# 2022/23

### February 14, 2023, 14:30-15:40(SCI 103)

Özlem Ejder (Bahçeşehir University)

Galois Theory of the iterates of a polynomial

Let a_0 be an integer, let f be a polynomial, and consider the sequence a_n = f(a_{n-1}). It is a
natural question to ask whether there are infinitely many primes in this sequence. One quickly
decides that there are not enough tools at hand to deal with this question and one asks instead about the primes dividing at least one term of the sequence. It turns out that the symmetries of the preimages of a_0 under the iterates of f play an essential role in the solution of this density question. Motivated by the prime density questions, we study the Galois theory of the iterates of a polynomial (or a rational function). We see fruitful connections between geometry, arithmetic, and group theory in this subject. Some of the results presented in this talk are joint work with I. Bouw, V. Karemaker, Y. Kara, E. Ozman.

### February 10, 2023, 11:00-12:10(online)

Marc Kegel (Humboldt University)

The search for alternating surgeries

A slope r is called alternating for a knot K in the 3-sphere if the r-surgery along K is diffeomorphic to the double branched cover branched over an alternating link in the 3-sphere. First, we will explain these notions and their relevance in detail. Then we will discuss that the set of alternating slopes for a given knot K is computable. For concreteness, we will algorithmically classify for interesting concrete example knots all their alternating slopes and discuss these results. For that, we will use SnapPy with the help from sage, regina, and KLO. This talk is based on joint work with Ken Baker and Duncan McCoy.

### February 9, 2023, 10:30-11:40(online)

Han Liu (NYU, Abu Dhabi)

On the long time behavior of Alfven waves

This talk shall present a few recent results concerning the stability of sheared
equilibria in ideal magnetohydrodynamics (MHD) equations. In particular, for a general
class of non-monotone shears with non-degenerate critical points, linear damping for the
vertical components of the velocity and magnetic field, and depletion for the horizontal
components are mathematically justified. In addition, a particular case with Alfven speed
of the form y2 + k; k > 0; is studied, for which the behavior as k -> 0+ is captured.

### February 7, 2023, 10:30-11:40(SCI 103)

Türkü Özlüm Çelik (Boğaziçi University)

Algebraic Curves, Integrable Systems, and Computer Algebra

Recent advances in theoretical mathematics and computer algebra systems have stimulated the use of algebraic geometry on problems arising from the sciences. This ranges from physics to statistics, from cryptography to optimization. Conversely, developments in these fields inspire new questions and algorithms in foundational mathematics and manifest connections between branches of mathematics, such as algebraic geometry, number theory, combinatorics and many more. In this talk, we focus on the importance of modern computational tools in the study of algebraic curves and their applications in integrable systems. This talk touches on several computational perspectives: symbolic, numerical, and combinatorial. Through these vistas, we exhibit the above phenomena using the fundamental objects of algebraic geometry and an essential notion from mathematical physics. More precisely, smooth algebraic curves give rise to solutions of the Kadomtsev-Petviashvili hierarchy, which is a universal integrable system. These solutions derive from Riemann’s theta function, and fascinatingly characterize the Jacobian varieties of the curves among abelian varieties, thus providing an answer to the Schottky problem. We, here, are in the land of transcendental algebraic geometry. Passing to combinatorial algebraic geometry, the theta function and the solution get degenerated when the curve degenerates to a singular curve. For instance, a smooth plane quartic degenerating to a union of one conic and two lines makes the Riemann theta function to be Scherk’s minimal surface. If the degeneration gets more singular, the theta function attains algebraic hypersurfaces. This is reflected in the solutions as soliton solutions, further lump solutions. The Grassmannian classifies the soliton solutions, which labels the solutions via the combinatorial objects indexing cells of the Grassmannian. All the solutions are encoded by Sato’s Grassmannian. Exploring all these classical objects from the lens of computational algebraic geometry, we present new results and open problems. This is joint work with Daniele Agostini (University of Tuebingen), and Bernd Sturmfels (UC Berkeley & MPI MIS).

