Department of Mathematics

Mathematics Colloquium


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06 January, 2022

Giray Ökten, Florida State University    

Polynomial chaos and Monte Carlo  

Abstract: We will discuss some ideas on how to use polynomial chaos to obtain improved Monte Carlo estimators, and present some applications to global sensitivity analysis and computational finance.

13 January, 2021

John Pike, Bridgewater State University    

Double jump phase transition in a soliton cellular automaton  

Abstract: I will discuss joint work with Lionel Levine and Hanbaek Lyu on the soliton cellu- lar automaton with random initial conditions. This CA is a discrete-time dynamical system which models the behavior of certain traveling wave packets arising in various areas of math and physics. After explaining the basics of the model, I will describe connections with a variety of combina- torial objects like pattern-avoiding permutations, Young tableaux, and Motzkin paths. I will then turn to the random setting where one can frame things in terms of probabilistic constructs like renewal processes, birth-and-death chains, Brownian motions, and Galton-Watson forests. Using these perspectives, I will present some limit theorems which establish a ’double jump phase tran- sition’ for certain statistics of the system analogous to that found by Erdos-Rényi in their seminal ̋ study of random graphs.

11 March, 2020

Richard Gonzales, Pontificia Universidad Católica del Perú    

Equivariant Operational Theories and the Localization Principle 

Abstract: For a complete nonsingular variety with a torus action, the localization principle asserts that one can read-off the equivariant K-theory and Chow cohomology of the variety from that of fixed point subscheme, modulo certain relations given by the fixed loci of codimension-one subtori. For singular varieties, however, such method quite often does not apply. Our goal is to show that in the setting of equivariant operational theories there is a version of the localization principle that works perfectly well for both singular and nonsingular varieties. For instance, if X is any complete variety where a torus acts with finitely many fixed points and invariant curves, then the equivariant operational K-theory of X is a ring of piecewise exponential functions (a version of GKM theory). Some relations to Chow cohomology, via the Riemann-Roch theorem, will be discussed too.

9 January, 2020

Mehmet Kıral, Sophia University    

Kloosterman Sums for SL3 long word element 

Abstract: Using the reduced word decomposition of the long word element of the Weyl group element of SL3, we give a nice expression for the long word Kloosterman sum. First classical Kloosterman sums, their importance, and matrix formulation will be introduced. This is joint work with Maki Nakasuji of Sophia University (Tokyo).

25 December, 2019

Özgür Bayındır, University of Haifa    

DGAs with polynomial homology 

Abstract: Differential graded algebras (DGAs) are one of the most important objects of study in homological algebra. These are chain complexes with an associative and unital multiplication. Examples of DGAs include cochain complexes of topological spaces equipped with the cup product. In fact, there are interesting interactions between DGAs and stable homotopy theory. I will present these connections, and show how they can be used to classify DGAs with polynomial homology. The talk will be accessible to a general audience.

4 December, 2019

Mohammad Sadek, Sabancı University    

Counting (hyper)elliptic curves with a given discriminant 

Abstract: The discriminant of a (hyper)elliptic curve encodes the primes of bad reduction of the curve. The number of isomorphism classes of (hyper)elliptic curves over a number field with the same dis- criminant is known to be finite. A more involved task is to count, if not list, all such isomorphism classes. In this talk, we survey old and recent results for elliptic curves. We also explain how one may tackle the same question for genus 2 curves.

20 November, 2019

Olcay Coşkun, Boğaziçi University    

The functor of p-permutation modules 

Abstract: Let G be a finite group and k be a field of characteristic p. A kG-module M is called a p-permutation module if its restriction to a Sylow p-subgroup S of G is a permutation kS-module. In this talk, we introduce the biset functor of p-permutation modules and describe its composition factors. This is a joint work with R. Boltje and Ç. Karagüzel.

13 November, 2019

Mohan Ravichandran, Boğaziçi University    

Generalized Permutohedra: Ehrhart Positivity and Minkowski Additive Valuations 

Abstract: The Ehrhart Polynomial counts the number of lattice points in integer dilates of a lattice polytope and is of fundamental importance in polyhedral combinatorics and toric geometry. The question of which polytopes have Ehrhart polynomials with non-negative coefficients is a well studied albeit poorly understood problem. I’ll talk about a recent result of mine, joint with Katharina Jochemko which shows that the linear coefficient of the Ehrhart polynomial of generalized permutohedra, an important class of polytopes introduced by Postnikov are indeed positive. We are also able to characterize all Minkowski additive functions on this class of polytopes, which includes several important classes of polytopes such as matroid and independent set polytopes. I’ll end with several open problems, some of which might be of interest to graduate or even undergraduate students.

