University, Department of Industrial Engineering)

Title: Extremal triangle-free graphs

Abstract: In this talk, we will address an extremal problem from two perspectives: structural graph theory
and Integer Programming (IP). We will determine the maximum number of edges in a triangle-free
graph when its maximum degree is at most d and its matching number is at most m for two given
integers d and m. We will derive structural properties of extremal triangle-free graphs, which will
allow us i) to provide a formula for this maximum number which is valid in most cases, ii) to develop
an IP formulation for constructing extremal graphs, which has surprising links with the classical
Knapsack problem. We conjecture that our formula giving the size of extremal triangle-free graphs
is also valid for all the open cases and endorse this conjecture by solving the IP formulation for
some additional cases.

Speaker: Berkay Anahtarcı (Özyeğin University)

Title: Learning Mean-Field Games

Abstract: This talk explores Mean-Field Games (MFGs), a framework for analysing how large populations make strategic decisions. The emphasis will be on integrating reinforcement learning with MFGs, offering insights into the dynamic processes through which agents learn and adapt their policies in a mean-field environment.