### February 6, 2023, 10:30-11:40(online)

Alisa Sedunova (CRM, Montreal)

Analytic number theory: distribution of primes and primes of a special form from both analytic and geometric perspectives

One of the most classical results in analytic number theory is the prime number theorem and its version for arithmetic progressions. From the famous result of Gauss we know that the number of points with integer coordinates inside a circle of a radius x behaves as pi x^2, but what about the number of points with prime coordinates? Using the prime number theorem along with combinatorial sieve methods one immediately shows that this amount behaves as expected from the usual heuristics, namely, the number around n is a prime with probability 1/log n. However, this question can be seen from both analytic and geometric points of view. We will discuss this sort of questions and approaches towards answering them along with the difficulties arising in the process.

### January 19, 2023, 14:30-15:40(online)

Doğa Can Sertbaş (Çukurova University)

On hyperharmonic integers

For any positive integer n, the n-th harmonic number hn is defined as the sum of reciprocals of the first n positive integers. These numbers share several arithmetic properties. For instance, it was shown by Theisinger in 1918 that hn is not an integer, when n>1. Later on, a generalization of harmonic numbers was introduced by Conway and Guy. They defined the n-th hyperharmonic number of order r as hn(r) and it is equal to the sum of the hyperharmonic numbers hk(r-1) where runs over 1 to (here, n,r are positive integers, r>1 and hn(1) = hn). Inspired by Theisinger’s result, Mező conjectured that there are no hyperharmonic integers except 1. In this talk, we first deal with the bounds on the number of lattice points in the first quadrant that give hyperharmonic integers. As a result, we will deduce that the probability of choosing an (n,r) tuple in [1,x] x [1,x] satisfying the property that hn(r) is an integer, tends to 0, as x tends to infinity. On the contrary, we will show that there are infinitely many different hyperharmonic integers. In particular, with the aid of a computer algebra toolbox, we will obtain a special set of (n,r) tuples which refute Mező’s conjecture.

### January 16, 2023, 11:00-12:10(SCI 103)

Selin Aslan (Argonne National Laboratory)

Innovative Strategies in Computational Imaging

Computational imaging is a rapidly growing interdisciplinary field that aims to overcome the traditional limitations in imaging instruments by using digital computation instead of optical processes. While computational imaging has a long history and enabled advances in a broad range of disciplines from microscopy and radiology to satellite imagery and astronomy, it has particularly become a developing trend in the past decade for large-scale challenging 3D imaging problems as driven by the advances in computing and machine learning. During my graduate and postdoctoral training, I have focused on developing novel and efficient computational imaging methods to address imaging under heavily under-sampled and corrupted data by combining algorithmic and computational advances. These developed methods can have ubiquitous scientific and commercial applications involving large-scale image datasets in various applied fields such as chemistry, biology, medicine, astrophysics and the defense industry. In this talk, I will discuss my research program on integrative strategies in computational imaging including randomization, reduced order models and machine learning.

### January 12, 2023, 14:30-15:40(SCI 103)

New Approach in Profile Analysis with
High-Dimensional Data Using Scores

In profile analysis, there exist three tests: test of parallelism, test of levels and test of
flatness. In this talk, these tests have been studied. Firstly, a classical setting, where the
sample size is greater than the dimension of the parameter space, is considered. The hypotheses
have been established and likelihood ratio tests have been derived. The distributions of these
test statistics have been given. In the latter stage, all tests have been derived in a high dimensional
setting, where the number of parameters exceeds the number of sample size. Such
settings have become more common due to the advances in computer technologies in the last
decades. In high-dimensional data analysis, several issues arise with the dimensionality and
different techniques have been developed to deal with these issues. We propose a dimension
reduction method using scores that was first proposed by Lauter et al. (1996). To be able
to find the specific distributions of the test statistics of profile analysis in this context, the
properties of spherical distributions are utilized.