6 November, 2019

Matías Courdurier, Pontificia Universidad Católica de Chile    

An Inverse Problem in Flurescence Microscopy 

Abstract: In this talk we will start by presenting a brief description of what is a mathematical inverse problems. We will then describe the particular case of the biomedical imaging technique of fluorescence microscopy, and the issue we want to tackle in light-sheet fluorescence microscopy. We will proceed to present the mathematical model we propose for this setting and an injectivity result for the corresponding measurement operator. We will conclude by showing some numerical experiments that explore the validity and improvements of our approach. The talk is planned to be a general talk for an audience with mathematical and/or biomedical imaging interests.

30 October, 2019

Murat Tuncalı, Nipissing University    

Countable Rank Maps 

Abstract: A motivation for this study comes from numerous topo- logical constructions where countable rank maps are used. In this talk, we will present some in- teresting results concerning such maps and their projective classes. For instance: for nice spaces, they behave like monotone maps. This is a joint work with Paweł Krupski (Wroclaw University of Science and Technology, Poland).

23 October, 2019

Yasemin Kara, Boğaziçi University    

Asymptotic Generalized Fermat's Last Theorem over Number Fields 

Abstract: In this work, combining their techniques we show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6. This is joint work with Ekin Özman.

16 October, 2019

José Luis Cisneros-Molina, UNAM    

Invariants of Hyperbolic 3-manifolds in Relative Group Homology 

Abstract: Let G be a discrete group. It is well known that the homology groups of G have both algebraic and topological definitions. Now consider a subgroup H of G. In the literature there are two versions of relative homology groups for the pair (G,H), the theory by Adamson generalizes in a natural way the algebraic definition, while the theory by Takasu generalizes in a natural way the topological definition. In the first part of the talk, we present both theories, we give simple examples that show that these theories does not coincide in general, and we give a sufficient condition on the subgroup H in order that the two theories coincide. In the second part of the talk, we define invariants of complete, orientable, non-compact hyperbolic 3-manifolds of finite volume, in Adamson relative homology groups. We explain the relation between these invariants and the elements in Takasu relative homology groups given by Zickert, which are used to define and invariant of such hyperbolic 3-manifolds in the extended Bloch group defined by Neumann. This is joint work with José Antonio Arciniega Nevárez.

7 August, 2019

Izzet Coskun, University of Illnois at Chicago    

The geometry of moduli spaces of sheaves on surfaces 

Abstract: Moduli spaces of sheaves on surfaces are higher rank analogues of Hilbert schemes of points and play a central role in mathematics with applications ranging from algebraic geometry to Donaldson’s theory of 4-manifolds, from combinatorics to mathematical physics. In this talk, I will describe some recent developments on our understanding of the geometry of these moduli spaces, concentrating on the Brill-Noether problem and the cohomology of these moduli spaces. This talk will be based on joint work with Jack Huizenga and Matthew Woolf.

10 July, 2019

Özlem Ejder, Colorado State University    

Sporadic Points on Modular Curves 

Abstract: The points on the modular curve X1(n) roughly classifies the pairs (E,P) (up to isomorphism) where E is an elliptic curve and P is a point of order n on E. We call a closed point x on X1(n) sporadic if there are only finitely many closed points of degree at most deg(x); hence classifying sporadic points on X1(n) is closely related to determining the torsion subgroups of elliptic curves over a degree d field. When d = 1 or 2, Mazur and Kamienny's work show that there are no sporadic points of degree d on X1(n). In this talk, I will discuss the sporadic points of arbitrary degree. This is joint with A. Bourdon, Y. Liu, F. Odumudu and B. Viray.

12 June, 2019

Saadet Öykü Yurttaş, Dicle University    

Algorithms for braids and multicurves with Dynnikov coordinates 

Abstract: This talk is divided into two parts. In the first part, I will talk about a fast algorithm to compute the dilatation and invariant measured foliations of a given pseudo–Anosov braid. In the second part two algorithms regarding multicurves on surfaces are presented. The first one calculates the number of connected components of a multicurve; and the second one calculates the geometric intersection number of two multicurves. The algorithms are illustrated with Dynnikov Coordinates on the finitely punctured disk.

29 May, 2019

Deniz Ali Kaptan, Alfred Renyi Institute    

Small Gaps Between Primes in Arithmetic Progressions 

Abstract: I will give a brief overview of the history of the methods and results culminating in the Maynard-Tao result on bounded gaps in primes, then describe how the method can be adapted to the setting of primes in arithmetic progressions, looking for bounds (in terms of the moduli M) that are uniform over ranges of M.

