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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 discretetime 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 patternavoiding 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, birthanddeath chains, Brownian motions, and GaltonWatson 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 ErdosRényi 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 readoff the equivariant Ktheory and Chow cohomology of the variety from that
of fixed point subscheme, modulo certain relations given by the fixed loci of codimensionone
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 Ktheory of X is a ring of piecewise exponential functions (a version
of GKM theory). Some relations to Chow cohomology, via the RiemannRoch 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 ppermutation modules
Abstract: Let G be a finite group and k be a field of characteristic p. A kGmodule M is called a ppermutation module if its restriction to a Sylow psubgroup S of G is a permutation kSmodule.
In this talk, we introduce the biset functor of ppermutation 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 nonnegative 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 lightsheet 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 CisnerosMolina, UNAM

Invariants of Hyperbolic 3manifolds 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, noncompact hyperbolic
3manifolds 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 3manifolds 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 4manifolds, 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 BrillNoether 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
MaynardTao 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 pseudorotations via symplectic topology
Abstract: Ever since the ConleyZehnder 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 twosphere admits Hamiltonian diffeomorphisms with finitely many periodic orbits — the socalled pseudorotations — 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 pseudorotations, 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 pseudorotations.

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 4manifolds, and recent progresses in 4manifolds 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 onevariable 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 0level 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 pseudodifferential 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 highfrequency 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 taufunction  have been the guiding force behind several themes in Number Theory. They continue to tantalize us with easytounderstand 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, nontrivial 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 Gspaces and Gequivariant maps admits a
model
structure in which the weak equivalences (resp. fibrations) are defined
as Gmaps that induce weak
equivalences (resp. fibrations) on Hfixed point spaces for every H
< G. This is a standard way
to study equivariant homotopy theory. The fibrantcofibrant objects in
this model category are
GCWcomplexes. A weak equivalence between GCWcomplexes is a
Ghomotopy equivalence.
Such a map induces weak equivalences on Horbits for every H < G.
The converse, however,
is not always true. It is natural to ask when a map inducing weak
equivalences on Horbits for
every H < G induces weak equivalences on Hfixed point spaces. To
answer this question, we
construct a new model structure on the category of Gspaces in which
the weak equivalences
and cofibrations are defined as maps inducing weak equivalences and
cofibrations on Horbits 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 noncommutative rings flourishing from the fact
that free modules over some noncommutative
rings can have two bases with different cardinality.
Surprisingly
enough not only noncommutative 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, Ktheory
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