### January 5, 2023, 14:30-15:40(SCI 103)

Kerem Uğurlu (Nazarbayev University)

A new coherent multivariate average-value-at-risk

A new operator for handling the joint risk of different sources has been presented and its various properties are investigated. The problem of risk evaluation of multivariate risk sources has been studied, and a multivariate risk measure, so-called multivariate average-value-at-risk, mAVaRα, is proposed to quantify the total risk. It is shown that the proposed operator satisfies the four axioms of a coherent risk measure while reducing to one variable average-value-at-risk, AVaRα, in case N = 1. In that respect, it is shown that mAVaRα is the natural extension of AVaRα to the N-dimensional case maintaining its axiomatic properties. The framework is applicable for Gaussian mixture models with dependent risk factors that are naturally used in financial and actuarial modelling. Examples with numerical simulations are also illustrated throughout.

### January 3, 2023, 14:30-15:40(SCI 103)

Şefika Kuzgun (University of Rochester)

Normal approximations for spatial averages of stochastic heat equation

In this talk, we will firstly introduce the stochastic heat equation. Secondly, we will present the so-called Malliavin-Stein method, which is a technique for finding normal approximations. Finally, we will mention how this technique is applied in our recent joint work with David Nualart to obtain normal approximations for spatial averages of the solution of stochastic heat equation.

### December  30, 2022, 11:00-12:00(CASE Z24)

Levent Alpöge (Harvard)

Integers which are(n’t) the sum of two cubes

Fermat identified the integers which are a sum of two squares, integral or rational: they are exactly those integers which have all primes congruent to 3 (mod 4) occurring to an even power in their prime factorization —a condition satisfied by 0% of integers! What about the integers which are a sum of two cubes? 0% are a sum of two integral cubes, but…  Main Theorem: 1. A positive proportion of integers aren’t the sum of two rational cubes, 2. and also a positive proportion are! (Joint work with Manjul Bhargava and Ari Shnidman.)

### December  29, 2022, 14:30-15:40(SCI 103)

Emre Sertöz (Leibniz University of Hannover)

Separating period integrals of quartic surfaces

Periods are a countable set of complex numbers that extend the algebraic numbers by adding many transcendental constants of nature. It is of great scientific importance to be able to effectively compute with periods. Applications range from number theory to biological systems, decidability problems, and econometrics. However, it is currently unknown whether it is even possible to algorithmically verify an equality between two integral expressions. Hodge theory offers hope by linking algebraic geometry to the transcendental number theory of periods. Indeed, the Grothendieck-Kontsevich-Zagier period conjecture implies, and is roughly equivalent to, the effectivity of the ring of periods, but this conjecture remains wide open. I will introduce an approach that brings transcendental numerical methods to algebraic geometry in order to prove identities involving period integrals. Together with my collaborators Costa (MIT), Lairez (Inria), and Movasati (IMPA), we developed this approach as a proof-of-concept to verify identities involving periods of quartic surfaces (K3 surfaces) in 3-space.

### December  27, 2022, 14:30-15:40(SCI 103)

Sümeyra Sakallı (University of Arkansas)

Exotic 4-manifolds and complex surfaces, algebraic fibrations, singularities and knots

Exotic manifolds are smooth manifolds which are homeomorphic but not diffeomorphic to each other. After Donaldson’s construction of exotic CP^2 # 9 \barCP^2, constructing exotic 4-manifolds has been an active research area in symplectic and low dimensional topology in the last 35 years, with a lot of progress as well as many open problems. In this talk, I will describe my research on exotic 4-manifolds in relation to complex surfaces, algebraic fibrations, singularities and knots. In particular, I will present the smallest exotic 4-manifolds with nonnegative signatures, deformation and splitting singularities in algebraic fibrations, and generalized chain surgeries which is a symplectic operation exchanging two Stein fillings. I will also discuss slicing knots in definite 4-manifolds. Some of these are joint with Akhmedov, Karakurt, Kjuchukova, Miller, Park, Ray, Van Horn-Morris and Yeung.

### December  22, 2022, 14:30-15:40(online)

Fırtına Küçük (University College Dublin)

Factorization of algebraic p-adic L-functions attached to adjoint representations of Coleman families

Artin formalism dictates a factorization of L-functions when the associated motive decomposes. The p-adic variant of the Artin formalism turns out to be a highly non-trivial problem in the absence of critical values. This arises, for example, when the motive in question is the one that comes attached to a self-Rankin-Selberg product of a modular form. In this scenario, Dasgupta and  Loeffler–Arlandini proved the Artin formalism for the relevant p-adic L-functions. To even make sense of these results, one needs to work with families of modular forms passing through the modular form in question. In this talk, I will give an overview of these concepts, and then discuss the algebraic counterpart of this problem in the p-non-ordinary setting and how one can resolve these.