20 May, 2019 r

Başak Gürel, University of Central Florida    

From Hamiltonian systems with infinitely many periodic orbits to pseudo-rotations via symplectic topology 

Abstract: Ever since the Conley-Zehnder proof of the Arnold conjecture for tori, the study of periodic orbits has arguably been the most important interface between Hamiltonian dynamical systems and symplectic topology. A general feature of Hamiltonian systems is that they tend to have numerous periodic orbits. In fact, for a broad class of closed symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many simple periodic orbits. There are, however, notable exceptions. Namely, an important class of symplectic manifolds including the two-sphere admits Hamiltonian diffeomorphisms with finitely many periodic orbits — the so-called pseudo-rotations — which are of particular interest in dynamical systems. Furthermore, recent works by Bramham (in dimension two) and by Ginzburg and myself (in dimensions greater than two) show that one can obtain a lot of information about the dynamics of pseudo-rotations, going far beyond periodic orbits, via symplectic techniques. In this talk I will discuss various aspects of the existence question for periodic orbits of Hamiltonian systems, focusing on recent higher dimensional results about pseudo-rotations.

8 May, 2019

E. Sercan Yılmaz,Boğaziçi University    

Counting Points on Curves and Irreducible Polynomials over Finite Fields 

Abstract: For any integers n ≥ 3 and r ≥ 1 we present formulae for the number of irreducible polynomials of degree n over the finite field F2^r where the coefficients of x^(n−1), x^(n−2) and x^(n−3) are zero. We will also apply our curved based techniques to some previous results and give them short proofs. Our proofs involve counting the number of points on certain algebraic curves over finite fields.

17 April, 2019

Ezgi Kantarcı,Sabancı University    

A Queer Crystal Structure on Shifted Tableaux 

Abstract: Crystal bases were introduced by Kashiwara in his study of the representation theory of quantized universal enveloping algebras. Tableaux are objects that are central in combinatorial representation theory, especially in relation to Lie groups. In this talk, we will combine the two to give a connected queer crystal structure on semistandard shifted tableaux of a given shape.

3 April, 2019

Sümeyra Sakallı, Max Planck Institute for Mathematics    

Surgeries and Exotica in Dimension Four 

Abstract: Exotic manifolds are smooth manifolds which are homeomorphic but not diffeomorphic to each other. In this talk, after reviewing main concepts, we will begin with the question “Why dimension four?”. Then we will go over some construction techniques of exotic 4-manifolds, and recent progresses in 4-manifolds theory. If time permits we will also discuss some open problems in the field. No background is required.

27 March, 2019

Ömer Küçüksakallı, Middle East Technical University    

Arithmetically exceptional polynomial mappings 

Abstract: A polynomial with integer coefficients is called (arithmetically) exceptional if it induces a permutation over infinitely many residue fields. The classification of one-variable exceptional polynomials is finished; such a map is a composition of linear polynomials, monomials and Chebyshev polynomials. Lidl and Wells have generalized Chebyshev polynomials to several variables and shown that certain such maps are exceptional. One can show that their polynomials, with n variables, correspond to a family of maps associated with the simple complex Lie algebras An. In this talk, we will focus on the multivariate polynomial mappings that are associated with the other simple complex Lie algebras. We will give an easy to check condition for these multivariate maps to induce permutations over finite fields. Using this result, we will show that there exist infinitely many exceptional polynomials for each simple complex Lie algebra.

20 March, 2019

Ali Devin Sezer, Middle East Technical University    

Approximation of the Exit Probability of a Stable Markov Modulated Constrained Random Walk 

Abstract: Let X be the constrained random walk on Z 2 + having increments (1,0), (−1,1), (0,−1) with jump probabilities λ(Mk), μ1(Mk), and μ2(Mk) where {Mk} is an irreducible aperiodic finite state Markov chain. X represents the lengths of two tandem queues with arrival rate λ(Mk), and service rates μ1(Mk), and μ2(Mk). We assume that the average arrival rate with respect to the stationary measure of M is less than the average service rates, i.e., X is assumed stable. Let τn be the first time X hits the line ∂An = {x : x(1) + x(2) = n}, i.e., the first time the sum of the components of X equals n. Let Y be the random walk on Z × Z+ (i.e., constrained only on ∂2 = {y ∈ Z × Z+ : y(2) = 0}) again modulated by M and having increments (−1,0), (1,1), (0,−1) with probabilities λ(Mk), μ1(Mk), and μ2(Mk). Let B = {y ∈ Z 2 : y(1) = y(2)} and let τ be the first time Y hits B. Let Tn : Z 2 7→ Z 2 be the affine map y 7→ (n − y(1), y(2) and let m denote the initial point of M. For x ∈ R 2 +, x(1) + x(2) < 1, x(1) > 0, and xn = bnxc, we show that P(Tn(xn),m) (τ < ∞) approximates P(xn,m) (τn < τ0) with exponentially vanishing relative error as n → ∞. For the analysis we define a characteristic matrix in terms of the jump probabilities of (X,M). The 0-level set of the characteristic polynomial of this matrix defines the characteristic surface H ⊂ C 2 for the problem. Conjugate points on H and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation / approximation of P(y,m) (τ < ∞).