### December  20, 2022, 14:30-15:40 (SCI 103)

Giray Ökten (Florida State University)

Global active subspace method

The active subspace method is a popular dimension reduction method used in problems from mathematical sciences and engineering, public health, quantitative finance, and economics. As a numerical tool it can be used in uncertainty quantification, sensitivity analysis, and optimization. The method uses the gradient information of a function to find the directions along which the function changes the most. The function is then approximated by one that depends only on the “few” important directions. The active subspace method, however, is prone to instability in the presence of noise. I will discuss a generalization of the method called the global active subspace method, which is more stable in the presence of noise. The new method uses finite-differences of function values as opposed to gradients, and has theoretical connections to global sensitivity analysis.

### December  13, 2022, 14:30-15:40 (SCI 103)

Keremcan Doğan (Istanbul Technical University)

Exceptional Geometries for M Theory

Usual differential geometric structures that play a fundamental role in the constructions of general relativity and standard model seem to be inadequate for quantum gravity theory candidates. In particular, string and M theories suggest that underlying geometric framework should be lifted to algebroids. Recent advancements in double and exceptional field theories shed light on geometrization of T- and U-dualities, which are absent in the geometry of particle fields. In this talk, necessary algebroid structures for M theory will be explained. The focus will be on T-duality whereas different formulations of T-duality and their relations to Courant algebroids, Drinfeld doubles and Poisson structures will be analyzed. Moreover, U-duality together with its relation to higher Courant algebroids, exceptional Drinfeld doubles and Nambu-Poisson structures will be also discussed.

### December  6, 2022, 14:30-15:40 (SCI 103)

Mutlu Koçar (Koç University)

Algebraic functions in terms of generalised hypergeometric functions

In this talk, we will first introduce Pochhammer hypergeometric functions and review a few basic concepts related to them. Then we will examine a particular class of algebraic equations and their solutions via the aforementioned hypergeometric functions. We will also briefly go over the so-called Newton-Puiseux algorithm, which can be considered as a method used to obtain solutions to algebraic equations under our consideration. And then, we show the two concepts coincide. Finally, we will describe braid depiction of roots to a slightly modified trinomial algebraic equation in making use of hypergeometric function theory.

### November  24, 2022, 14:30-15:40 (SCI 103)

Ahmet Berkay Kebeci (Koç University)

Geometry of configurations and dilogarithms

One expects that the Hopf algebra of mixed Tate motives is isomorphic to the bi-algebra of Aomoto polylogarithms. In this talk, our aim is to understand the 2nd grade of this algebra, namely Aomoto dilogarithms. First we will give a short introduction to periods. Then, we will give an idea about mixed motives and talk about Tannakian formalism. Finally, we will show that dilogarithmic configurations give an isomorphism between the Bloch group and Aomoto dilogarithms modulo prisms.

### October  25, 2022, 14:30-15:40 (SCI 103)

Selin Aslan (Argonne National Laboratory)

Randomization for the Efficient Reduced Order Models in Diffuse Optical Tomography

Nonlinear inverse problems appear in many applications for identification and localization of anomalous regions, such as finding tumors in the body, luggage screening, and finding contaminant pools in the earth. In this work, we focus on diffuse optical tomography (DOT) in medical image reconstruction. DOT presents huge computational challenges since it requires at least one forward and adjoint PDE solve for each source and detector at each optimization step. As rapid advances in technology allow for large numbers of sources and detectors, these problems become computationally prohibitively expensive. In the past, the use of reduced order models (ROM) has been proposed to drastically reduce the size of the linear systems solved in each optimization step in DOT, while still solving the inverse problem accurately. However, interpolatory model reduction requires the solution of large linear systems for all sources and frequencies as well as for all detectors and frequencies for each interpolation point in parameter space, followed by an expensive rank-revealing factorization to reduce the dimension. Hence, as the number of sources and detectors increases, even the construction of the ROM bases still incurs a substantial cost in the offline stage. In this talk, we propose to employ randomization to reduce the number of large linear solves for constructing a ROM global basis.

### October  20, 2022, 14:30-15:40 (SCI 103)

Umut Varolgüneş  (Boğaziçi University)

From classical mechanics to symplectic rigidity (and back?)

Consider a particle moving in Euclidean space under the influence of a Hamiltonian energy function. All possible trajectories of this particle define a flow on the phase space R2 x …x R2, where we paired each position coordinate with its corresponding momentum coordinate. One can assign to each (oriented) patch of surface in the phase space its symplectic area: add up the signed areas of the projections to each R2 factor. The birth of symplectic geometry is the observation that any Hamiltonian flow preserves these symplectic areas. A symplectic manifold is a generalization of this phase space structure to spaces with more interesting topology, e.g. on a three holed torus a symplectic structure is equivalent to an area form. I will outline some recent results (including some of mine) in symplectic geometry, restricting myself to phase spaces and surfaces.

### April 19, 2022, 14:30-15:40

Tobias Fritz  (University of Innsbruck, Austria)

Categorical probability and the de Finetti theorem

Probability theory and statistics are usually developed based on Kolmogorov’s axioms of probability space as a foundation. In this talk, I will present an alternative foundation that is currently under development, and I will argue that it improves over the traditional one in various respects. This approach is formulated in terms of category theory, and it makes Markov kernels instead of probability spaces into the fundamental primitives. Its abstract nature also implies that no measure theory is needed. After introducing the approach itself, I will summarize our proof of the de Finetti theorem in terms of it.

### April 26, 2022, 14:30-15:40

Alp Eren Yilmaz (Koç University)

Steiner triple systems

A Steiner triple system or order v; STS(v); is a pair (S; T) where S is a v element set and T is a set of 3 element subsets of S (called block) with the property that each pair of elements from S is contained in exactly one block. Steiner triple systems are widely used as experimental designs and are closely related to many other mathematical structures like error-correcting codes, graph decompositions, and affine and projective planes. In this talk, we will give the necessary and sufficient conditions for the existence ofSteiner triple systems. Kirkman proved in 1847 that Steiner triple system exists if and only if v ≡ 1 or 3 (mod 6). In this talk,  we will present the proofs given by Bose in 1939 for the case v ≡ 3 (mod 6) and by Skolem in 1958 for the case v ≡ 1 (mod 6).

### May 10, 2022, 14:30 – 15:40

Celal Umut Yaran (Koç University)

Weil Reciprocity on Compact Riemann Surfaces

The Weil reciprocity law is a result of Andre Weil for algebraic curves over algebraically closed
fields. We can generalize this result on compact Riemann surfaces by using Tate symbols. In this
talk, for each pair of meromorphic functions f and g on a Riemann surface X, we will associate a
complex line bundle (f, g) with connection. Then we will built a correspondence between the
holonomy effect on these complex line bundles and associated moderate symbols on (f, g). After
introducing the prerequisites and the statement of the law, we will summarize a geometric proof
for Weil reciprocity law on compact Riemann surfaces which depends on this correspondence and
the principal construction ”le symbole modere” due to P. Deligne.

### May 17, 2022, 14:30 – 15:40

Nihan Tanısalı (Koç University)

Szemerédi’s theorem has been strengthened in many different directions. In this presentation, we will consider a version of it on a finite vector space. We  will  use some of the important tools in Additive combinatorics, such as discrete Fourier analysis and Gowers norms. We will explain their connections, and discuss how they are used to prove the version.

### May 24, 2022, 14:30 – 15:40

Tolga Temiz (Koç University)

Exotica Problem for 4-manifolds with boundary via Knot Traces

Some difficult problems for closed smooth 4-manifolds become rather tractable when posed about smooth 4-manifolds with boundary. One such problem is the problem of exotica, and in this talk, I will talk about a specific type of 4-manifolds called knot traces and a related result regarding the exotica problem together with a sketch of its proof.