26 December, 2018

Souaad Lazergui, University of Mostaganem (UMAB), Algeria    

High frequency asymptotic of the Kirchhoff amplitude for convex obstacles 

Abstract: In this talk, we are concerned with diffraction of waves around a strictly convex obstacle. Our objective is to produce the high frequency asymptotic expansion of the amplitude of the Helmholtz equation solution. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator DtN. In our work, we use first and second order approximations of the DtN operator so as to derive new asymptotic expressions of the normal derivative of the total field. The resulting expansions can be used to appropriately choose the ansatz in the design of high-frequency numerical solvers, such as those based on integral equations, in order to produce more accurate approximation of the solutions of the Halmhotz equation around the shadow and the deep shadow regions than the ones based on the usual ansatz.

31 October, 2018

H. Nurettin Ergün, Marmara University    

The Reed & Zenor Theorem 

Abstract: The following question, asked in 1920s, is of fundamental importance. Question: Which normal Hausdorff spaces are metrizable?

12 October, 2018

Eknath Ghate, Tata Institute of Fundamental Research, School of Mathematics, Mumbai    

The Tau of Ramanujan 

Abstract: The sequence of numbers 1,−24,252,−1472,4830,... was extensively studied by the great Indian mathematician Ramanujan a century ago. These numbers - the values of Ramanujan’s tau-function - have been the guiding force behind several themes in Number Theory. They continue to tantalize us with easy-to-understand problems some of which are still open today. This talk will be a relaxed introduction to these numbers and will lead up to some of our own results about them.

10 October, 2018

Christian Houdre, Georgia Institute of Technology    

On the Limiting Shape of Young Diagrams Associated With Markov Random Words 

Abstract: Motivated by a conjecture on the asymptotic behavior of the length of the longest increasing subsequences of Markov random words, over a totally ordered finite alphabet of fixed size, using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest (weakly) increasing subsequence of the word, the corresponding limiting law follows. Under a cyclic condition, a spectral characterization of the Markov transition matrix precisely characterizes when the limiting shape is the spectrum of the m×m traceless GUE and so is akin to the iid framework. For each m ≥ 4, this characterization identifies a proper, non-trivial class of cyclic transition matrices producing such a limiting shape. However, for m = 3, all cyclic Markov chains have such a limiting shape, a fact previously only known for m = 2. For m arbitrary, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. This is joint work with Trevis Litherland.

26 September, 2018

Mehmet Akif Erdal, Bilkent University    

Equivariant model structures via orbit spaces 

Abstract: Let G be a group. The category of G-spaces and G-equivariant maps admits a model structure in which the weak equivalences (resp. fibrations) are defined as G-maps that induce weak equivalences (resp. fibrations) on H-fixed point spaces for every H < G. This is a standard way to study equivariant homotopy theory. The fibrant-cofibrant objects in this model category are G-CW-complexes. A weak equivalence between G-CW-complexes is a G-homotopy equivalence. Such a map induces weak equivalences on H-orbits for every H < G. The converse, however, is not always true. It is natural to ask when a map inducing weak equivalences on H-orbits for every H < G induces weak equivalences on H-fixed point spaces. To answer this question, we construct a new model structure on the category of G-spaces in which the weak equivalences and cofibrations are defined as maps inducing weak equivalences and cofibrations on H-orbits for each H < G. We show that a weak equivalence between objects that are fibrant in this new model structure is a weak equivalence in the fixed point model structure. This is a joint work with Asli Güçlükan İlhan.

2 May, 2018

Müge Kanuni, Düzce University    

Leavitt path algebras: A taste for all? 

Abstract: We will give a survey of the last 15 years of research done in a particular example of non-commutative rings flourishing from the fact that free modules over some non-commutative rings can have two bases with different cardinality.

Surprisingly enough not only non-commutative ring theorists, but also C*-algebraists gather together to advance the work done. The interplay between the topics stimulates interest and many proof techniques and tools are used from symbolic dynamics, ergodic theory, homology, K-theory and functional analysis. Open problem pages are put up and research schools are organized throughout the world. Over 150 papers have been published on this structure, so called Leavitt path algebras, which is constructed on a directed graph.


25 April, 2018

Sibel Şahin, Mimar Sinan Fine Arts University