### June 21, 2022, 14:30 – 15:40

Gregory Seregin (Oxford University)

Axisymmetric Solutions to the Navier-Stokes Equations

I’ll discuss recent regularity results for axisymmetric solutions to the Navier-Stokes equations.Among them are slightly supercritical assumptions providing smoothness, which, for example, exclude Type I blowups of axisymmetric solutions.

### June 28, 2022, 14:30 – 15:40

Sebastián Donoso (University of Chile)

Multiplicative actions and applications

In this talk, I will discuss recurrence problems for actions of the multiplicative semigroup of integers.  Answers to these problems have consequences in number theory and combinatorics, such as understanding whether Pythagorean trios are partition regular. I will present in general terms the questions, strategies from dynamics to address them and mention some recent results we obtained. This is joint work with Anh Le, Joel Moreira, and Wenbo Sun.

### July 26, 2022, 14:30 – 15:40

Henrik Kreidler (University of Wuppertal)

Pure Koopmanism: An operator theoretic approach to dynamical systems

Dynamical systems play a key role in modern mathematics and
come in many variations, e.g., in the form of continuous
transformations on topological spaces or measure-preserving maps on
probability spaces. In this talk we discuss a simple, but effective
operator theoretic approach to such systems known as the Koopman
linearization.

# Fall 2021-22

 October 19, 2021, 14:30-15:40 Asgar Jamneshan  (Koç University) Extensions of the ergodic Roth and Szemeredi theorems and aspects of uniformity I will give an introduction to ergodic Ramsey theory and then present some recent results. October 26, 2021, 14:30-15:40 Türkü Özlüm Çelik  (Boğaziçi University) Integrable systems from computational algebro-geometric perspective The Kadomtsev-Petviashvili (KP) hierarchy is a universal one among all integrable systems. This talk aims to make an excursion into the study of KP hierarchy with a view through computational algebraic geometry. The emphasis will be on exploiting modern tools in symbolic, numerical and combinatorial algebraic geometry to investigate solutions of the differential equations. November 02, 2021, 17:30 – 18:40 Pieter Spaas (UCLA) Group actions and equivalence relations Group actions of a countable group on a measure space give rise to interesting dynamical systems, and studying the associated orbit equivalence relations has been the subject of a lot of study over many years. In this talk we will consider some natural constructions and discuss classification results for such equivalence relations. In particular, we will study how the behavior of certain central sequences implies structural results on the involved equivalence relations and their products. All necessary notions will be introduced during the talk. This is partially based on joint work with Adrian Ioana. November 09, 2021, 14:30-15:40 Haydar Göral (İzmir Institute of Technology) Classification of Quadratic Number Fields Using Diophantine Equations In this talk, we consider some Diophantine equations over number fields. Recall that a number field is a finite field extension of the field of rational numbers. We show that the finiteness of the set of all solutions of certain Diophantine equations determines quadratic number fields among all number fields. Along the way, we explain our motivation, we give many examples and relate the results to nonstandard analysis and arithmetic geometry. November 23, 2021, 14:30-15:40 Or Shalom (Einstein Institute of Mathematics) A structure theorem for Gowers-Host-Kra seminorms for non-finitely generated countable abelian groups of unbounded torsion Furstenberg’s famous proof of Szemeredi’s theorem leads to a natural question about the convergence and limit of some multiple ergodic averages. In the case of $\mathbb{Z}$-actions these averages were studied by Host-Kra and Ziegler. They show that the limiting behavior of such multiple ergodic average is determined on a certain factor that can be given the structure of an inverse limit of nilsystems (i.e. rotations on a nilmanifold). This structure result can be generalized to $\mathbb{Z}^d$ actions (where the average is taken over a Folner sequence), but the non-finitely generated case is still open. The only progress prior to our work is due to Bergelson Tao and Ziegler, who studied actions of the infinite direct sum $\mathbb{Z}/p\mathbb{Z}$. In our work we generalize this further to the case where the sum is taken over different primes (the most interesting case is when the multiset of primes is unbounded). We will explain how this case is significantly different from the work of Bergelson Tao and Ziegler by describing a new phenomenon that only happens in these settings. Moreover, we will discuss a generalized version of nilsystems that plays a role in our work and some corollaries. If time allows we will also discuss the group actions of other abelian groups. November 30, 2021, 14:30-15:40 Özge Ülkem (Galatasaray University) Generalized D-elliptic sheaves and their moduli space Elliptic curves play a fundamental role in algebraic number theory. In the 1970’s Drinfeld defined analogues of elliptic curves in the function field setting, which are now called Drinfeld modules. Later on he defined a categorically equivalent notion, called elliptic sheaves, and studied their moduli space to prove Langlands correspondence. Since then many generalizations of Drinfeld modules and elliptic sheaves have been worked out. In the first part of this talk we will form the function field and classical setting and discuss similarities between them. Then, we define Drinfeld modules, discuss the analogy between elliptic curves and Drinfeld modules. In the second part we will define a new generalization of elliptic sheaves, called generalized D-elliptic sheaves and talk on their moduli space and of the uniformization of the latter if time permits. December 07, 2021, 16:00-17:10 Özlem Ejder (Boğaziçi University) Isolated Points on Modular Curves One of the oldest areas of mathematics is the study of integer or rational solutions to polynomial equations with integer coefficients and it remains active till today. The most natural question we can ask about such an equation is whether its set of rational solutions is finite or infinite. This can be determined by the genus of the curve defined by such equations. In particular, if the genus is greater than one, there are finitely many rational points on a curve.  What happens when one allows for solutions involving  square-roots of integers or cubic roots? Perhaps in general all complex numbers that are roots of a degree d polynomial? We call such solutions of degree 2,3 or d in general. In this talk, we will discuss when a curve has infinitely many degree d points focusing particularly on points on modular curves. December 14, 2021, 14:30-15:40 Luka Milićević (Mathematical Institute of the Serbian Academy of Sciences and Arts) Bilinear Bogolyubov Argument in Abelian Groups Bilinear Bogolyubov argument states that if we start with a dense set A of a product U x V of finite vector spaces and carry out sufficiently many steps where we replace every row or every column of A by the set difference of it with itself, then inside the resulting set we obtain a bilinear variety of codimension bounded in terms of the density of A. In this talk, we focus on a generalization of this theorem to arbitrary finite abelian groups. Along the way we also discuss analogues of a few linear-algebraic facts as well as quasirandomness of analogues of bilinear varieties in the setting of finite abelian groups. December 21, 2021, 14:30-15:40 Faruk Temur (İzmir Institute of Technology) Discrete fractional integrals, lattice points on thin arcs, and diophantine approximation  The study of discrete fractional integral operators began with an article ofArkhipov and Oskolkov on boundedness of certain multipliers, and for over thirty years concentrated on cases with translation invariant or quasitranslation invariant phase polynomial, as these cases are amenable to the Fourier transform and the Hardy-Littlewood circle method. Recently in joint work with E. Sert, we introduced methods from arithmetic to study discrete fractional integral operators along quadratic bivariate polynomials in their full generality. This is achieved by combining the information regarding distribution of lattice points on conics gleaned via classical theory of binary quadratic forms and sieving with very careful partitioning of sums under question. This effort as a side benefit yields new results on various conjectures on concentration of lattice points on conics and makes connections to diophantine approximation.  In this talk we will give a brief summary of these results on fractional integrals and lattice points, together with their mathematical context and main ideas of their proofs. December 28, 2021, 14:30-15:40 Oğuz Şavk (Boğaziçi University) Homology 3-spheres bounding contractible 4-manifolds and homology 4-balls A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results. Our approach is based on Mazur’s famous argument and its generalization, they together provide a unification of all recognized results. January 04, 2022,16:00-17:10 Önder Türk (Institute of Applied Mathematics at METU) Modal analysis of linear elasticity equations in the incompressiblelimit A modal analysis approach for approximating the linear elasticity equations with the incompressibility constraint will be presented. The spectrum of the linear elasticity operator is approximated using an optimally convergent stabilized finite element method that is based on a variational multiscale approach. The convergence of the approximate solutions to the true ones in appropriate norms is proved both theoretically and numerically.